# Lesion Detection on Leaves using Class Activation Maps

Enes Sadi Uysal
Huawei R&D Turkiye
<EMAIL_ADDRESS>
Deniz Sen
Huawei R&D Turkiye
<EMAIL_ADDRESS>
Ahmet Haydar Ornek
Huawei R&D Turkiye
<EMAIL_ADDRESS>
Ahmet Emin Yetkin
Huawei R&D Turkiye
<EMAIL_ADDRESS>

###### Abstract

Lesion detection on plant leaves is a critical task in plant pathology and
agricultural research. Identifying lesions enables assessing the severity of
plant diseases and making informed decisions regarding disease control
measures and treatment strategies. To detect lesions, there are studies that
propose well-known object detectors. However, training object detectors to
detect small objects such as lesions can be problematic. In this study, we
propose a method for lesion detection on plant leaves utilizing class
activation maps generated by a ResNet-18 classifier. In the test set, we
achieved a 0.45 success rate in predicting the locations of lesions in leaves.
Our study presents a novel approach for lesion detection on plant leaves by
utilizing CAMs generated by a ResNet classifier while eliminating the need for
a lesion annotation process.

_Keywords_ Class Activation Maps $\cdot$ Lesion Detection $\cdot$ CNN

##  1 Introduction

Lesion detection on plant leaves is a critical task in plant pathology and
agricultural research. Plant diseases, caused by various pathogens such as
fungi, bacteria, and viruses, can significantly impact crop yield, quality,
and overall agricultural productivity. Early and accurate detection of lesions
on plant leaves plays a crucial role in disease management and prevention,
allowing farmers and researchers to implement timely interventions and
minimize the spread of infections.

Lesions are visible abnormalities or discolorations on plant leaves, which can
manifest as spots, blotches, necrotic areas, or other irregular patterns. They
are often indicators of disease presence or environmental stressors, such as
nutrient deficiencies, chemical damage, or physical injuries. Identifying and
analyzing these lesions enable scientists and growers to assess the severity
of plant diseases, monitor their progression, and make informed decisions
regarding disease control measures and treatment strategies.

Traditionally, lesion detection has relied on manual visual inspection by
trained experts, which can be time-consuming, labor-intensive, and prone to
subjective interpretation. However, recent advancements in imaging
technologies, machine learning algorithms, and computer vision techniques have
opened up new possibilities for automating and enhancing lesion detection
processes. By leveraging these technological tools, researchers are able to
develop efficient and accurate methods for lesion identification and
classification, aiding in the early diagnosis and management of plant
diseases.

ResNet He et al. (2015), short for Residual Neural Network, is a deep learning
architecture known for its exceptional performance in image classification
tasks. It incorporates skip connections, or residual connections, which help
alleviate the vanishing gradient problem and enable the network to learn more
effectively. By using a pre-trained ResNet model as a feature extractor, we
can leverage the knowledge learned from vast amounts of labeled data,
enhancing the overall performance and robustness of the lesion detection
system. In this study, we propose a method for lesion detection on plant
leaves utilizing class activation maps generated by a ResNet-18 classifier.
Class activation maps (CAMs) Zhou et al. (2016) are powerful visualizations
that highlight the regions in an image that contribute most to a particular
class prediction made by a convolutional neural network (CNN). By leveraging
CAMs in combination with the ResNet-18 architecture, we aim to accurately
identify and localize lesions on plant leaves, providing a cost-efficient tool
for plant pathology and agricultural research. We reduce the complexity of the
problem from detection to classification considering both training and
computational costs of classifiers are lower than object detector models. In
addition, we get rid of drawing separate bounding boxes for each lesion.

##  2 Literature Review

There have been many approaches to plant disease detection; specifically,
image processing, machine learning, and deep learning algorithms were highly
investigated to solve the related problems. Segmentation via color channel
thresholding is a basic, yet effective technique in image processing, and
Dhaygude and Kumbhar (2013) utilized it to retrieve the most relevant
segments, which are then put through statistical analysis. Similarly, Ngugi et
al. (2021) created a pipeline that involves Otsu segmentation Otsu (1979) on 3
different color spaces and an empirically adapted rule mechanism. Apart from
the traditional image segmentation techniques, Singh et al. (2015) used the
genetic algorithm to separate the diseased parts of the leaf.

Nowadays as the data got denser, deep feature extraction is preferred over
traditional methods due to their complexity and generalization capability.
Korkut et al. (2018) compared the performances of different classification
algorithms which were trained over the deep features of a pretrained
InceptionV3 feature extractor. As the amount of data increases, imbalance
problems between the classes can occur, which is a problem Ahmad et al. (2021)
addressed by data augmentation and the introduced stepwise transfer learning
scheme. Ashwinkumar et al. (2022) proposed a multistage deep classifier that
took advantage of the efficiency of MobileNet in the feature extraction step.
Wang et al. (2021) introduced a trilinear CNN model architecture that
separates the crop and disease identification processes which leads the
network to learn more distinct and informative features. Atila et al. (2021)
compared the state-of-the-art deep classifier performances over the standard
and augmented PlantVillage dataset Hughes and Salath’e (2015). Following the
developments in the singular model configurations, Bansal et al. (2021)
adapted an ensemble model scheme which is constituted from DenseNet121 Huang
et al. (2017), EfficientNetB7 and EfficientNet Noisy Student models Tan and Le
(2020). Furthermore, Li et al. (2022) trained a YOLOV5s-based object detector
to localize and classify multiple diseased plants inside a scene.

Besides supervised object detectors such as YOLO, classification models can be
used to detect objects by utilizing activation maps and this concept is known
as Weakly Supervised Object Detection Yang et al. (2019). When a
classification model is trained, its last convolutional layers learn high-
level features related to desired classes. CAM is a technique that highlights
regions of desired classes in images using the last convolutional layers.
Chattopadhay et al. (2018) uses a combination of first-order and second-order
gradients to calculate the importance weights of each feature map. This allows
for more accurate localization of desired classes in images. Cheng et al.
(2020) combines selective search and gradient-based CAM (GradCAM) to achieve a
higher object detection performance using IoU. When PASCAL VOC 2007 and 2012
dataset and MS COCO dataset are used, their method significantly improves
their IoU. Li et al. (2019) proposes a generative adversarial segmentation
module to obtain more precise localization including segmentation tasks.

##  3 Materials and Method

###  3.1 Dataset

The final dataset consists of the alteration and combination of several public
datasets. PlantVillage is an image classification dataset containing 54304
samples and 38 classes; the classes are crops taken from 14 species, each
having images from healthy and diseased lab-controlled samples. The dataset
severely suffers from class imbalance, such that the class with the most
samples is constituted of 5507 images, whereas the one with the least number
of samples includes 152 images.

The second dataset we used in this work is the Leaf Disease Segmentation
Dataset Alam (2021) which includes 588 images of diseased leaves and 588
binary masks segmenting the diseased parts of the corresponding leaf. The
samples are also augmented using basic transformations such as rotation and
flipping. Using this dataset, we created an object detection dataset where
each lesion segment is replaced with its smallest enclosing rectangle, which
became the ground truth bounding box.

Finally, as we train both image classifiers and object detectors for plant
lesions, we combined our 2 datasets to obtain a new image classification
dataset; we counted each image of the newly obtained lesion detection dataset
as a positive sample, on the other hand, we took the healthy plant images from
the PlantVillage dataset and counted each one as a negative sample, as they do
not include any lesions.

###  3.2 Class Activation Mapping (CAM)

To explain the classification process of a CNN model, visual explanation
techniques such as CAM are used. The CAM techniques aim to emphasize
significant areas of provided images by utilizing the final convolutional
layers of the models, which include high-level features. The initial CAM
technique needs to be adjusted by eliminating the fully-connected layer,
incorporating the Global Average Pooling layer, and retraining the model. Once
this is done, the activation map related to training can be generated using
Eq. 1.

$L^{c}=\sum w^{c}A$ (1)

Where $L^{c}$ represents the activation map for class $c$, $w^{c}$ represents
the importance of the convolutional layer for class $c$, and $A$ represents
the convolutional layer. As the model’s performance decreases when the fully-
connected layer is eliminated, GradCAM techniques have been introduced.
GradCAM eliminates the need for layer removal and retraining. To determine the
weights for GradCAM, Eq. (2) is used. Once the weights are calculated, the
activation maps are generated through the Eq 1.

$w^{c}=\frac{1}{N}\sum\frac{\partial Y^{c}}{\partial A}$ (2)

###  3.3 Model Architecture and Training

We have chosen ResNet-18 as the diseased plant classifier, thanks to its
relatively low parameter count, and proven learning capability.

The ResNet-18 He et al. (2015) architecture consists of 18 layers, including a
convolutional layer, a max pooling layer, and several residual blocks. Each
residual block contains two convolutional layers and a shortcut connection
that bypasses the convolutional layers. The shortcut connection allows the
gradient to flow directly through the network, which helps to prevent the
vanishing gradient problem that can occur in very deep networks.

The first layer of the network is a 7x7 convolutional layer with stride 2,
followed by a 3x3 max pooling layer with stride 2. This reduces the spatial
dimensions of the input by a factor of 4. The next 4 layers each contain a
single residual block, with 2 convolutional layers and a shortcut connection.
The final layer of the network is a fully connected layer with 1000 output
units, which corresponds to the number of classes in the ImageNet dataset.

The ResNet-18 architecture has been shown to achieve state-of-the-art
performance on a variety of image classification tasks, including the ImageNet
dataset Deng et al. (2009). It is also relatively lightweight compared to
other deep neural network architectures, making it well-suited for deployment
on resource-constrained devices.

We trained the model for 4 epochs on the final lesion dataset with a batch
size of 128 and input size of 224x224. We used Adam optimizer with an initial
learning rate of 0.001, $\beta_{1}$, and $\beta_{2}$ parameters of 0.9 and
0.999 respectively.

###  3.4 Extracting The Bounding Boxes from Activation Map

The present study utilizes the Class Activation Mapping (CAM) technique to
extract the activation map. In order to segment the highly attended regions, a
combination of OTSU and binary thresholding methods is employed, resulting in
the creation of binary maps. Subsequently, an opening operation is applied
with a 3 by 3 kernel and 3 iterations to separate the lesions from each other.
Finally, the contours are detected using the chain approximate method on the
binary activation map, which allows for the drawing of bounding boxes around
each contour. The bounding boxes are then eliminated based on a size
threshold, thus finalizing the detection process.

##  4 Results

|  |
---|---|---
(a) | (b) | (c)

|
---|---
(d) | (e)

Figure 1:  (a) Diseased leaf image with lesions. (b) Ground truth lesion
bounding boxes. (c) GradCAM output of the lesion classifier (d) Segments
obtained from the GradCAM output (e) Predicted bounding boxes

The results of our experiments can be seen on Table 1. The train dataset
consists of 2100 images. We obtained an mAP score of 0.1967 with the IoU
threshold being 0.001; it should be noted that GradCAM outputs have
considerably low resolution, therefore it can be expected that the predicted
bounding boxes are considerably larger than the ground truth boxes.
Furthermore, these large boxes can cover multiple ground truth boxes and in
such a situation, mAP tends to generate false negative predictions, even
though the true predictions are included inside the large bounding box. For
that, we also crafted a metric(which is referred to as the success rate) where
we count the ground truth boxes that are covered with a single large bounding
box as true positive predictions. The success rate of capturing a lesion
inside of a predicted bounding box is 0.4519. In the test dataset that
consists of 400 images, we obtained mAP 0.1702 with 0.001 IoU. The success
rate of capturing a lesion inside of a predicted bounding box is 0.4518.

