# ProtoKD: Learning from Extremely Scarce Data for Parasite Ova Recognition

Shubham Trehan1, Udhav Ramachandran2, Ruth Scimeca2, and Sathyanarayanan N.
Aakur1
1 Department of Computer Science and Software Engineering, Auburn University,
Auburn, AL, 36849
2 Department of Veterinary Pathobiology, Oklahoma State University,
Stillwater, OK, 74078

###### Abstract

Developing reliable computational frameworks for early parasite detection,
particularly at the ova (or egg) stage, is crucial for advancing healthcare
and effectively managing potential public health crises. While deep learning
has significantly assisted human workers in various tasks, its application in
diagnostics has been constrained by the need for extensive datasets. The
ability to learn from an extremely scarce training dataset, i.e., when fewer
than 5 examples per class are present, is essential for scaling deep learning
models in biomedical applications where large-scale data collection and
annotation can be expensive or not possible (in case of novel or unknown
infectious agents). In this study, we introduce ProtoKD, one of the first
approaches to tackle the problem of multi-class parasitic ova recognition
using extremely scarce data. Combining the principles of prototypical networks
and self-distillation, we can learn robust representations from only one
sample per class. Furthermore, we establish a new benchmark to drive research
in this critical direction and validate that the proposed ProtoKD framework
achieves state-of-the-art performance. Additionally, we evaluate the
framework’s generalizability to other downstream tasks by assessing its
performance on a large-scale taxonomic profiling task based on metagenomes
sequenced from real-world clinical data.

###### Index Terms:

Learning from Extremely Scarce Data, Ova Detection, Microscope Image Analysis

##  I Introduction

Parasitic infections pose a significant threat to human and animal health,
often leading to severe illness and even fatalities. These infections can be
transmitted through various means, including contaminated food and water
sources and common disease vectors such as mosquitoes. For instance, certain
zoonotic diseases can be transmitted to humans through the consumption of
infected livestock, such as cows and pigs. Many infections, particularly
gastrointestinal, can cross the species barrier, and can further amplify the
public health risks associated with them. Detecting these parasites early,
especially at the ova stage, is crucial for preventing outbreaks of parasitic
diseases. Parasitic ova (eggs or cysts) have unique characteristics that allow
them to be a distinguishing factor between different kinds of parasitic
infections. The typical identification process requires the isolation of the
egg from fecal samples collected from the infected host which are subsequently
analyzed by a parasitologist under a microscope. Genus-level identification
can enable the development of treatments to prevent severe infections and
large-scale outbreaks.

Figure 1: The overall architecture of the proposed ProtoKD is illustrated.
Based on prototypical networks and self-distillation, a two-stage process
captures the intra-class variations that can occur in biological data in a
robust representation.

Deep learning has enabled the development of “smart” systems that have made
immense progress in many fields. However, enormous amounts of historical data
(hundreds if not thousands of samples per class) have been necessary to derive
insights for downstream applications. Advances in disease diagnostics have
been sparse due to the need to learn associations from highly limited,
constrained data. Acquiring clinical data to train such deep learning models
can be expensive, both in terms of the data acquisition cost and the time and
effort of highly skilled human users to provide high-quality, large-scale
annotations, which curbs the applications of deep learning models trained
under a traditional supervised setting. Transfer learning [1] and few-shot
learning [2, 3, 4] have somewhat alleviated this dependency but still require
significant amounts of quality data. Generative models such as diffusion
models [5] and generative adversarial networks (GANs) [6] show potential in
generating synthetic samples for training data augmentation but have been
prone to memorizing and replicating training data [7], and prone to
hallucinating artifacts [8]. Hence, their application in biomedical settings
is inhibited due to privacy concerns [9], where patient identity and data
integrity could be compromised.