Table 1: Experimental Results on The Defined Metrics | Train Set (2100 image) | Test Set (400 image)
---|---|---
mAP (iou_th=0.001) | 0.1967 | 0.1702
Custom IoU Metric | 0.4519 | 0.4518

##  5 Discussion

In this study, we proposed a method for lesion detection on plant leaves using
CAMs generated by a ResNet-18 classifier. By leveraging the capabilities of
deep learning and CAMs, we aimed to accurately identify and localize lesions,
providing a valuable tool for plant pathology and agricultural research.

While our study provides a promising approach for lesion detection on plant
leaves, there are several avenues for future research and improvement.
Firstly, the method can be further optimized and fine-tuned to achieve even
higher accuracy and robustness. Exploring different CNN architectures and
investigating the impact of hyperparameters can contribute to refining the
lesion detection system. Additionally, expanding the dataset used for training
the CNN classifier can enhance the model’s ability to generalize across
different plant species, diseases, and environmental conditions. In addition,
layer selection for generating class activation maps is vital. We obtained the
best results with intermediate layers. Lastly, research efforts should be
directed towards the validation and evaluation of the proposed method on
large-scale field trials and diverse plant species. Assessing its performance
under different environmental conditions.

In conclusion, our study presents a novel approach for lesion detection on
plant leaves by utilizing CAMs generated by a ResNet-18 classifier. The
significance of early and accurate lesion detection in plant pathology and
agriculture cannot be overstated. The proposed method offers a valuable tool
for researchers and farmers, enabling timely disease management and
contributing to improved crop health and productivity. Future research can
focus on optimizing the method, exploring additional data sources, and
developing practical applications to further enhance its impact and usability.

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Throughout this note we restrict attention to the generic case. A generic
property is one that holds for almost all systems. This means that the systems
for which the property holds form an open and dense subset of the relevant
function space. We will be a little cavalier in not mentioning the function
spaces we are using but generally it should be clear.

##  1 Morse-Smale Systems

The time integration of a differential equation $\dot{x}=X(x)$ or,
equivalently the vector field $X$ defines a flow
$x\rightarrow\varphi_{X}^{t}(x)$ on the phase space $P$. If $x\in P$ then
$\varphi^{t}(x)$ is the solution with initial condition $x$.

Throughout we shall be considering vectorfields $X$ and their associated flows
$\varphi_{X}^{t}(x)$ on a domain $P\subset\mathbb{R}^{n}$ in which there is a
globally attracting ball $B$ i.e. a region $B$ which is a $n$-dimensional ball
(i.e. diffeomorphic to $\\{x\in\mathbb{R}^{n}:||x||<1\\}$) with a boundary
$\partial B$ that is a smooth111Throughout smooth means C2. sphere (i.e.
diffeomorphic to $\\{x\in\mathbb{R}^{n}:||x||=1\\}$) such that every
trajectory of $X$ starting in $P$ outside $B$ enters $B$ eventually and $X$ is
transverse to $\partial B$.

A point $x$ of $P$ is called chain recurrent for the flow provided that for
any $T,\epsilon>0$ there are points $x_{i}\in P$ and real numbers $t_{i}>T$,
$i=0,1,\ldots,N$, such that $x=x_{0}=x_{N}$, and
$d(\varphi^{t_{i}}(x_{i}),x_{i+1})<\epsilon$.

A Morse-Smale (MS) flow is one satisfying the following properties:

  1. (a)

the set of chain recurrent points consists of a finite number of hyperbolic
rest points and hyperbolic closed orbits (see [guckenheimer2013nonlinear] for
definitions);

  2. (b)

the unstable manifold of any closed orbit or rest point has transversal
intersection with the stable manifold of any closed orbit or rest point.

We study generic parameterised families in which the structurally stable
systems are MS flows $X$ that only have rest points. We call these _gradient-
like_ Morse-Smale flows because by [Smale61gradient] they always have a smooth
Liapunov function i.e. a function $L:P\rightarrow\mathbb{R}$ such that its
derivative is zero at rest points and $X(L)<0$ otherwise, where $X(L)=dL\cdot
X$.

###  Saddle-attractor configurations

Consider for a moment the planar case where $n=2$ and the above conditions
apply so that the boundary of $P$ is a smooth simple closed curve and the flow
is inwardly transverse to it. Then one can always find a smooth simple closed
curve $\gamma$ such that the flow is inward transverse to $\gamma$ and so that
the interior $\Omega$ of $\gamma$ contains all the $m$ attractors of the
system, $m-1$ saddles and no other rest points. Moreover, $\gamma$ can be
chosen so that the stable manifolds of the saddles inside $\gamma$ intersect
the boundary in two points and do so transversally. They thus divide $\Omega$
into $m$ components each of which is the basin of one of the attractors and,
moreover, the unstable manifold of each saddle intersects the basin of two
attractors. The possibilities with $m=3,4$ are shown in Fig. 1(A-C) and we see
that the $m=3$ case is particularly simple.

[width=0.80]../figures/connectors_extend_2

Figure : (A-C) These illustrate some of the different possibilities (the most
important for us) for the configuration of the stable manifolds of the saddles
in the attractor disk region $R$ defined above which is represented by the
interior of the circle. All cases for 3 and 4 attractors are shown. (D-E)
These illustrate the smooth (D) and cuspoidal (E) versions of the connector
(blue curve) which is made up of the unstable manifolds of the two saddles in
each case. Other non-smooth configurations are possible such as with a spiral
due to non-real eigenvalues at the central attractor.

There is a corresponding decomposition in $n$ dimensions. In Appendix
LABEL:sect:decision we explain the following result: For a gradient-like MS
system on a region $P$ in $\mathbb{R}^{n}$ with attracting ball $B$ as above
one can always find an attracting $n$-dimensional disk $R$ in $P$ with smooth
spherical boundary $\partial R$ that contains all the $m$ attractors plus
$m-1$ index 1 saddles but no other rest points. It captures almost all the
trajectories in the following sense: if $W^{s}$ is the union of all stable
manifolds of rest points outside $R$ and $x\in R\setminus W^{s}$ then for $t$
large $\varphi^{t}(x)\in R$. Moreover, $R$ has the property that the stable
manifolds of the index 1 saddles inside $R$ transversally intersect the
spherical boundary of $R$ in disjoint spheres of dimension $n-2$ and divide
$R$ into regions each of which is the basin of attraction of one of the
attractors.

We are interested in regions like $R$ above because in applications to
developmental biology the index 1 saddles play a particularly important role:
their unstable manifolds determine the escape routes by which cells change
their state by escaping one attractor and moving to another.

As a corollary to the above result, for such systems with 3 attractors,
associated with the 3 attractors are two saddles $s_{1}$ and $s_{2}$ with a
disposition as in Fig. 1D,E i.e. we can label the attractors $A$, $B$ and $C$
so that $s_{1}\succ A,B$ and $s_{2}\succ B,C$ where $s\succ X$ means that the
unstable manifold of $s$ intersects the stable manifold of $X$.

Thus the unstable manifolds of these two saddles join at one of the
attractors. The attractor at the joint is called the _central attractor_ and
the other attractors are said to be _peripheral_. The union of the parts of
the unstable manifolds which connect the attractors is called the _connector_.
The joint can either be smooth (Fig. 1D) so that the connector is a smooth
curve or it can be non smooth (e.g. as in Fig. 1E).

Consider such a situation where we have two index 1 saddles $s_{1}$ and
$s_{2}$ whose unstable manifolds intersect the stable manifold of an
attracting rest point at $x$. If the latter is close to undergoing a fold or
cusp bifurcation then there is a line $\ell$ through $x$ and a
$n-1$-dimensional submanifold $W^{ss}$ (the strong stable manifold)
intersecting $\ell$ tranversally at $x$ such that all trajectories starting
near $x$ and not starting in $W^{ss}$ approach $x$ in a direction tangent to
$\ell$. Consequently, it is generic that the unstable manifolds of the two
saddles in the connector approach $x$ tangent to $\ell$. If they approach in
opposite directions relative to $\ell$ then the connector is smooth (Fig. 1D)
but if they approach from the same direction relative to $\ell$ then the
connector is not smooth and has what we call a _cuspoidal_ structure (Fig.
1E).

If the attractor $x$ is not close to a fold then there can be other non-smooth
connectors. One obvious example is where the linearisation of the system about
$x$ has eigenvalues with non-zero imaginary part so that the unstable
manifolds spiral into $x$. The reader should note that in order to keep our
figures to a minimal size we never illustrate such cases but we do take them
into account in our arguments.

In Lemma LABEL:lem:nofold of Appendix LABEL:App:no_fold we show that if the
joint is cuspoidal at an attractor then that attractor cannot undergo a fold
bifurcation involving either of the saddles $s_{1}$ and $s_{2}$. For the other
non-smooth cases this is obvious. Thus such a joint involving $s_{1}$, $s_{2}$
and $x$ can only undergo a fold bifurcation involving $x$ and $s_{1}$ or
$s_{2}$ if the joint is smooth.

##  2 Fold and cusp bifurcations

We consider the bifurcations between gradient-like MS systems in families of
differential equations given by a family of vector fields $X_{\theta}$
parameterised by $\theta\in\mathbb{R}^{c}$. Throughout this discussion we
restrict to the case $c=2$.

When $c=2$ for gradient-like systems the only local bifurcations that can
occur generically are the _fold_ (or _saddle-node_) and the _cusp_
bifurcation. There are other codimension 2 bifurcations for dynamical systems
[arnold2013dynamical, guckenheimer2013nonlinear] such as the Takens-Bogdanov
bifurcation but these introduce periodic orbits and therefore move us outside
gradient-like systems that we focus on here.

At such a fold or cusp bifurcation point $\theta$ there is an invariant
1-dimensional smooth center manifold $W^{c}(x_{0})$ through the point $x_{0}$
in phase space where the bifurcation is taking place and the system on this
submanifold may be transformed into one of the following families via a
$C^{N(r)}$ change of coordinates ([arnold2013dynamical] Chap. 1, Sect. 3 and
Chap. 2, Sect. 5.7):

$\text{(fold)}\dot{x}=\pm x^{2}+\theta_{1}$ (1)

or

$\text{(cusp)}\dot{x}=\pm x^{3}+\theta_{1}x+\theta_{2}.$ (2)

In fact, if the original system is $C^{\infty}$ then for each $r>1$ one can
find a neighbourhood $U$ of $x$ such that there is a $C^{r}$ center manifold
of $x$ in $U$.

If the system is $n$-dimensional then the dynamics near $x_{0}$ are
topologically conjugate to (fold) & ˙x=x^2 + ε_1(θ);
˙v=-v; ˙w=w; or (cusp) & ˙x= ±x^3 + ε_1(θ) x +ε_2(θ);
˙v=-v; ˙w=w where $v\in\mathbb{R}^{n_{s}}$, $w\in\mathbb{R}^{n_{u}}$ and
$n_{s}$ (resp. $n_{u}$) is the number of eigenvalues of the linearisation of
the system at $x_{0}$ with negative (resp. positive) real part.

Consider the set of eigenvalues $\lambda_{i}$, $i=1,\ldots,n$, of the
linearisation $\dot{x}=Ax$ of the system $\dot{x}=X(x)$ at an equilibrium
$x_{*}$ and let $r_{i}$ be the real part of $\lambda_{i}$. A _resonance_ is a
relation of the form $r_{s}=\sum_{i}m_{i}r_{i}$ with the $m_{i}$ positive
integers with $\sum_{i}m_{i}\geq 2$. We say this equilibrium is non-resonant
if no such resonance exists.

Consider a fold $x_{0}$ as above in $n$ dimensions. Then if it is non-resonant
for $r>2$ one can find $C^{r}$ coordinates
$(x,y)\in\mathbb{R}\times\mathbb{R}^{n-1}$ on some neighbourhood $U$ of the
fold point in which the differential equation takes the form ˙x&=±x^2 +
a(θ)x^3+θ_1
˙y=A(x,θ)y ([arnold2013dynamical] Part I, Chap. 2, Sect. 5.7). Without loss of
generality we can assume that the bifurcation takes place at $\theta=0$. In
the systems we study the $y$ direction is stable and the matrix $A(0,0)$ has
all its eigenvalues with negative real parts.