In this work, we present ProtoKD, a framework designed to work with extremely
scarce data, e.g., where less than $5$ training samples are present per class.
Such settings are common in biomedical and diagnostic applications where
labeled training data can be expensive. There is a need for rapid learning
from a few samples for novel or unseen classes of interest. An illustration of
the approach is shown in Figure 1. Our approach was based on the idea that
learning robust representations of limited training data and using domain-
specific augmentations to create an auxiliary dataset that can, together,
capture intra-class variation. Through a cyclical, two-phase process, we aim
to align representations from original and augmented images to capture
variations in the decision boundary across closely related classes. First, a
matching loss is introduced to learn robust representations to distinguish
between classes using a prototypical network. Second, a self-distillation loss
is introduced to help capture the intra-class variations in the data by
presenting the network with heavily augmented data and training with pseudo-
labels generated by the prototypical network. This step has a two-fold effect:
(i) it adds a level of regularization that prevents the networks from
overfitting, and (ii) it allows us to introduce other learning losses that
help discriminate between fine-grained representations. By extending the idea
of prototypical networks and self-distillation, we learned robust
representations from extremely sparse data that was capable of capturing the
intra-class variations through domain-specific data augmentation.

The contributions of our work are three-fold: (i) we present one of the first
works to address the problem of multi-class parasitic ova recognition from
microscopic images, (ii) we develop a framework to learn from extremely scarce
data (from a single example per class) in a multi-class classification
setting, and finally (iii) we demonstrate its generalization to other
biomedical tasks by evaluating on metagenome profiling.

###  I-A Related Work

There have been very few automatic parasitic ova detection frameworks explored
in literature. Supervised transfer learning has been explored in detecting and
classifying seven species of Eimeria spp. in chickens [10] through the
analysis of curated large-scale microscopic imagery [11, 12]. Data
augmentation techniques, such as image flipping, adding Gaussian noise and
histogram normalization, and transfer learning from ImageNet [13] pre-trained
models have enabled the training of large deep learning models for ova
detection. However, the dependency on large-scale training data was not
alleviated, which limits their generalization to other biomedical
applications. Weakly supervised approaches such as those based on Multiple
Objects Feature Fusion (MOFF) [14] and traditional image processing techniques
[15] have been used to reduce the dependency on densely annotated data, yet
still assume access to large-scale datasets for learning associations between
the input and target classifications. Advances in generative models such as
GANs provided a viable mechanism to generate additional training data for
learning from limited data. DADA [16] explores training with small samples
from Cifar-10 and SVNH. They use GANs to augment the original dataset for more
diversity with the same labels. Barz et al. [17] showed that the Cosine Loss
is better than categorical cross-entropy whenever only a few samples are
present per class. Ishikawa et al. [18] use conditional GAN for augmentation
to improve efficiency in generating training samples but increase the
computational cost. Brigato et al. [19] use Auxialiary-Classifier GANs for
image synthesis and classification in low data settings. Meta-learning
approaches such as MAML [2] and Prototypical Networks [4, 20] have enabled
few-shot learning where only a few samples per class are required for making
inferences. However, such or similar approaches [20, 21, 22] assume a
reasonably large training corpus exists to create “meta-tasks” for learning
robust representations for downstream classification.

##  II ProtoKD: Learning from Scarce Data

Problem Statement. In this work, we consider the task of learning from
extremely scarce samples $n\leq 5$ per class. During training, the model can
access a set of $N$ training examples $X{=}\\{x_{1},x_{2},x_{3},\ldots
x_{N}\\}$ drawn from $C$ classes with $n$ samples each. The model is presented
with samples from any of the $C$ classes at test time. In contrast, meta-
learning approaches [4, 3, 2] consider the training and evaluation phases to
consist of $c$-way classification tasks, where $c\ll C$. Our setup is more
challenging since we have a $C$-way classification task and have access to
extremely scarce data.

|  |  |  |  |  |
---|---|---|---|---|---|---
Capillarids | Cystoiso- | Dipylidium | Eimeria | Giardia | Moniezia | Nematodirus
(190) |  spora sp.(76) |  caninum (26) | sp. (65) | sp. (149) | sp.(134) | sp.(53)
|  |  |  |  |  |
Parascaris | Strongyles | Taeniid | Toxascaris | Toxocara | Trichostron- | Trichuris
sp. (173) | (84) | eggs (90) |  leonina (32) | sp. (201) | gyles (264) | sp. (36)
Figure 2: Dataset Characteristics. Our parasite ova dataset has $1573$
instances across $14$ ova classes, imaged from $594$ clinical samples. An
exemplar image from each class and the number of samples in each class are
presented. It is highly imbalanced, with high inter- and intra-class
variation.

####  II-1 Prototypical Networks for Scarce Data

Prototypical networks aim to construct a $D$-dimensional vector representation
of each class, called the prototype, that captures its underlying
characteristics. Ideally, each prototype $p_{j}$ represents the typical
example from that class and captures the intra-class variations through an
embedding function $f_{\theta}:x_{i}\longrightarrow F_{i}$ that projects a
sample $x_{i}$ to a $D$-dimensional vector $F_{i}$. The prototype is computed
as the mean representation of all $n$ training samples in the class $c_{j}\in
C$. In a typical meta-learning setup, a support set $S$ consisting of $k$
examples from $c$ classes is sampled to construct these prototypes. A function
$\phi$ provides the distance between each example in the query set $Q$ and
each prototype. A softmax function over these distances provides a probability
distribution to label each example $x_{i}\in Q$. This setup assumes that (i)
each sample in the query comes from the $c$ classes sampled in the episode and
(ii) there exists a reasonably large dataset to sample the support and query
sets during training to learn good representations for $c$-way classification.
However, in biomedical applications, the number of training samples can be
sparse due to the high data acquisition cost or the need to adapt rapidly to a
changing scenario. Similarly, assuming a smaller subset of classes at
inference is unrealistic since this requires prior knowledge about each test
sample.

We extend the prototypical network formulation to the extremely scarce data
regime to overcome these limitations by proposing to learn the embedding
function $f_{\theta}$ through domain-specific data augmentation. Given a
support set $S_{i}$, each example $x_{i}$ is augmented through a controlled,
pre-determined scheme to create $k$ samples
$\\{x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},\ldots x^{\prime}_{k}\\}$
that become part of the query set. Hence, each minibatch consists of examples
from all $C$ classes, with the prototypes constructed from original,
uncorrupted samples and the query samples providing samples that cover the
plausible intra-class variations for each class. The training objective is to
minimize the negative log probability of each query sample $x^{\prime}_{i}$
belonging to its true class $c_{j}$ represented by its prototype $p_{j}$.
Hence, we minimize the matching loss ($\mathcal{L}_{m}$) given by

$-logp(y=j|x^{\prime}_{i},\theta)=-log\left(\frac{-exp(\phi(f_{\theta}(x^{\prime}_{i},p_{j})))}{\sum^{C}_{k}{exp(\phi(f_{\theta}(x^{\prime}_{i},p_{k}))}}\right)$
(1)

where $x^{\prime}_{i}$ is an example from the query set $Q$; $\phi(\cdot)$ is
a distance function that provides the distance (in the range $[0,+\infty)$)
between each sample $x^{\prime}_{i}$ and a prototype $p_{j}$; and $f_{\theta}$
is an embedding function, defined as a Wide ResNet [23], and $\phi(\cdot)$ is
the Euclidean distance. The data augmentation scheme is specific to each use
case and must be designed based on prior, domain-specific knowledge. In our
experiments with the parasite ova data, we apply augmentation mechanisms such
as random zoom, rotation, contrast change, flip, shear, and solarization.
However, these are not plausible augmentation schemes for the genome data to
capture the intra-class variations. Hence, we design genome-specific
augmentation schemes based on base flipping to simulate observation error
[24]. We add noise drawn from a normal distribution (with $0$ mean and
variance of $1$) to $5\%$ of the pixels randomly selected symmetrically along
the diagonal. This augmentation mechanism mimics the observation errors
commonly found in genome sequencing [24, 25] and provides a natural
augmentation scheme to capture the intra-class variation and additionally
helps preserve the symmetry of the pseudo-image-based k-mer representations.
We refer the reader to [26] for more details on the pseudo-imaging for
genomics.