###  Center manifolds

Center manifolds will play a key role in our considerations. Our use of the
term _center manifold_ will be a little more general than usual as normally it
is discussed when the systems is at a bifurcation and we will want to use it
when we are only near a bifurcation. For example, we want to be able to
associate a center manifold to an attractor that is close to undergoing a fold
bifurcation. Also we need to consider the smoothness of the variation in the
center manifold as parameters are changed.

For relevant information about center manifolds see [hirsch2006invariant]
Sect. 5A. In particular note that by Theorem 5A.3 of [hirsch2006invariant], if
$W^{c}$ is a center manifold through a rest point $x$ and $W$ is a backward
invariant set containing $x$ then, near $x$, $W$ is contained in $W^{c}$.
Thus, for example, if the unstable manifold of a saddle is asymptotic to a
fold point, then close to the fold point it is in the center manifold. Center
manifolds are not necessarily unique but their tangent space is. We will use
this fact below.

We now consider what we call _pseudo-hyperbolic_ rest points $x$. At such rest
points $x$ there is $a>b>0$ such that the Jacobian of the vector field at $x$
has eigenvalues $\lambda$ that either have their real part $\leq-a$ or
$\geq-b$. Then pseudo-hyperbolic index 1 saddles and attractors have
1-dimensional center manifolds $W^{c}(x)$ that vary smoothly with parameters
(Sect. 5 [hirsch2006invariant], especially Theorems 5.1, 5.5 and 5A.1). If
$\varphi^{t}$ is the flow, this manifold is characterised by the fact that
$z\in W^{c}(x)\iff||\varphi^{-t}(z)-x||/e^{ct}\rightarrow 0$ as
$t\rightarrow\infty$ for any $c$ with $a>c>b$. There is a complementary
submanifold $W^{ss}(x)$ transversal to $W^{c}(x)$ at $x$ characterised by
$z\in W^{ss}(x)\iff||\varphi^{t}(z)-x||/e^{-ct}\rightarrow 0$ as
$t\rightarrow\infty$ for such a $c$. This we call the _strong stable
manifold_. Note that our use of the term _center manifold_ is a little more
general than usual as in that case one commonly takes $b=0$.

Index 1 saddles are always pseudo-hyperbolic and attractors are if they are
close to having a fold bifurcation. For an index 1 saddle, part of the
unstable manifold containing the saddle can be taken for a center manifold.

According to Theorems 5.1 of [hirsch2006invariant], $W^{c}(x)$ has $C^{r}$
dependence upon parameters provided $e^{jb-a}<1$ for $1\leq j\leq r$. Thus the
center manifold for saddles always is smooth and that for attractors is smooth
provided they are close enough to having a fold bifurcation. The later point
is true because the closer an attractor is to being a fold, the closer one can
take $b$ to zero.

##  3 Catastrophe Manifold

[width=.75]../figures/cusp_blue

Figure : The catastrophe manifold $\mathcal{M}$ for the cusp catastrophe.
Notice that although the bifurcation set has a cusp and hence is not smooth
the corresponding fold curve in $\mathcal{M}$ is smooth.

We now consider a family of flows parameterised by $\theta\in\mathbb{R}^{c}$
given by the family of vector fields $X_{\theta}=X_{\theta}(x)$. The
catastrophe manifold is defined as

$\mathcal{M}=\\{(x,\theta):x\text{isarestpointof}X_{\theta}\\}.$

Generically, (see [arnold2013dynamical], I Sect. 1.4), this is a
$c$-dimensional submanifold of $\mathbb{R}^{n}\times\mathbb{R}^{c}$. We denote
by $\chi:\mathcal{M}\rightarrow\mathbb{R}^{c}$ the projection
$\mathbf{x}=(x,\theta)\mapsto\theta$ and let $\mathcal{S}$ denote the set of
singularities of $\chi$ i.e. the set of points $\mathbf{x}\in\mathcal{M}$
where $\chi$ is not a local diffeomorphism. Then
$\mathcal{B}=\chi(\mathcal{S})$ is the set of local bifurcation points. In the
generic case this consists of fold and cusp points and the only other
bifurcation points are where heteroclinic connections occur. The latter are
considered to be global bifurcations because their effect is not restricted to
a small neighbourhood.

Since in our discussion $c=2$, $\mathcal{M}$ will be a surface in
$\mathbb{R}^{n+2}$ and $\chi$ will be a projection into the plane. Fig. 3 is
useful to see the form of cusps and folds in this case.

If $\theta\in\mathbb{R}^{2}$ is a fold bifurcation point and
$x\in\mathbb{R}^{n}$ is the corresponding rest point then
$\mathbf{x}=(x,\theta)$ is in $\mathcal{M}$ and, by [arnold2013dynamical], I
Sect. 1.4 (cf. Theorem 8.1 of [trotman_zeeman]), generically there is a smooth
curve $C$ in $\mathcal{M}$ through $\mathbf{x}$ that consists of fold points.
At $\mathbf{x}$ there are local coordinates $(x_{1},x_{2})$ on $\mathcal{M}$
and $(\theta_{1},\theta_{2})$ on $\mathbb{R}^{2}$ such that $\chi$ is locally
given by $(\theta_{1},\theta_{2})=(x_{1}^{2},x_{2})$.

A universal unfolding222See [trotman_zeeman] for a definition of universal
unfolding. of the cusp is $\dot{x}=\pm x^{3}+\theta_{1}x+\theta_{2}$ and, in
the minus sign case the catastrophe manifold $\mathcal{M}$ is given by
$x^{3}-\theta_{1}x-\theta_{2}=0$. Therefore, the map
$(\theta_{1},x)\mapsto(x,\theta_{1},\theta_{2}=-\theta_{1}x+x^{3})$ from
$\mathbb{R}^{2}$ to $\mathcal{M}$ parameterises $\mathcal{M}$ in terms of $x$
and $\theta_{1}$. Thus, in this parameterisation $\chi$ is given by
$(\theta_{1},x)\mapsto(\theta_{1},\theta_{2}=-\theta_{1}x+x^{3})$ and this is
singular when $\theta_{1}=3x^{2}$ which defines a smooth curve $C$ in
$\mathcal{M}$. The bifurcation set $B_{C}$ is its image under $\chi$, which is
the set of points given by $\theta_{1}=3x^{2},\theta_{2}=-2x^{3}$ i.e.
$4\theta_{1}^{3}=27\theta_{2}^{2}$. The $+$ case is entirely analogous.

In Appendix LABEL:app:2 we prove the following result telling us that in a
broad range of cases of interest to us the catastrophe manifold has trivial
topology. On the other hand, as we will demonstrate, the mapping $\chi$ has a
highly non-trivial structure.

Suppose there are no bifurcation points near the boundary of the parameter
domain $P$ and that $P$ is diffeomorphic to a 2 dimensional disk. Then the
connected components of $\mathcal{M}$ are diffeomorphic to the disk $D^{2}$.

Standing hypothesis 1. We therefore assume throughout that the connected
components of $\mathcal{M}$ are diffeomorphic to 2 dimensional disks. This is
the case for a far more general set of circumstances than those given in Prop.
3.

It follows from this hypothesis that any simple closed curve $\Gamma$ in one
of these components $\mathcal{M}_{0}$ separates $\mathcal{M}_{0}$ into two
connected components, one of which contains the boundary of $\mathcal{M}_{0}$.
This component we call the _exterior_ of $\Gamma$ and the other component the
_interior_.

In the discussion below we often lift to the catastrophe manifold
$\mathcal{M}$ smooth curves of the form $\theta=\gamma(t)$, $0\leq
t\leq\tau_{0}$ in the parameter space that do not meet any bifurcation points.
A lift is a curve $\mathbf{x}=\Gamma(t)$ such that $\chi(\Gamma(t))=\gamma(t)$
and to ensure it is uniquely determined we will need to fix a point
$\Gamma(t_{0})$ such that $\chi(\Gamma(t_{0}))=\gamma(t_{0})$. Often the curve
will be closed so that $\gamma(0)\equiv\gamma(\tau_{0})$ and then $\Gamma$ can
be taken to be closed although one may have to extend the domain of $t$
because the period of $\Gamma$ is a multiple of $\tau_{0}$. The lifts exist
because $\gamma$ meets no bifurcation points.

In some other cases we are able to lift curves from parameter space to
$\mathcal{M}$ even though they intersect bifurcation curves. An example is the
lift of curves around a cusp in Sect. 4, p. 4 and Fig. 4. These lifts can be
carried out because the process is local and we can use the universal
unfolding of the bifurcation to check that the lift is possible.

###  Fold curves and bifurcation sets

It follows from the above (cf. [trotman_zeeman], Theorem 8.1,
[arnold2013dynamical], I Sect. 1.4) and the fact that our parameters are
2-dimensional that generically the set of singularities of $\chi$ (i.e.
$\mathcal{S}$) consists of a finite number of 1-dimensional submanifolds of
$\mathcal{M}$. Moreover, if $C$ is one of these submanifolds then every point
on $C$ is either a fold point or a cusp point. We call these submanifolds
_fold curves_ and those that are circles _fold circles_. Clearly, since the
closure of $\mathcal{M}$ is compact, a fold curve that does not meet the
boundary of $\mathcal{M}$ must be a fold circle.

Standing hypothesis 2. We always assume that our parameterised system is
generic in the sense discussed above i.e. that the fold curves are smooth
curves in $\mathcal{M}$.

Generically, if $C$ is a fold circle then the associated bifurcation set
$B_{C}=\chi(C)$ is a closed curve without any self crossings and
$\chi|C:C\rightarrow B_{C}$ has degree one. It is smooth except possibly at a
finite number of cusp points and $B_{C}$ separates $\mathbb{R}^{2}$ into
exactly two components, one of which (denoted $B_{C}^{\text}{int}$) is a disk.

Proof. See Appendix LABEL:app:1 for details.

The _bifurcation set_ is the subset in parameter space of all bifurcation
points. A _bifurcation curve_ is any curve contained in the bifurcation set. A
_fold curve_ is a smooth curve of fold points in the catastrophe manifold
$\mathcal{M}$ and this is called a _fold circle_ iff it is a closed curve. A
_fold bifurcation curve_ is any set of the form $B_{C}=\chi(C)$ where $C$ is a
fold curve. A _bifurcation segment_ or _segment of a bifurcation curve_ is a
connected smooth curve contained in a set of the form $B_{C}=\chi(C)$ where
$C$ is a fold curve.

A very useful construct associated with fold curves and circles that we will
use a number of times involves an analysis of their neighbourhoods. Suppose
$C$ is a fold circle. Consider a small tubular neighbourhood $N$ of $C$. By
definition of a tubular neighbourhood there is a retraction $q:N\rightarrow C$
making $(q,N,C)$ a vector bundle whose zero section is the inclusion
$C\rightarrow N$. Then $N\setminus C$ has two components one of which is
contained in the interior of $C$. We denote this component by
$N^{\text}{int}_{C}$. If $N$ is sufficiently small then $N^{\text}{int}_{C}$
meets no other fold circles and $\chi|N^{\text}{int}_{C}$ is injective. This
injectivity follows from the fold structure given by equation (2). Let
$N^{\text}{ext}_{C}$ denote the other component of $N\setminus C$.

The following result follows from this discussion.

If $C$ is a fold circle then $\chi$ restricted to the interior $\breve{C}$ of
$C$ has degree 1 and at all points in $\breve{C}$ near $C$, $\chi$ is
injective.

###  Heteroclinic connections and flips

We will be interested in systems where there are two or more index 1 saddles.
In such a system a codimension 1 global bifurcation that can occur involves a
heteroclinic connection. This occurs when the unstable manifold of one of the
saddles $s_{1}$ intersects the stable manifold of another saddle $s_{2}$. In
this case we call $s_{2}$ the _target saddle_ and $s_{1}$ the _source saddle_.
If this occurs at a parameter $\theta$ then we call this parameter value a
_flip point_ and we refer to such bifurcations as _flip bifurcations_ or just
flips.

Generically, in 2-parameter families these bifurcations occur on smooth curves
which we call _flip curves_ and these curves end on fold bifurcation curves in
ways described below in Sect. 5.

##  4 Counting cusps on fold curves & circles

###  The fold orientation

Consider the fold equation (1). Close to $x=0$ the $\dot{x}$ is always $\geq
0$ or $\leq 0$. Thus close to a fold point there is a definite direction of
flow on a center manifold and this induces an orientation on the center
manifold and its tangent space $\ell(\mathbf{x})$. Moreover,
$\ell(\mathbf{x})$ varies smoothly with $\mathbf{x}\in C$.

Using this smoothness we have that this orientation is locally consistent on
any segment of a fold curve $C$ which contains no cusp points but it switches
at cusp points as can be verified by looking at the normal form (2). Thus we
have the following proposition.

Consider a fold curve $C$ whose endpoints are not cusps. Then, for a generic
parameterised system, the number of cusps in $C$ equals the number of
orientation switches in $C$.

###  Saddle & near-fold bundles

Now consider a smooth closed curve $\Gamma$ in $\mathcal{M}$ such that, for
each point $\mathbf{x}=(x,\theta)\in\Gamma$, $x$ is either an index 1 saddle
point or a pseudo-hyperbolic attracting rest point (Sect. 2). In this case we
say that such a curve $\Gamma$ is _pseudo-hyperbolic_.

Then we can associate to such a point $\mathbf{x}$ the 1-dimensional tangent
space $E^{c}(x)$ to the center manifold $W^{c}(x)$ of the rest point at $x$.
These vary smoothly with $\mathbf{x}$ and so we have a bundle
$\pi:E^{c}_{\Gamma}\rightarrow\Gamma$ over $\Gamma$ with fibre the line
$E^{c}(x)$. Clearly this bundle $E^{c}_{\Gamma}$ must be either a cylinder or
a Möbius band and the topology of this bundle encodes important information.

###  Fold circle bundle

For the special case where $\Gamma$ is a fold circle $C$ we call the bundle
$E^{c}_{C}$ the _fold circle bundle_ and denote it $\mathcal{C}_{C}$.

Now if $C$ is a fold circle and we start at a point $\mathbf{x}\in C$ which is
not a cusp point and traverse the circle, when we arrive back at $\mathbf{x}$
the orientation will have changed a number of times. If this number is odd
when we arrive back at $\mathbf{x}$, the orientation will be the opposite to
when we started out. Thus for the fibres of $\mathcal{C}_{C}$ to glue smoothly
at $\mathbf{x}$ and for the orientation to be constant on fibres near to
$\ell(\mathbf{x})$ (since $\mathbf{x}$ is not a cusp point), $\mathcal{C}_{C}$
must be a Möbius band. Conversely, if $\mathcal{C}_{C}$ is a Möbius band then
the number of switches must be odd. Thus, for a generic parameterised system,
we have the following.

If $C$ is a fold circle, $\mathcal{C}_{C}$ is a Möbius band if and only if $C$
contains an odd number of cusps.

We call such fold circles _Möbius_ fold circles.

###  Conditions for a fold circle to be Möbius

We now investigate the immediate neighbourhood of a Möbius fold circle in
$\mathcal{M}$. In particular, we demonstrate that a simple closed curve
$\Gamma$ in $\mathcal{M}$ has the bundle $E^{c}_{\Gamma}$ a Möbius band if and
only if it contains an odd number of Möbius fold circles, and we show that, if
$C$ is a Möbius fold circle, then on the exterior of $C$ close to $C$, $\chi$
has degree two. As a corollary we see that although a fold circle can contain
simple fold circles, it cannot contain a Möbius one.

Suppose that $\Gamma$ is a smooth simple closed pseudo-hyperbolic curve in
$\mathcal{M}$ with $E^{c}_{\Gamma}$ a Möbius band. Then in the generic
situation, $\Gamma$ contains an odd number of Möbius fold circles in its
interior. Moreover, if $C$ is one of these Möbius fold circles one can find
another such curve $\Gamma^{\prime}$ such that $C$ is the only fold circle
contained in the interior of $\Gamma^{\prime}$ except possibly for simple fold
circles contained in the interior of $C$.

Proof. See Appendix LABEL:sect:E.

Now consider a Möbius fold circle $C$. Then $C$ contains an odd number of
cusps. Suppose that the folds in $C$ involve a saddle and an attractor with
the interior of $C$ consisting of points $\mathbf{x}=(x,\theta)$ where $x$ is
an attractor. Let $B_{C}^{\text}{int}$ be the open disk bounded by $B_{C}$.

Now consider a closed curve $\gamma$ that has one self intersection inside
$B_{C}$ near each cusp (temporarily called nodes), loops around the cusps as
shown in Fig. 4C, and otherwise lies entirely in the interior
$B_{C}^{\text}{int}$ of $B_{C}$ near to $B_{C}$. Provided it is sufficiently
close to $B_{C}$ this curve can be lifted to a smooth curve $\Gamma$ in
$\mathcal{M}$ so that (i) $\chi(\Gamma)=\gamma$, and (ii) if
$(x,\theta)\in\Gamma$, $x$ is an attractor of $X_{\theta}$ which we denote by
$x(\theta)$.

[width=]../figures/around_cusp_big

Figure : This shows how the disposition of the rest points involved in a
standard cusp vary as one follows the projection into parameter space under
$\chi$ of a closed curve (red) of attractors in $\mathcal{M}$ which
approximates the fold curve giving the cusp. Since the red curve is close to
the bifurcation set there is a center manifold through the red attractor. The
grey arrow shows the orientation on the center manifold. The cusp shown is a
standard one and going round the loops swaps the attractors; for a dual cusp
the roles of saddles and attractors are interchanged.

Since the curve $\gamma$ lies close to $B_{C}$, the attractor $x(\theta)$,
$\theta\in\gamma$, lies on a smoothly changing center manifold. Since the
center manifold varies smoothly, we can give it a consistent orientation
$\mathcal{O}$ for all $\theta\in\gamma$. At each node and at all points of
$\gamma$ in $B_{C}^{\text}{int}$, the unstable manifold of the saddle
intersects the stable manifold of the attractor created in the fold
bifurcation of $B_{C}$.

We define the _relative orientation_ of the saddle at $\theta\in\gamma\cap
B_{C}^{\text}{int}$ to be the relative orientation of the saddle to the
attractor using the orientation $\mathcal{O}$. Note that although the
orientation $\mathcal{O}$ is constant the relative orientation of the saddle
can change. In fact, this relative orientation is constant along any segment
of $\gamma$ wholly inside $B_{C}^{\text}{int}$ but flips as $\theta$ moves
around the loops going from the node around the cusp (Fig. 4A). As a
consequence, if there are an odd number of cusps then starting from a node and
doing one revolution around $\gamma$ back to that node results in a flip in
the relative orientation. But at the node $\theta$, since it is in
$B_{C}^{\text}{int}$ and is close to the cusp, the system has two attractors
and a saddle. Thus traversing $\gamma$ results in the attractors swapping.

A similar picture holds for dual cusps with the roles of the attractors and
saddles interchanged and with the unstable manifold of the saddle replacing
the role of the center manifold.

Now let $U_{C}$ be an open set in $\mathcal{M}$ that contains $C$ in its
interior and meets no other fold curves in the exterior of $C$ and consider
$\chi_{1}=\chi|U_{C}$. The choice of nodes above was arbitrary apart from
being in $B_{C}^{\text}{int}$ close to a cusp. Consequently, we deduce that
there is a neighbourhood $U$ of the node such that $\chi_{1}^{-1}(U)$ consists
of two disks on each of which $\chi_{1}$ is orientation preserving (for
reasons given below). Consequently, $\chi_{1}$ has degree 2. This gives the
following lemma and its very useful corollary.

Consider the following conditions on a fold circle $C$. (i) $C$ is a Möbius
fold circle; (ii) $C$ contains an odd number of cusps; (iii) there is a simple
closed pseudo-hyperbolic curve $\Gamma$ in $\mathcal{M}$ that contains in its
interior $C$ and no other fold circles that are in the exterior of $C$ such
that (a) under $\chi$, $\Gamma$ double covers a curve $\gamma$ in the
parameter space and (b) $E^{c}_{\Gamma}$ is Möbius; and (iv) $C$ is contained
in an open disk $\mathcal{G}$ in $\mathcal{M}$ on which $\chi$ has degree 2.
Generically, (i) $\iff$ (ii) $\iff$ (iii) $\Rightarrow$ (iv).

Proof. We already proved (i) $\iff$ (ii) in Prop. 4. That (i) implies (iii)
follows from the above discussion. The curve $\Gamma$ discussed there double
covers the curve $\gamma$ (cf. Fig. 4). Moreover, since $\gamma$ can be taken
arbitrarily close to $B_{C}$, $\Gamma$ can be taken arbitrarily close to $C$.
Thus the fibres of the bundles $\mathcal{C}_{C}$ and $E^{c}_{\Gamma}$ will be
close and consequently $\mathcal{C}_{C}$ and $E^{c}_{\Gamma}$ will both have
the same Möbius band topology.

To see that (iii) implies (i) we need to use Prop. 4. This asserts that we can
modify the curve $\Gamma$ in (iii) so that it contains in its interior the
fold curve $C$ and no other fold points in the exterior of $C$. Then the
modified $\Gamma$ is homotopic to curves $\Gamma^{\prime}$ arbitrarily close
to $C$ but exterior to $C$. Since they are connected by a homotopy,
$E^{c}_{\Gamma}$ and $E^{c}_{\Gamma^{\prime}}$ are homeomorphic, and since
$\Gamma^{\prime}$ is close to $C$, $E^{c}_{\Gamma^{\prime}}$ and
$\mathcal{C}_{C}$ are homeomorphic.

To see that (iii) implies (iv) note that $\chi$ is a local diffeomorpism at
points of $\Gamma$ and, moreover, since there are no fold points on $\Gamma$
the sign of the Jacobian of $\chi$ at two points on $\Gamma$ is the same. It
follows immediately that if $\Gamma^{\prime}$ is any simple closed curve
containing $\Gamma$ in its interior then $\chi$ has degree 2 on the interior
of $\Gamma^{\prime}$ which is a disk.

If $C$ is a fold circle then there are no Möbius fold circles contained in its
interior.

Proof. This follows from part (iv) of the Theorem and the fact that the
restriction of $\chi$ to the interior of a fold circle has degree 1 by Lemma
3.

More generally, if $\mathcal{G}$ is a subset of $\mathcal{M}$ diffeomorphic to
a disk and $\chi|\mathcal{G}$ has degree 1 then $\mathcal{G}$ contains no
Möbius fold circles.

###  Cusps and bifurcation diagrams

In applications it is common to plot bifurcation diagrams in which the
position of the rest points is plotted against a single bifurcation parameter
$\theta_{1}$. For example for bistable systems one obtains a S or Z shaped
multivalued graph over the parameter. If there is another parameter
$\theta_{2}$ involved in the system it is then of interest to know whether the
2-parameter bifurcation system contains cusps.

Now we extend the cusp index for use in such a situation to determine the
number of cusps. We restrict to fold bifurcations between attractors and index
1 saddles but it will be clear that it applies to other fold bifurcations.