####  II-2 Learning Intra-class Variations with Self-Distillation

The second step is to enhance the prototypes $P{=}\\{p_{1},p_{2},p_{3},\ldots
p_{C}\\}$ to capture the intra-class variations. First, we generate pseudo-
labels for each query sample $x^{\prime}_{i}\in Q$ using the probability
distribution defined in Equation 1 to create labels for the augmented samples.
This self-labeled data is then used to fine-tune the encoder network using a
distillation loss given by

$\mathcal{L}_{s}=D_{KL}(p_{t}(y=j|x_{i},\theta_{k})/\tau,p_{s}(y=j|x_{i},\theta_{s}))$
(2)

where $D_{KL}(\cdot)$ is the Kullback–Leibler divergence [27] between the
probability distributions of the prototypical network defined in Section II-1
and a linear layer trained on top of representations $F_{i}$ from
$f_{\theta}$; $\tau$ is a temperature parameter that controls how closely the
linear layer’s output should match that of the prototypical network; and
$\theta_{t}$ and $\theta_{s}$ are the trainable parameters of the prototypical
network (i.e., the embedding function $f_{\theta}$) and the linear layer,
respectively. We set $\tau$ to 5, chosen based on a grid search between $0$
and $10$. This formulation could be extended to leverage unlabeled data into a
semi-supervised learning setting in future works. We leave that exploration to
future work and focus only on the extremely scarce data setting.

In addition to the distillation loss described above, we introduce a simple
discriminative loss to help increase the separability of the decision boundary
between the classes. To this end, we employ a similarity-based contrastive
loss that reduces the difference between the features of each query sample
$x^{\prime}_{i}$ and its corresponding prototype $p_{j}$ while increasing the
distance to other prototypes. The intuition behind this loss is that
increasing the distance between the query sample and other prototypes can
capture the variability within each class since we widen its decision boundary
based on its prototype $p_{j}$. We define this to be a discriminative loss
given by

$\mathcal{L}_{d}=-log\left(\frac{exp(p_{j}^{T}\cdot
F_{i})}{\sum^{C}_{k=1}{1\\!\rm l}_{j\neq k}exp(p_{k}^{T}\cdot F_{i})}\right)$
(3)

where $F_{i}$ is the feature representation of a query sample
$x^{\prime}_{i}$; ${1\\!\rm l}\in\\{0,1\\}$ indicates whether the sample is
from its true class $c_{j}$; and both $p_{j}$ and $F_{i}$ are $\ell_{2}$
normalized vectors. This formulation allows us to leverage our proposed
domain-specific augmentation setup for a contrastive learning mechanism
without having to sample new positive and anchor examples or triplet mining,
as with other contrastive learning mechanisms such as SimCLR [28] or triplet
losses [29].

TABLE I: Quantitative results on the parasite ova dataset. One example from
each class was randomly sampled for training. Average performance from $10$
runs is reported.

| Supervised | ProtoNet | ProtoKD
---|---|---|---
| Precision | Recall | Precision | Recall | Precision | Recall
Average | 0.436 | 0.448 | 0.472 | 0.494 | 0.523 | 0.544
Capillarids | 0.558 | 0.468 | 0.554 | 0.680 | 0.730 | 0.460
Cystoisospora spp. | 0.166 | 0.140 | 0.174 | 0.188 | 0.360 | 0.260
Dipylidium caninum | 0.116 | 0.424 | 0.236 | 0.448 | 0.080 | 0.180
Eimeria spp. | 0.292 | 0.152 | 0.120 | 0.222 | 0.480 | 0.290
Giardia spp. | 0.908 | 0.978 | 0.934 | 0.918 | 0.945 | 1.000
Moniezia spp. | 0.528 | 0.580 | 0.492 | 0.528 | 0.515 | 0.575
Nematodirus spp. | 0.272 | 0.414 | 0.352 | 0.464 | 0.250 | 0.790
Parascaris spp. | 0.768 | 0.746 | 0.858 | 0.716 | 0.755 | 0.785
Strongyles | 0.268 | 0.464 | 0.362 | 0.484 | 0.565 | 0.650
Taeniid spp. | 0.574 | 0.334 | 0.650 | 0.320 | 0.785 | 0.715
Toxascaris leonina | 0.226 | 0.670 | 0.342 | 0.834 | 0.255 | 0.630
Toxocara spp. | 0.738 | 0.438 | 0.790 | 0.592 | 0.795 | 0.610
Trichostrongyles | 0.562 | 0.174 | 0.560 | 0.180 | 0.665 | 0.265
Trichuris spp. | 0.150 | 0.300 | 0.184 | 0.322 | 0.140 | 0.400