[width=]../figures/along_bcurve

Figure : The black and red curve is a curve in the catastrophe manifold such
as that over the line AB in Fig. 3. The red part corresponds to saddles. The
left hand fold point determines the orientation on the center manifold shown
using the hollow arrow. This induces the corresponding orientation on the
saddle points in the systems corresponding to the points along the red curve.
Then all the systems on the red curve close to the right hand fold point have
the opposite orientation to that fold point. This implies that on any fold
curve joining the two fold points shown there are an odd number of cusps.

Suppose that $\Gamma(t)$, $0\leq t\leq 1$ is a smooth curve in $\mathcal{M}$
that connects two fold points $\mathbf{x}_{i}$, $i=1,2$, and except at its
end-points consists of index 1 saddles. Suppose moreover that the tangent
$E^{u}(\mathbf{x})$ to the unstable manifold of these saddles $\mathbf{x}$
varies smoothly as $t$ varies.

Given this, we can compare the orientation of $\ell(\mathbf{x}_{1})$ with that
of $\ell(\mathbf{x}_{2})$. Use the orientation of $\ell(\mathbf{x}_{1})$ at
the fold point $\mathbf{x}_{1}$ to put an orientation on $E^{u}(\Gamma(t))$
for $t$ close to $0$ using the fact that $E^{u}(\Gamma(t))$ converges to
$\ell(\mathbf{x}_{1})$ as $t$ tends to $0$ and extend this to all $0\leq t\leq
1$ by continuity. Since $E^{u}(\Gamma(t))$ converges to $\ell(\mathbf{x}_{2})$
as $t$ tends to $1$ this produces an orientation $\mathcal{O}$ on
$\ell(\mathbf{x}_{2})$ and the question is whether this is the same as that
$\mathcal{O}^{\prime}$ induced by the fold at $\mathbf{x}_{2}$ (see Fig. 4).

[width=]../figures/cuspexists_bifn_diag

Figure : Over part of the boundary $\partial P$ of a 2-dimensional domain of
parameter space $P$ we assume we have an S-shaped bistable section and that
the system is monostable over the rest of $\partial P$. Then if Condition (*)
holds it follows that inside $P$ there must be at least one cusp. This is
because there must be a fold curve joining the two folds shown and on this
fold curve the two folds shown have opposite orientations.

If $\,\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ are on a common fold curve $C$ and
the orientations $\mathcal{O}$ and $\mathcal{O}^{\prime}$ of
$\ell(\mathbf{x}_{2})$ are incoherent then there is an odd number of cusps on
$C$ between $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$.

In a number of applications of CT (e.g. [zeeman1977CT]) one is given the
behaviour on the boundary of the parameter domain and one wants to deduce the
bifurcation behaviour inside. By far the most considered situation in this
regard is where the boundary behaviour is as in Fig. 4(Left) i.e. one has a
parameter space $P$ which is a 2-dimensional disk-like region and the
catastrophe manifold over the boundary $\partial P$ is bistable over part of
$\partial P$. In this case one would like to know if there is a cusp inside
$P$.

Suppose the boundary $\partial P$ is given by $\theta=\theta(s)$, $0\leq s\leq
1$, and that the interval where there is bistable behaviour is given by
$\theta(s)$, $s_{1}\leq s\leq s_{2}$, so that there are folds at $s=s_{1}$ and
$s=s_{2}$. To obtain the required result about cusps we need the following
condition on the two attractors involved in the two folds: (*) for at least
one of these attractors, the attractor varies smoothly with $s$ in
$(s_{1},s_{2})$ and for all $s$ in $(s_{1},s_{2})$ the unstable manifold of
the saddle point intersects the basin of this attractor. This condition is
automatic if the state space is 1-dimensional but not so in higher dimensions.
To understand why see Fig. LABEL:fig:bistable_but_no_switch in Appendix
LABEL:sect:just. Note that we are not assuming that the saddle and attractors
that are involved in the fold bifurcations are all the rest points in the
system. If they are then Condition (*) is redundant. An example of a bistable
1-parameter system that swaps orientation but where the two folds are not on a
common fold curve is given by taking a straight line in the parameter space in
Fig. 6 which is roughly parallel to one of the boundary curves and does not
meet the deltoid $\mathcal{B}$. Then we have the following theorem:

If the two fold points in $\mathcal{M}$ corresponding to the two folds at
$\theta=\theta(s_{1})$ and $\theta=\theta(s_{2})$ are on a common fold curve
in $\mathcal{M}$ and condition (*) holds then generically there are an odd
number of cusps in $P$.

Proof. By Condition (*), as $s$ moves between the folds at $s_{1}$ and $s_{2}$
the unstable manifold of the saddle must intersect the basin of the smoothly
varying attractor. The identity of this attractor remains constant for all $s$
and we call it $A$. The unstable manifold can also be involved in heteroclinic
bifurcations but, in generic systems, for all but a finite number of $s$
values, it will also intersect the basin of a different attractor of the
system at $\theta(s)$ which we denote by $B(s)$. At these finite number of $s$
values the identity of $B(s)$ may change. In the generic case when $s$ in
$(s_{1},s_{2})$ is close to $s_{1}$ or $s_{2}$ the attractors $A$ and $B(s)$
vary smoothly and they are connected by the unstable manifold of the saddle.
If for $\mathbf{x}_{1}$ in Prop. 4 we take the fold involving $A$, then
$\mathcal{O}$ points away from $A$ when $s$ is near the bifurcation value.
Since the unstable manifold of the saddle and $A$ vary smoothly, $\mathcal{O}$
remains doing this for all $s\in(s_{1},s_{2})$ and therefore is incompatible
with the orientation at the fold involving $B(s)$ since this orientation
points away from $B(s)$. Thus the odd number of cusps follows from Prop. 4.

Historical Note. Results similar to Theorem. 4 for the case where $n=1$ were
obtained by Stewart in [stewart1980catastrophesup] using entirely different
methods based on singularity theory. Ian Stewart pointed out to us a paper of
Tim Poston [poston1978deducing] that discusses the problem in Theorem. 4 and
mentions a preprint by E. C. Zeeman (_A boundary value problem involving
cusps_) in which this problem is treated. However, in Poston’s discussion the
need to check the incompatibility of the orientations $\mathcal{O}$ and
$\mathcal{O}^{\prime}$ is not mentioned. If it existed this manuscript seems
to have disappeared and is not listed in the Zeeman Archive
(www.lms.ac.uk/2015/zeeman_archive). One of us (DAR) recalls Zeeman mentioning
that someone else had proved this result but we cannot trace such a paper.

##  5 Characterising 3-attractor MS components

[width=0.6]../figures/cusp_cusp_noattrs

Figure :

Now we consider 3-attractor MS components $\Omega$ in 2-parameter systems with
compact dynamics. Our aim is to understand the bifurcations in the boundary of
$\Omega$ and we will do this by deducing a set of rules for the way in which
cusps, crossings and flip end points are disposed around the boundary.

We denote the attractors by $A,B$ and $C$. Since they do not bifurcate in
$\Omega$ they vary smoothly with the parameters and maintain their identity.

As one proceeds around the boundary $\partial\Omega$ of $\Omega$ there is a
sequence of codimension 2 points $\theta_{1}\,\ldots\theta_{N}$ with
$\theta_{N}=\theta_{1}$ as described in Fig. 2 of I, namely cusps, dual cusps,
and crossings (see Fig. 5). (The other relevant codimension 2 points, namely,
the two types of endpoints of flip curves are dealt with in Sect. 5.) If we
know these then we know all the bifurcation structures associated with
$\partial\Omega$. The points $\theta_{i}$ and $\theta_{i+1}$ are joined by a
smooth fold bifurcation curve segment $\ell_{i}$. The bifurcations on this
segment involve one of the attractors $X\in\\{A,B,C\\}$ and we denote this by
$\ell_{i}\in\mathcal{L}_{X}$ or say that the attractor identity of $\ell_{i}$
is $X$. For a sequence $\ell_{i},\ldots,\ell_{i+j}$ with $\ell_{k}$ having
attractor identity $X_{k}$ we write
$\ell_{i},\ldots,\ell_{i+j}\in\mathcal{L}_{X_{i},\ldots,X_{i+j}}$. There are
strong constraints on what sequences can occur as we now describe.

###  Crossings of bifurcation curves

In general when two bifurcation curves cross, the two bifurcations may be
completely independent. However, this is not the case when the corresponding
rest points are both on the same smooth connector and are adjacent. This is
because, when it is smooth, the connector is contained in any center manifold
of a fold point on the connector and hence the fold orientations (Sect. 4) of
the two folds provide orientations for the connector that must be the same for
the simultaneous bifurcations to be allowed. See Figs. 5 and 5.

Now consider the case where the system has 3 attractors $P_{1}$, $P_{2}$ and
$X$, and two saddles, where $X$ is the central attractor. Assume all these are
on a smooth connector.

Firstly, note that there is only one possibility for the orientation of a fold
involving a peripheral attractor. The vector field near the fold point must
point along the connector towards the other rest points on the connector. In
particular, the orientations $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ of folds
involving the two peripheral attractors $P_{1}$ and $P_{2}$ are opposite.
Given this, when considering a fold curve or fold bifurcation curve involving
the central attractor $X$ we say _the fold of $X$ is compatible with_ $P_{i}$
(with $i=1$ or $2$) to indicate that the orientation of the fold involving $X$
is $\mathcal{O}_{i}$.

Now consider two smooth segments $\ell_{1}$ and $\ell_{2}$ of fold bifurcation
curves that cross each other. Suppose that the segment $\ell_{i}$, $i=1,2$,
involves the attractor $A_{i}\in\\{P_{1},P_{2},X\\}$. Then the following lemma
follows immediately from the above discussion.

[width=0.9]../figures/crossings_compat

Figure : (A) If at the central attractor the connector has a cuspoidal
structure then a fold bifurcation of this attractor is impossible. This is
proved in Lemma LABEL:lem:nofold Appendix LABEL:App:no_fold and basically
follows from the incoherence of the two arrows on the branches approaching the
fold point. (B-D) This illustrates examples of what dynamics on the connector
are possible and impossible at points where bifurcation curves cross. The
orientations of two folds at adjacent rest points must be the same. There are
no restrictions on the way the blue connector approaches the red attractor.
(B) allowed since the arrows are consistent, (C) allowed since the folds are
not adjacent, (D) not allowed as center and bifurcating peripheral have
opposite orientations.

The only restrictions on crossings concern the case where one of the $A_{i}$
is the central attractor $X$. In this case the folds on $\ell_{1}$ and
$\ell_{2}$ all have the same orientation.

####  Crossings near cusps

A key fact in this section is that since a bifurcation curve $B_{C}$ is the
image of a single fold curve in $\mathcal{M}$, as one moves along $B_{C}$,
either the attractor or the saddle that is bifurcating remains constant.

The bifurcation curve through a cusp is the projection under $\chi$ of a fold
curve in $\mathcal{M}$. Near the cusp point the bifurcation curve in parameter
space $\Omega$ consists of two smooth segments $\ell$ and $\ell^{\prime}$
joined at the cusp. In a neighbourhood of the cusp point in $\Omega$ there is
a smooth center manifold containing the rest points involved in the
bifurcation because at the cusp point the central manifold is normally
hyperbolic and hence varies smoothly with the parameters and contains the
bifurcating rest points.

[width=]../figures/cuspexists_final

Figure : Illustration of Lemma 5 and Theorem 5. A. The orientation of the fold
bifurcation involving the central attractor determines the possible crossings
of the fold curve. B. The peripheral attractor involved in the fold constrains
the possible crossings of the fold curve. C. Standard cusp. Fold curves
crossing the cusp’s branches satisfy the conditions in (A) and (B). . D. Dual
cusp. Fold curves crossing the cusp’s branches satisfy the conditions in (A)
and (B) and the crossings imply the existence of the cusp. E. The orientations
of the folds of the central attractor imply the existence of a cusp. F.
Certain consecutive crossing are not possible. Legend: A diamond shape denotes
a fold bifurcation of the specified attractor. Bifurcation curves are coloured
accordingly.