Implementation Details. The framework is trained end-to-end in two alternating
phases. Every epoch consists of $10$ iterations of training only with the
matching loss, followed by $10$ iterations of training using both distillation
and discriminative losses. The matching loss cycle allows us to build
prototypes of each class, while the self-distillation phase allows us to
refine the prototypes by capturing the intra-class variation explicitly. We
use a Wide-ResNet [23], trained from scratch, as the backbone for our mapping
function, with a depth of $28$ layers, a width of $2$, and a dropout rate of
$0.3$. The features are projected to a $128$-dimension vector using a linear
layer. The images are resized to $128\times 128$ and pre-processed as done in
ResNet [30]. All networks are trained for 100 epochs with a support set size
of $1$ and a query set of $5$ and converge in 90 minutes. All experiments were
conducted on a workstation server with an AMD ThreadRipper CPU with 64 cores,
128 GB RAM, and an NVIDIA Titan RTX (24GB).

##  III Experimental Evaluation

In this section, we present the experimental setup and evaluation results for
the proposed ProtoKD approach. We begin by describing the data collection
process to curate the parasitic ova recognition dataset, one of the first
attempts at a comprehensive benchmark for the task. We discuss the metrics and
baselines used for evaluation and present the quantitative results. We
conclude by demonstrating the generalization capabilities of the proposed
ProtoKD framework to other biomedical applications, such as genome
classification.

TABLE II: Ablation studies to determine the impact of each component on the performance of ProtoKD on the parasite ova recognition task. $\mathcal{L}_{m}$ | $\mathcal{L}_{s}$ | $\mathcal{L}_{d}$ | Precision | Recall | F1-Score
---|---|---|---|---|---
✓ | ✗ | ✗ | 0.472 | 0.494 | 0.483
✗ | ✓ | ✗ | 0.436 | 0.448 | 0.442
✓ | ✓ | ✗ | 0.497 | 0.501 | 0.499
✓ | ✗ | ✓ | 0.508 | 0.516 | 0.512
✓ | ✓ | ✓ | 0.523 | 0.544 | 0.533

Data Collection. We collected 1573 ova examples, curated from 594 clinical
samples at a local [redacted for anonymity] diagnostics laboratory. Each type
of egg was collected by performing a centrifugal fecal flotation on samples
from various hosts infected with different parasites and were subsequently
observed under a microscope by a certified parasitologist. Following
identification, images were captured using the Olympus BX43 microscope (Tokyo,
Japan) and the Olympus Cell Sens Entry software v1.18. Based on their
frequency of occurrence in samples received at both local and national
diagnostics labs, the following 14 parasitic ova were considered: Capillarids,
Cystoisospora spp., Dipylidium caninum, Eimeria spp., Giardia spp., Moniezia
spp., Nematodirus spp., Parascaris spp., Strongyles, Taeniid eggs, Toxascaris
leonina, Toxocara spp., Trichostrongyles, and Trichuris spp. Figure 2 provides
examples and statistics. Each parasite and its relative size determined where
the magnification would be 40x, 20x, or 10x objective. Most of the images
captured are at 10x objective magnification which is available on a large
majority of microscopes. Clinical samples were collected until each class had
at least 25 examples to provide a comprehensive benchmark for evaluating
machine learning frameworks for ova recognition with extremely scarce data.

Metrics and Baselines. Due to the highly imbalanced nature of the data, we
choose precision and recall per class as our evaluation metric since accuracy
can be highly skewed towards classes with more examples. We evaluate the
performance of each algorithm with a training set with $1$ example per class
and report the mean results from $10$ random trials to avoid conflating the
results due to the choice of the example from each class. We choose a fully
supervised Wide-ResNet as our backbone network for all baselines, including a
fully supervised one, the modified ProtoNet from Section II-1, and ProtoKD.
All hyperparameters for each baseline are kept constant for each trial and
trained for $100$ epochs with early stopping based on a validation set of $5$
samples per class.