For a dual cusp the folds along both of the segments $\ell$ and
$\ell^{\prime}$ involve the central attractor and they have opposite
orientations on the two segments (Fig. 5D). For a standard cusp the two smooth
segments either side of the cusp correspond to a common saddle and two
different attractors. One of them is the central attractor $X\in\\{A,B,C\\}$
and the other a peripheral attractor $P\in\\{A,B,C\\}$ (Fig. 5C).

As a corollary of Lemma 5, if $C$ is a fold curve with such a standard cusp
involving $P$ and $X$ compatible with the other peripheral attractor
$P^{\prime}$, then a fold curve crossing the $P$ side involves the bifurcation
of $P^{\prime}$ or $X$ compatible with $P$, and a fold curve crossing the $X$
side involves the bifurcation of $P^{\prime}$ (Fig. 5C).

Now consider a fold curve $C$ in $\mathcal{M}$ such that $B_{C}=\chi(C)$ is
contained in the boundary of a 3-attractor MS component and which is such that
moving out of the component across $B_{C}$ involves the loss of an attractor.
Suppose that the attractors in the MS component are (central) $X$,
(peripheral) $P_{1}$ and $P_{2}$. Suppose that on $B_{C}$ there are two
crossing points at $\theta_{1}$ and $\theta_{2}$ and none in between them and
suppose the fold bifurcation curves which cross at these points are $\ell_{1}$
and $\ell_{2}$ respectively.

1\. $\ell_{1}$ and $\ell_{2}$ are associated with different peripheral
attractors if and only if $C$ is a fold curve for $X$ and it contains an odd
number of dual cusps (Fig. 5D).
2\. If the folds in $\ell_{1}$ at $\theta_{1}$ and in $B_{C}$ at $\theta_{2}$
involve the central attractor $X$ and the former (resp. latter) have
orientation compatible with $P_{1}$ (resp. $P_{2}$) then there is an odd
number of standard cusps in $B_{C}$ between $\theta_{1}$ and $\theta_{2}$ and
the fold in $\ell_{2}$ at $\theta_{2}$ involves $P_{2}$ (Fig. 5E).
3\. It is not possible that the fold in $\ell_{1}$ at $\theta_{1}$ involves
$X$ with orientation compatible with $P_{1}$ and the one in $\ell_{2}$ at
$\theta_{2}$ involves $P_{1}$ (Fig. 5F).

Proof. Proof of 1. If $C$ is a fold curve for $X$ and it contains an odd
number of dual cusps, then the orientation of the $X$ fold is different at
$\mathbf{x}_{1}=(x,\theta_{1})$ and $\mathbf{x}_{2}=(x,\theta_{2})$. Hence, by
lemma 5 the crossing curves involve the bifurcation of different peripheral
attractors.

Conversely, assume that the crossing curves involve different peripheral
attractors and in particular that $\ell_{1}$ at $\theta_{1}$ involves $P_{1}$.
Then, $B_{C}$ at $\theta_{1}$ involves either $X$ compatible with $P_{1}$ or
$P_{2}$. Assume that it is $P_{2}$. Since the fold in $\ell_{2}$ at
$\theta_{2}$ involves $P_{2}$ there must be a cusp in $C$ that changes the
bifurcating attractor. This cusp is a standard cusp that involves $X$
compatible with $P_{1}$, which again can not cross $\ell_{2}$ at $\theta_{2}$.
Therefore, $C$ at $\theta_{1}$ involves $X$ compatible with $P_{1}$.
Similarly, $C$ at $\theta_{2}$ involves $X$ compatible with $P_{2}$.
Therefore, $C$ is a curve of $X$ folds with different orientations at
$\theta_{1}$ and $\theta_{2}$ and therefore it contains an odd number of cusps
between these two points.

Proof of 2. By lemma 5 the fold along $B_{C}$ near $\theta_{1}$ involves
$P_{1}$ and the fold along $\ell_{2}$ near $\theta_{2}$ involves $P_{2}$.
Therefore $B_{C}$ contains an odd number of dual cusps between these two
points.

Proof of 3. The condition at $\theta_{1}$ implies that at that point in
$B_{C}$ the saddle between $P_{1}$ and $X$ bifurcates with $P_{1}$. Thus there
cannot be a crossing by $P_{1}$ at $\theta_{2}$ as this would also have to
involve the same saddle to create the additional fold.

###  Heteroclinic bifurcations (flips) and flip curves

Before proceeding we need to discuss flip curves and how they end. This
relates to the question of whether or not the connector is smooth. As one
moves along a bifurcation curve such smoothness can change. This happens at
certain endpoints of flip curves.

[width=]../figures/fold+flip_point

Figure : This shows how a generic flip curve (black) meets a fold curve
(yellow) when the fold occurs at the central attractor. See [Vegter82].

####  Target saddles

The first way that flip curves can end is by a bifurcation of the
corresponding target saddle.

Consider a smooth fold bifurcation segment $\ell$ involving the attractor
$X\in\\{A,B,C\\}$. Suppose that at a point $\theta_{0}$ on $\ell$ the
connector changes from smooth to non-smooth. Then, generically, $X$ is central
at $\theta_{0}$ which is the end-point of a flip curve where the target saddle
is destroyed in a fold bifurcation with $X$. There is a neighbourhood $U$ in
$\ell$ of such a point $\theta_{0}$ such that for $\theta$ in one component of
$U\setminus\\{\theta_{0}\\}$ the connector is smooth and $X$ is central, while
for $\theta$ in the other component the connector is non-smooth and another
attractor $Y$ is central. The third attractor $Z$ is the source attractor. In
this situation the only fold bifurcation curves that can cross the
$U\setminus\\{\theta_{0}\\}$ must involve the attractor $Z$.

Proof. For the first part of the proposition see Theorem I.3 and Corollary I.4
of [Vegter82]. Strictly speaking the results quoted are for gradient systems
but it is clear that the arguments used apply to our situation. According to
this, the generic situation is as follows. On a neighbourhood of $\theta_{0}$
there are coordinates $(\mu_{1},\mu_{2})$ such that $\theta_{0}$ is given by
$\mu_{1}=0,\mu_{2}=0$, the fold bifurcation curve through $\theta_{0}$ is
given by $\mu_{2}=0$ and the flip curve is given by $\mu_{2}=\mu_{1}^{2}$,
$\mu_{1}\geq 0$. Along $\mu_{2}=0$ the connector is smooth for $\mu_{1}>0$ and
non-smooth otherwise.

To see the result about crossing fold bifurcation curves note that in the
component of $U\setminus\\{\theta_{0}\\}$ where the connector is smooth, the
orientation at the fold is compatible with $Z$ and not $Y$. In the other
component, the central attractor is $Y$ which is therefore not able to undergo
a fold bifurcation. Thus any crossing must be by a curve involving $Z$.

It follows that if there are no flip curves then the connector is always
smooth or always non-smooth. Also a corollary is that in terms of its
connections with crossing points and cusps such a bifurcation segment with a
single such target flip curve endpoint behaves exactly as one that has a
smooth connector all along its length.

The above result raises the question of how the transition from a smooth to a
cuspoidal connector takes place in the region where there are three
attractors. If the attractors involved are as in Fig. 5 then the transition
involves the yellow attractor i.e. the attractor that is close to bifurcation.
When the system is sufficiently close to the fold curve the yellow attractor
$x$ has a center manifold and a strong stable manifold $W^{ss}(x)$ as
described in Sect. 2. The transition takes place when the unstable manifold
from the source saddle $s_{S}$ intersects $W^{ss}(x)$ and flips from joining
smoothly at $x$ to the unstable manifold of the other saddle, to having a
cuspoidal joint to it at $x$.

####  Source saddles

The other way a flip curve can end is via a fold bifurcation that destroys the
source saddle. This is more straightforward and needs no special
consideration. It meets the fold curve transversally.

####  Crossings sequences

Recall that if the connector is not smooth at the point where two fold
bifurcation curves cross then the two crossing curves must involve both
peripheral attractors as a fold bifurcation of the central attractor is
impossible in this case (see Appendix LABEL:App:no_fold).

Consider an adjacent pair of crossing points involving the smooth segments
$\ell_{i-1}$, $\ell_{i}$ and $\ell_{i+1}$ where these respectively involve the
attractors $Y$, $X$ and $Z$. Assume that for each segment, if it contains more
than one target flip curve endpoint then they all involve the same target
saddle. Then if $X$ is central at some point on $\ell_{i}$, $Y=Z\neq X$.

Proof. Firstly, suppose that there are no flip curves along $\ell_{i}$. Then
as you move along the fold curve $\ell_{i}$ the central attractor remains the
same and the orientation $\mathcal{O}$ of the fold at the central attractor
stays constant. At a crossing of $\ell_{i}$ by a fold bifurcation curve
corresponding to $P$ the orientation induced by the bifurcation of $P$ must be
compatible with $\mathcal{O}$ and this is the case for only one peripheral
attractor. If there is a flip curve ending then the result follows from Prop.
5.

With the same assumption as in the previous lemma, suppose that
$\ell_{i},\ldots,\ell_{i+j}$ is a sequence of crossing points and
$X_{i},\ldots,X_{i+j}$ is the corresponding sequence of attractors. Then if we
replace all triples of the form $PXP$ (where $X$ is central and $P$
peripheral) by $P$, the resulting sequence just alternates between $P$ and
$P^{\prime}$ where $P^{\prime}$ is the other peripheral attractor.

Proof. This follows from the fact that by Lemma 5 if $\ell\in\mathcal{L}_{X}$
has $X$ central for some $\theta\in\ell$ then the crossings are of the form
$PXP$. Thus replacing the $PXP$ triples by $P$s removes all $X$s. Moreover, a
$PXP$ triple is adjacent either to a $XP$ pair or a $P^{\prime}$. So repeating
this process produces a sequence alternating between $P$ and $P^{\prime}$.

###  Simplest/minimal MS components

Given the constraints outlined above it is clear that for a given
configuration of cusps the first four of the following give the minimal
complexity examples. The fourth is more complex (see Sect. 6). In the first
three the connector is always smooth and there are no flip curves. The first
two can be obtained as a 2 dimensional section through the butterfly
catastrophe.

  1. 1.

With a dual cusp and no other cusps: I Fig. 5A.

  2. 2.

With a dual cusp and a standard cusp: I Fig. 5A(inset).

  3. 3.

With a standard cusp and no other cusps: I Fig. 5B.

  4. 4.

With a standard cusp and a flip curve: I Fig. 5C.

  5. 5.

With no cusps and only smooth bifurcation segments: I Fig. 5D & see Sect. 6
below.

##  6 Topological charge and the elliptic umbilic

Our goal in this section is to explain why the boundary behaviour in Fig. 6
necessitates the complex dynamics inside. This is a particularly striking
boundary value problem and we introduce the notion of a phase or _topological
charge_ into the analysis. In physics an optical vortex has a topological
charge. This is observed by interfering an optical vortex with a plane wave of
light. This reveals the spiral phase of the vortex as concentric spirals. The
number of arms in the spiral equals the topological charge.

In our case the topological charge of a fold circle is the number of cusps it
contains. It is topological because when measured mod 2 the even and odd cases
correspond to the topology of the various line bundles we have introduced. One
can recognise the presence or not of mod 2 charge by topology. Lastly, non-
zero mod 2 charge is rigid and cannot be perturbed or homotoped away while
zero charge fold curves can.

We are concerned with a system where the boundary behaviour is as shown in
Fig. 6, and described as follows:

Compact Elliptic Umbilic boundary behaviour assumption is that there are three
smooth curves $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ which intersect as shown
in Fig. 6 and on which the indicated attractor undergoes a fold bifurcation
that destroys the attractor. On a neighbourhood of the three curves we have
the bistable and monostable behaviour shown in Fig. 6.

We denote by $\Delta$ the open region enclosed by $\ell_{A}$, $\ell_{B}$, and
$\ell_{C}$. Clearly, it is diffeomorphic to the 2-dimensional disk. Since we
are interested in characterising 3-attractor MS components we also assume that
inside $\Delta$ we have three attractors $A$, $B$ and $C$ that have smooth
dependence upon the parameters.

We show that given this boundary behaviour the system in $\Delta$ contains all
the ingredients of the compact elliptic umbilic (CEU) that we now introduce.

###  The compact elliptic umbilic

[width=]../figures/CEU

Figure : The compact elliptic umbilic. See text for description.

This is the set of gradient systems associated with the potential

$f_{\theta}(x_{1},x_{2})=x_{1}^{4}+x_{2}^{4}+x_{1}^{3}-2x_{1}x_{2}^{2}+\theta_{3}(x_{1}^{2}+x_{2}^{2})+\theta_{1}x_{1}+\theta_{2}x_{2}.$
(3)

These parameterised families of gradient systems are given by
$\dot{x}_{i}=-\sum_{j}g^{ij}{\partial f_{\theta}}/{\partial x_{j}}$ where
$g_{ij}$ is a Riemannian metric and $(g^{ij})=(g_{ij})^{-1}$.

We illustrate the case where $\theta_{3}$ is fixed at 0.1 but we will see this
provides a ubiquitous landscape structure. There are three curves $\ell_{A}$,
$\ell_{B}$, and $\ell_{C}$ in the parameter space $\mathbb{R}^{2}$ of fold
bifurcations as shown in Fig. 6. For $X\in\\{A,B,C\\}$ the curve $\ell_{X}$ is
where the attractor $X$ is destroyed in a fold bifurcation. These curves
enclose a triangular shaped region which we denote by $\Delta$. There is a
hypocycloid $\mathcal{B}$ with three cusps (a _deltoid_) inside $\Delta$ such
that for $\theta$ inside $\Delta$ but outside $\mathcal{B}$ the system has
three attractors that we label $A$, $B$, and $C$ plus two saddles. Inside
$\mathcal{B}$ it has an extra saddle and either an extra attractor (if
$\theta_{3}>0$) or an extra repellor.

For a given Riemannian metric $g_{ij}$ the corresponding gradient system also
has three flip curves in $\Delta$ as shown on Fig. 6 on which heteroclinic
bifurcations occur. These start on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ and
end on the hypocycloid $\mathcal{B}$. The configurations shown are for a
generic metric; for the standard one the situation is not generic and the
curves end at a cusp point [Khesin90sup].

Let $\mathcal{M}$ be the catastrophe manifold over $\Delta$ i.e.
$\mathcal{M}=\\{\mathbf{x}=(x,\theta):x\text{acriticalpointof}f_{\theta},\theta\in\Delta\\}$.
Since the attractors $A$, $B$ and $C$ vary smoothly, the components
$\mathcal{M}_{A}$, $\mathcal{M}_{B}$ and $\mathcal{M}_{C}$ of $\mathcal{M}$
corresponding to them are graphs over $\Delta$ that form disjoint components.
Thus we focus on $\mathcal{M}_{0}$ which is the remaining component of
$\mathcal{M}$ after $\mathcal{M}_{A}$, $\mathcal{M}_{B}$ and $\mathcal{M}_{C}$
have been removed. Let $\chi_{0}=\chi|\mathcal{M}_{0}$.

If $\mathcal{B}^{\text}{int}$ denotes the interior of $\mathcal{B}$ then
$\mathcal{M}_{1}=\chi_{0}^{-1}(\Delta\setminus\mathcal{B}^{\text}{int})$ is an
annulus that under $\chi_{0}$ double covers the annulus
$\Delta\setminus\mathcal{B}^{\text}{int}$ i.e $\chi_{0}|\mathcal{M}_{1}$ is
equivalent to the map:
$\mathbb{S}^{1}\times(-1,1)\ni(\theta,t)\mapsto(2\theta\text{mod}2\pi,t)\in\mathbb{S}^{1}\times(-1,1)$.

To obtain $\mathcal{M}_{0}$ we take the inner boundary of $\mathcal{M}_{1}$
which is a circle and glue in to it a disk correponding to the new non-saddle
rest point that is created by the fold bifurcation that occurs on
$\mathcal{B}$. This $\mathcal{M}_{0}$ is diffeomorphic to a disk but the
mapping $\chi_{0}$ is non-trivial.

###  Boundary conditions

Now we show that if we have the same boundary conditions as for the CEU then
we have dynamics closely similar to those of the CEU inside. Indeed the only
change is the possible addition of one or more simple fold circles inside the
Möbius fold circle corresponding to $\mathcal{B}$ that then add extra rest
points to the dynamical system that bifurcate in and out maintaining the
architypal CEU structure.

For the rest of this section we assume the CEU boundary behaviour defined
above in Sect. 6.

Since there is a neighbourhood $U$ of $\partial\Delta$ that contains no
bifurcation points in $U\cap\Delta$, then by Proposition 3, each connected
component of $\mathcal{M}$ over $\Delta$ is diffeomorphic to a disk. As above,
since the attractors $A$, $B$ and $C$ vary smoothly, the components
$\mathcal{M}_{A}$, $\mathcal{M}_{B}$ and $\mathcal{M}_{C}$ of $\mathcal{M}$
corresponding to them are graphs over $\Delta$ that form separate components.
We denote by $\mathcal{M}_{0}$ the remaining component of $\mathcal{M}$ after
$\mathcal{M}_{A}$, $\mathcal{M}_{B}$ and $\mathcal{M}_{C}$ have been removed.
Let $\chi_{0}=\chi|\mathcal{M}_{0}$. This is the connected component of the
catastrophe manifold that involves the saddles.

In the following sections we will deduce the following results which mirror
similar results for the CEU. We now show that close to $\partial\Delta$ it has
the same double covering annular structure as this CEU. This will involve
analysis of the flip curves in $\Delta$. From this we will deduce the
existence of a Möbius fold circle $C$ in $\mathcal{M}$. Further analysis of
the flip curves will show that $C$ is unique and that $B_{C}=\chi(C)$ plays
the role of the deltoid $\mathcal{B}$ of the CEU and has a 3-fold cusp
structure.

###  Using flip curves

####  Flips ending on $\ell_{A}$, $\ell_{B}$ and $\ell_{C}$

In the following subsections $X$, $Y$ and $Z$ are always distinct elements of
$\\{A,B,C\\}$.

The CEU boundary behaviour assumption implies that

  1. 1.

there is a neighbourhood $U$ of $\ell_{Y}\cap\ell_{Z}$ such that if $\theta\in
U\cap\Delta$ then $X$ is the central attractor.

  2. 2.

there is a neighbourhood $V$ of $\ell_{X}$ such that for all $\theta\in
V\cap\Delta$ either there is a heteroclinic connection or the system is MS
with $X$ a peripheral attractor.

Proof. Firstly, note that the CEU boundary behaviour assumption implies that
there is a neighbourhood $U$ of the boundary such that if $\theta\in
U\cap\Delta$ then $X_{\theta}$ has three attractors and two saddles all of
which vary smoothly with $\theta\in U\cap\Delta$.

Since the curves $\ell_{X}$, $\ell_{Y}$, and $\ell_{Z}$ are smooth and since
$\ell_{Y}$ meets both $\ell_{X}$, and $\ell_{Z}$ then $Y$ is peripheral along
$\ell_{Y}$ by Lemmas 5 and 5. Similarly, $Z$ is peripheral along $\ell_{Z}$.
Thus at $\ell_{Y}\cap\ell_{Z}$, $X$ is central. It follows that $X$ is also
central close to $\ell_{Y}\cap\ell_{Z}$ for $\theta\in\Delta$ and $X$ is
peripheral for all $\theta$ in $\Delta$ near to $\ell_{X}$.

The CEU boundary behaviour assumption implies that for $X\in\\{A,B,C\\}$ there
are generically an odd number of flip curves starting on the intersection of
$\ell_{X}$ with the closure of $\Delta$. On each of these the central
attractor changes from $Y$ to $Z$ or vice-versa.

Proof. The result follows because in moving along $\ell_{X}$ from
$\ell_{X}\cap\ell_{Y}$ to $\ell_{X}\cap\ell_{Z}$ is to move from $Z$ being the
central attractor to $Y$ being central.

####  Algebra of three-way flips

[width=]../figures/equilaterals3

Figure : A series of flips that results in the saddles swapping. The $r_{i}$
are all reflections in the thin blue line that is shown.

To each generic configuration of three attractors and two saddles as above we
can associate a triangle with vertices marked $A$, $B$ and $C$. We label the
saddles $g$ (green) and $y$ (yellow). See Fig. 6A. Then we associate to each
MS 3-attractor system the corresponding marked equilateral triangle as in Fig.
6B. The vertices correspond to the attractors $A$, $B$ and $C$ and we mark an
edge with green or yellow if the corresponding unstable manifold of $g$ or $y$
joins the corresponding two attractors. The meaning of the red arrows in Fig.
6 is explained below.

To each flip we can associate a corresponding reflection as shown in Fig. 6.
This is an element of the Dihedral group $D_{3}$ which can be represented as
the group of symmetries of the equilateral triangle consisting of reflections
and rotations through 2$\pi/3$ radians.

Thus, consider an oriented loop $\gamma(t)$ in $\Delta$ with a given
orientation and starting point (not in a flip curve), that does not meet any
fold or cusp points and which is transversal to the flip curves. To this we
can associate a sequence $\sigma_{\gamma}$ of elements of $D_{3}$. This is the
product $r_{k}r_{k-1}\cdots r_{1}$ where $r_{i}$ is the reflection that occurs
when the curve crosses the $i$th flip. The reduction of this product is
$r_{i_{r}}r_{i_{r-1}}\cdots r_{i_{1}}$ which is what is left when we
iteratively delete all products in the product of the form $r_{i}r_{i}^{-1}$
or $r_{i}^{-1}r_{i}$. It is then easy to see that the resulting product is

$r_{\sigma(1)}r_{\sigma(2)}r_{\sigma(3)}\cdots
r_{\sigma(1)}r_{\sigma(2)}r_{\sigma(3)}=(r_{\sigma(1)}r_{\sigma(2)}r_{\sigma(3)})^{k}$

for some cyclic permutation $\sigma$ of $\\{1,2,3\\}$ and some $k\geq 0$. This
is because the starting and ending configurations are the same and since once
all the terms $r_{i}r_{i}^{-1}$ or $r_{i}^{-1}r_{i}$ have been removed the
sequence of indices must be periodic of the form
$\cdots\sigma(1)\sigma(2)\sigma(3)\sigma(1)\sigma(2)\sigma(3)\cdots$ since
only two $r_{i}$s can be applied to any given configuration and if that
configuration has been achieved then the inverse of one of these was applied
to get it.

We call $(\sigma,k)$ the _flip index_ of $\gamma$. Note that $k$ does not
depend on the starting point and that, for $k$ even, when evaluated the
product is the identity and otherwise is a non-trivial reflection.

When the unstable manifolds of the saddles undergo the transformations shown
in Fig. 6 the unstable manifolds undergo interesting changes in orientation.
Similar things occur for the stable manifolds but we only consider unstable
manifolds here as we are interested in index 1 saddles and then the unstable
manifold is 1-dimensional even in $n$-dimensional systems.

Put an orientation on the two saddles. This is indicated in Fig. 6 by the red
arrow. As the flips occur this orientation is propagated as shown. (Note that
this is not what you get by applying the geometric reflections. Instead follow
what is happening in the top row.) After applying them once, the two saddles
have switched and one of the unstable manifolds has opposite orientation. From
this we see that applying the flips corresponding to
$r_{\sigma(1)}r_{\sigma(2)}r_{\sigma(3)}$ twice we reverse the orientation of
both unstable manifolds and the saddles switched twice. Thus we have the
following result.