Performance on Parasite Ova Data. We first evaluate and compare our approach
against baselines on the parasite ova dataset. Table I summarizes the results
when training with only one example per class. The proposed ProtoKD approach
performs well across classes, with an average precision of $0.523$ and recall
of $0.544$, outperforming the supervised and ProtoNet baselines. Of particular
interest is the performance of the baselines on the two classes with the least
intra-class variation (Giardia spp.) and the highest intra-class variation
(Nematodirus spp.). With Nematodirus spp., ProtoKD’s recall ($0.790$) was
almost $1.5$ times that of ProtoNet ($0.464$) and the supervised ($0.414$)
baselines, which indicates that the self-distillation and discriminative
losses played their role in learning robust prototypes. All three baselines
performed well on Giardia, with ProtoKD achieving $100\%$ recall with a high
precision of $0.945$. On average, we find that the ProtoKD achieves high
recall at the cost of precision, particularly in the case of classes with a
large number of samples, such as Parascaris spp. and Trichuris spp. We
hypothesize that this is an effect of the contrastive, discriminative loss
intended to expand each class’s decision boundary. Interestingly, the ProtoKD
severely fails on Dipylidium caninum. Upon close inspection, Dipylidium
caninum was highly confused with Tirchostrongyle eggs, which, though visually
similar (see Figure 2), are functionally different. The strong augmentation
scheme resulted in overlapping decision boundaries due to the extreme
variation in Trichostrongyle examples. We anticipate using super-resolution
mechanisms [31, 32] to enhance the images will help find better
representations.

TABLE III: Evaluation on metagenome classification. Host and average pathogen
f1-scores are used following prior work [26].

# Samples | MG-NET | ProtoNet | ProtoKD
---|---|---|---
Host F1 | Pathogen F1 | Host F1 | Pathogen F1 | Host F1 | Pathogen F1
1 | 0.230 | 0.026 | 0.596 | 0.108 | 0.699 | 0.121
5 | 0.330 | 0.029 | 0.690 | 0.127 | 0.804 | 0.139
10 | 0.320 | 0.035 | 0.840 | 0.081 | 0.780 | 0.129
15 | 0.360 | 0.047 | 0.850 | 0.109 | 0.824 | 0.127
20 | 0.370 | 0.051 | 0.780 | 0.147 | 0.799 | 0.151
25 | 0.371 | 0.056 | 0.720 | 0.200 | 0.775 | 0.193

Ablation Studies. We perform ablation studies to systematically evaluate the
impact of the different components of the approach on its performance. The
three loss functions, defined in Equations 1, 2, and 3, are the major
components of the approach. Hence, we ablate over the impact of using
$\mathcal{L}_{s}$ and $\mathcal{L}_{d}$ in combination with $\mathcal{L}_{m}$
and report results in Table II. Using only the matching loss (Equation 1), the
approach degenerates to a standard ProtoNet, one of our baselines (Row 1).
When $\tau$ in $\mathcal{L}_{s}$ (defined in Equation 2) is set to 1, it
becomes the standard cross-entropy loss and hence is equivalent to our fully
supervised baseline (Row 2). It can be seen that using knowledge distillation
loss ($\mathcal{L}_{c}$) alone or matching loss ($\mathcal{L}_{m}$) performs
reasonably well, although not as much as the proposed ProtoKD. Adding the
discriminative loss ($\mathcal{L}_{d}$) with the matching loss
($\mathcal{L}_{m}$) provides a higher increase in performance than combining
the matching loss ($\mathcal{L}_{m}$) with self-distillation
($\mathcal{L}_{s}$). Combining all three provides a higher increase overall,
indicating the subtle balance between learning inter-class and intra-class
variations provided by the alternating training methodology proposed in
ProtoKD. Note that this formulation can naturally be extended to semi-
supervised learning where the self-distillation loss ($\mathcal{L}_{m}$) can
be used to train on unlabeled data. We leave that to future work since our
focus is on tackling the problem of learning from scarcely available ($<5$
samples per class) training data.