Suppose that $\gamma$ is a closed curve in $\Delta$ that contains no fold or
cusp bifurcation points. Put an orientation on the unstable manifolds of the
two saddles. The flip index of $\gamma$ has $k$ odd if and only if this
orientation is reversed in one of the unstable manifolds when $\gamma$ is
traversed once.

Consider a simple closed curve $\gamma(t)$ in $\Delta$ that has a lift
$\Gamma(t)=(x(t),\theta(t))$ $0\leq t\leq 1$ to a smooth closed curve in
$\mathcal{M}_{0}$. Assume that it meets no fold points. Then the unstable
manifold $W^{u}(x(t))$ of the saddle $x(t)$ varies smoothly with $t$. Thus as
$t\rightarrow 1$ from below $W^{u}(x(t))$ converges as a point set to
$W^{u}(x(0))$. Thus if we fix an orientation on $W^{u}(x(0))$ and use the
smooth dependence of $W^{u}(x(t))$ on $t$ to propagate this orientation to all
$W^{u}(x(t))$ we can ask whether as $t\rightarrow 1$ from below the
orientation on $W^{u}(x(t))$ converges to that on $W^{u}(x(0))$. Clearly, this
is the case if and only if the saddle bundle of $\Gamma$ is a cylinder.

On the other hand, from the above discussion, this is not the case if the flip
index of $\Gamma$ has $k$ odd and is the case otherwise. Thus we deduce the
following result

The flip index of $\Gamma$ is odd if and only if the bundle $E_{\Gamma}^{c}$
is Möbius.

Proof. By the above, the flip index is odd if and only if a complete
revolution around $\Gamma$ causes any orientation put on the unstable
manifolds of the saddles to be reversed and this is the case if and only if
the bundle $E^{c}_{\Gamma}$ is Möbius.

####  Flips at $\partial\Delta$ imply a Möbius fold circle

Cor. 6 above tells us that there are an odd number of flip curves along each
of the edges $\ell_{X}$, $X\in\\{A,B,C\\}$, and these flip curves induce the
same element or its inverse when calculating the flip index of a simple closed
curve $\gamma$ close to $\partial\Delta$. We denote the corresponding element
$r_{X}$. If the curve is sufficiently close to $\partial\Delta$ then the
elements must alternate between $r_{X}$ and $r_{X}^{-1}$ with a total
contribution of $r_{X}$. Thus, in the notation analogous to Cor. 6, the
resulting product for the whole curve is of the form
$r_{\sigma(Z)}r_{\sigma(Y)}r_{\sigma(X)}$ and we deduce that $k=1$.

Since $\gamma$ is a closed curve we can parameterise it as $\theta=\gamma(t)$,
$t\in\mathbb{R}$, with $\gamma(t+1)\equiv\gamma(t)$. Since there are no
bifurcation points in $\Delta$ close to $\partial\Delta$, we can lift this to
a closed curve $\mathbf{x}=\Gamma(t)$ in $\mathcal{M}$ so that if
$\Gamma(t)=(x(t),\theta(t))$ then $x(t)$ is one of the two saddles. If $0\leq
t<1$ it follows from the above discussion that $\Gamma(t)$ and $\Gamma(t+1)$
are two distinct points in $\mathcal{M}$ that are interchanged when $t$ is
increased by one. This is because $k=1$ and therefore going once round
$\gamma$ flips the saddles as explained in Sect. 6. Let $\mathcal{M}_{0}$ be
the connected component of $\mathcal{M}$ containing $\Gamma$ and
$\chi_{0}=\chi|\mathcal{M}_{0}$.

There is a neighbourhood $U$ of $\partial\Delta$ such that $U\cap\Delta$ and
$\chi_{0}^{-1}(U\cap\Delta)$ are annuli and the latter is a double cover of
the former on which $\chi_{0}$ has degree 2.

Proof. Take a small tubular neighbourhood $N_{X}$ of each curve $\ell_{X}$,
$X\in\\{A,B,C\\}$ so that (i) there are no bifurcation points in $N$ other
than in $\partial\Delta$ and (ii) so that if $N=N_{A}\cup N_{B}\cup N_{C}$
then $U=N\cap\Delta$ is an annulus and such that for any point $\theta\in U$
there is a curve $\gamma$ as above with $\theta\in\gamma$. Then $\chi$ is a
local diffeomorphism and, from the above discussion every point in $U$ has two
preimages in $V=\chi_{0}^{-1}(U)$ under $\chi_{0}$. Put an orientation on $V$
so that the degree of $\chi$ is non-negative. Then if $\mathbf{x}_{1}$ and
$\mathbf{x}_{2}$ are the two preimages of $\theta$, since we can move from
$\mathbf{x}_{1}$ to $\mathbf{x}_{2}$ by moving along a closed curve in $U$,
$\chi_{0}$ is orientation preserving at both points. Thus the degree of
$\chi_{0}|V$ is two.

Now we can prove one of our main results.

Suppose that the configuration of the smooth bifurcation curves $\ell_{A}$,
$\ell_{B}$, and $\ell_{C}$ is as in the CEU boundary behaviour assumption (cf.
Fig. 6) and that $\Delta$ is the open region enclosed by these curves. Then
$\Delta$ contains $B_{C}=\chi(C)$ where $C$ is a Möbius fold circle.

Proof. Let $\gamma$ be a smooth simple closed curve in $\Delta$ close to the
boundary of $\Delta$. Lift $\gamma$ to a closed curve $\Gamma$ in
$\mathcal{M}_{0}$. Then by Cor. 6 the flip index of $\gamma$ is odd and
therefore the same is true for $\Gamma$. Therefore, by Prop. 6, the bundle
$E_{\Gamma}^{c}$ is Möbius. It follows by Theorem 4 that there is a Möbius
fold circle $C$ in the interior of $\Gamma$. Since $C$ is in the interior of
$\Gamma$, $B_{C}$ is contained in $\Delta$.

####  Flip curves in $\mathcal{M}$

In order to understand the way in which flip curves end on fold bifurcation
curves in the parameter space we need to lift the flips to $\mathcal{M}_{0}$.
Whereas in $\mathcal{M}$ a lift of a flip curve can only meet fold curves at
its end points, in parameter space a flip curve can meet a fold bifurcation
curve at other points (see e.g. Fig. 6).

If $\theta\in\Delta$ is a flip point then we associate to it two points
$\mathbf{x}=\mathbf{x}(\theta)=(x,\theta)\in\mathcal{M}_{0}$ where $x$ is
either the target saddle $x_{T}$ or the source saddle $x_{S}$. Thus a flip
curve $\theta=\varphi(t)$, $0\leq t\leq 1$, in $\Delta$ can be lifted to two
smooth curves $\mathbf{x}=\Phi_{T}(t)$ and $\mathbf{x}=\Phi_{S}(t)$ in
$\mathcal{M}_{0}$ that we respectively call _the target flip curve_ and
_source flip curve_.

If such a flip curve in $\mathcal{M}_{0}$ meets a fold curve in
$\mathcal{M}_{0}$ it involves a fold in which the target or source saddle is
destroyed. It follows that a flip curve in $\mathcal{M}_{0}$ only meets fold
curves at its end points. As we will see this is different from flip curves in
$\Delta$ which can cross fold bifurcation curves. We call a flip curve a
target-source one if one end point is a target one and the other a source one.
Source-source and target-target curves are possible but here we are only
interested in target-source ones

If $\Gamma$ is a lift to $\mathcal{M}_{0}$ of a closed curve $\gamma$ in
$\Delta$ then there is a 1-1 correspondence of points where they meet target
flip curves and the intersections are ordered identically as in $\gamma$.
Therefore, if one defines the flip index of such a closed curve $\Gamma$ in
$\mathcal{M}_{0}$ analogously to that of $\gamma$ but only using the target
flip curves in $\mathcal{M}_{0}$, then the flip indices of $\Gamma$ and
$\gamma$ are equal. Of course, instead of using target flip curves one could
use source flip curves.

Now suppose that the closed smooth curve $\Gamma^{\prime}$ is obtained from
$\Gamma$ by a smooth homotopy and that both are transverse to the flip curves.
Assume also that the homotopy does not cross any fold curves.

The reduced products $(r_{\sigma(1)}r_{\sigma(2)}r_{\sigma(3)})^{k}$ obtained
from $\Gamma$ and $\Gamma^{\prime}$ are the same.

Proof. The homotopy $\Gamma_{t}$ can be chosen so that the curve $\Gamma_{t}$
is always transversal to the flip curves except at a finite number of $t$ and
at these it touches a single flip curve quadratically. The only changes in the
sequence of flips occur at such $t$ but then this just adds or removes a term
of the form $r_{i}^{-1}r_{i}$ and therefore does not change the reduced form.

Consider a closed curve $\Gamma$ in $\mathcal{M}$ that contains a fold circle
$C$ in its interior. Suppose, moreover, that $C$ is the only fold circle
contained in the interior of $\Gamma^{\prime}$ except possibly for fold
circles contained in the interior of $C$. We call such a curve $\Gamma$ a
_primary flip container_ for $C$.

The fold circle $C$ is Möbius if and only if there is a primary flip container
$\Gamma$ for $C$ for which the flip index has $k$ odd.

Proof. Let $N$ be a tubular neighbourhood of a fold curve and $r$ be the
associated retraction. One can use a homotopy to move $\Gamma$ into
$N\setminus C$ so that the new curve $\Gamma^{\prime}$ has degree 1 with
respect to $r$. Moreover, by Prop. 6, $\Gamma^{\prime}$ has $k$ odd if and
only if $\Gamma$ does.

If $N$ is sufficiently small, $C$ is Möbius $\iff$ its cusp bundle is a Möbius
band $\iff$ $E_{\Gamma^{\prime}}^{u}$ is a Möbius band $\iff$ $k$ is odd.

Suppose that $\Gamma$ is a smooth simple closed curve in $\mathcal{M}$
consisting of index 1 saddles. If its flip index is odd then in the generic
situation, $\Gamma$ contains an odd number of Möbius fold circles in its
interior. Moreover, if $C$ is one of these Möbius fold circles one can find a
primary flip container $\Gamma^{\prime}$ for $C$ for which the flip index has
$k$ odd.

Proof. One uses a construction as in Appendix LABEL:sect:E to show that if
$\Gamma$ is a simple closed curve in $\mathcal{M}$ then it contains at least
one fold circle and that $\Gamma$ can be modified so that it only contains any
one of these that is Möbius. One firstly removes the fold circles with $k$
even and then pairs of fold circles with $k$ odd. Having modified $\Gamma$ in
this way one can use a homotopy to move it arbitrarily close to $C$ and use
Prop. 6.

Under the CEU boundary behaviour assumption, $\Delta$ contains a unique Möbius
fold circle.

Proof. By Cor. 6 the lift $\Gamma$ to $\mathcal{M}_{0}$ of any closed curve in
$\Delta$ close to $\partial\Delta$ has $k$ odd. Thus by Theorem 6 there is a
Möbius fold circle $C$ in the interior of $\Gamma$ and a primary flip
container $\Gamma^{\prime}$ for $C$ for which the flip index has $k$ odd. This
means that for each $X\in\\{A,B,C\\}$ there is a flip curve
$\ell_{X}^{\prime}$ connecting the boundary component $\ell_{X}$ to
$\Gamma^{\prime}$. If $C^{\prime}$ were another Möbius fold circle in the
interior of $\Gamma$ it would also have a primary flip container
$\Gamma^{\prime\prime}$ with similar connections $\ell_{X}^{\prime\prime}$ to
$\ell_{X}$. Since $\ell_{X}^{\prime}$ and $\ell_{X}^{\prime\prime}$ cannot
meet as they are flip curves, this is impossible. Thus $C$ is unique.