Extension to Other Biomedical Applications. In addition to our experiments on
parasite ova recognition, we evaluate the generalizability of the proposed
ProtoKD formalism to other biomedical applications by evaluating its ability
to learn representations from an entirely separate application: metagenome
sequences. We evaluate the ProtoKD framework on the data provided by MG-NET,
which has $31,580$ sequence reads across seven classes - Bovine (host), B.
trehalosi, H. somni, M. bovis, M. haemolytica, P. multocida and T. pyogenes.
Specifically, we use the pseudo-images generated by the MG-NET framework as
input and evaluate it by training with varying samples per class. The test set
was fixed with 8192 samples for a fair comparison with MG-NET in all the
scenarios. Average performance from 10 trials is reported. Table III
summarizes the result. We can see that the proposed ProtoKD framework and
ProtoNet outperform the supervised MG-NET at very low samples, i.e., less than
25 samples per class. It takes MG-NET at least 500 samples per class to
outperform ProtoKD with 25 samples, achieving an overall host F1-score of
$0.906$ and an average pathogen F1-score of $0.319$. Interestingly, the
performance initially reduces as the number of samples per class is increased.
We attribute this phenomenon to the fact that fine-grained recognition
requires highly distinct samples for learning robust features. Genomes between
closely related species have shown to have similar genome sequences [33];
hence, larger amounts of data do not necessarily translate into better
performance. For example, ProtoNet achieves a higher pathogen F1-score
($0.200$) than ProtoKD ($0.193$) at 25 samples. It has $100\%$ precision and
$0\%$ recall on two pathogen classes, indicating that the model does not make
balanced predictions and fails on edge cases. We anticipate that including
structural information [26, 33] and other metadata will improve the
performance.

##  IV Conclusion and Future Work

In this work, we presented ProtoKD, one of the first works to tackle the
problem of learning from extremely scarce training samples. Using a benchmark
dataset of parasitic ova, we demonstrate its strong ability to learn robust
representations from just one example per class. Experiments on large-scale
metagenome-based taxonomic profiling data demonstrated its generalizability to
other downstream applications. We anticipate using super-resolution to enhance
the images will help find better representations. We aim to extend this
framework for scaling deep learning frameworks to work with highly constrained
data typical in biomedical applications such as disease diagnostics and the
Internet of Medical Things.

Acknowledgement. This work was partially supported by the US National Science
Foundation (NSF) grant IIS 1955230.

## References

  * [1] M. Ghafoorian, A. Mehrtash, T. Kapur, N. Karssemeijer, E. Marchiori, M. Pesteie, C. R. Guttmann, F.-E. de Leeuw, C. M. Tempany, B. Van Ginneken _et al._ , “Transfer learning for domain adaptation in mri: Application in brain lesion segmentation,” in _Medical Image Computing and Computer Assisted Intervention- MICCAI 2017: 20th International Conference, Quebec City, QC, Canada, September 11-13, 2017, Proceedings, Part III 20_. Springer, 2017, pp. 516–524.
  * [2] C. Finn, P. Abbeel, and S. Levine, “Model-agnostic meta-learning for fast adaptation of deep networks,” in _International conference on machine learning_. PMLR, 2017, pp. 1126–1135.
  * [3] O. Vinyals, C. Blundell, T. Lillicrap, k. kavukcuoglu, and D. Wierstra, “Matching networks for one shot learning,” in _Advances in Neural Information Processing Systems_ , D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, Eds., vol. 29. Curran Associates, Inc., 2016. [Online]. Available: https://proceedings.neurips.cc/paper/2016/file/90e1357833654983612fb05e3ec9148c-Paper.pdf
  * [4] J. Snell, K. Swersky, and R. S. Zemel, “Prototypical networks for few-shot learning,” 2017. [Online]. Available: https://arxiv.org/abs/1703.05175
  * [5] J. Ho, C. Saharia, W. Chan, D. J. Fleet, M. Norouzi, and T. Salimans, “Cascaded diffusion models for high fidelity image generation.” _J. Mach. Learn. Res._ , vol. 23, no. 47, pp. 1–33, 2022.
  * [6] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial networks,” _Communications of the ACM_ , vol. 63, no. 11, pp. 139–144, 2020.
  * [7] C. A. Corneanu, S. Escalera, and A. M. Martinez, “Computing the testing error without a testing set,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 2677–2685.
  * [8] J. P. Cohen, M. Luck, and S. Honari, “Distribution matching losses can hallucinate features in medical image translation,” in _Medical Image Computing and Computer Assisted Intervention–MICCAI 2018: 21st International Conference, Granada, Spain, September 16-20, 2018, Proceedings, Part I_. Springer, 2018, pp. 529–536.
  * [9] M. Fredrikson, S. Jha, and T. Ristenpart, “Model inversion attacks that exploit confidence information and basic countermeasures,” in _Proceedings of the 22nd ACM SIGSAC conference on computer and communications security_ , 2015, pp. 1322–1333.
  * [10] P. He, Z. Chen, Y. He, J. Chen, K. Hayat, J. Pan, and H. Lin, “A reliable and low-cost deep learning model integrating convolutional neural network and transformer structure for fine-grained classification of chicken eimeria species,” _Poultry Science_ , vol. 102, no. 3, p. 102459, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0032579122007532
  * [11] K. Ray, S. Saharia, and N. Sarma, “Detection and identification of ascaris lumbricoides and necator americanus eggs in microscopic images of faecal samples of pigs,” _International Journal of Automation and Control_ , vol. 15, p. 378, 01 2021.
  * [12] Y. Wang, Z. He, S. Huang, and H. Du, “A robust ensemble model for patasitic egg detection and classification,” 2022. [Online]. Available: https://arxiv.org/abs/2207.01419
  * [13] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein _et al._ , “Imagenet large scale visual recognition challenge,” _International journal of computer vision_ , vol. 115, pp. 211–252, 2015.
  * [14] P. Manescu, C. Bendkowski, R. Claveau, M. Elmi, B. J. Brown, V. Pawar, M. J. Shaw, and D. Fernandez-Reyes, “A weakly supervised deep learning approach for detecting malaria and sickle cells in blood films,” in _Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part V 23_. Springer, 2020, pp. 226–235.
  * [15] J. E. Arco, J. M. Górriz, J. Ramírez, I. Álvarez, and C. G. Puntonet, “Digital image analysis for automatic enumeration of malaria parasites using morphological operations,” _Expert Systems with Applications_ , vol. 42, no. 6, pp. 3041–3047, 2015.
  * [16] X. Zhang, Z. Wang, D. Liu, and Q. Ling, “Dada: Deep adversarial data augmentation for extremely low data regime classification,” in _IEEE International Conference on Acoustics, Speech and Signal Processing_. IEEE, 2019, pp. 2807–2811.
  * [17] B. Barz and J. Denzler, “Deep learning on small datasets without pre-training using cosine loss,” in _Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision_ , 2020, pp. 1371–1380.
  * [18] T. Ishikawa and S. Stent, “Boosting supervised learning in small data regimes with conditional gan augmentation,” in _2022 IEEE International Conference on Image Processing (ICIP)_. IEEE, 2022, pp. 1351–1355.
  * [19] A. Dravid, F. Schiffers, Y. Wu, O. Cossairt, and A. K. Katsaggelos, “Investigating the potential of auxiliary-classifier gans for image classification in low data regimes,” in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 3318–3322.
  * [20] J. Li, P. Zhou, C. Xiong, and S. C. Hoi, “Prototypical contrastive learning of unsupervised representations,” _arXiv preprint arXiv:2005.04966_ , 2020.
  * [21] F. Sung, Y. Yang, L. Zhang, T. Xiang, P. H. Torr, and T. M. Hospedales, “Learning to compare: Relation network for few-shot learning,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 1199–1208.
  * [22] C.-Y. Chuang, J. Robinson, Y.-C. Lin, A. Torralba, and S. Jegelka, “Debiased contrastive learning,” _Advances in Neural Information Processing Systems_ , vol. 33, pp. 8765–8775, 2020.
  * [23] S. Zagoruyko and N. Komodakis, “Wide residual networks,” in _British Machine Vision Conference 2016_. British Machine Vision Association, 2016.
  * [24] T. Laver, J. Harrison, P. O’neill, K. Moore, A. Farbos, K. Paszkiewicz, and D. J. Studholme, “Assessing the performance of the oxford nanopore technologies minion,” _Biomolecular Detection and Quantification_ , vol. 3, pp. 1–8, 2015.
  * [25] V. Indla, V. Indla, S. Narayanan, A. Ramachandran, A. Bagavathi, V. L. Ramnath, and S. N. Aakur, “Sim2real for metagenomes: Accelerating animal diagnostics with adversarial co-training,” in _Pacific-Asia Conference on Knowledge Discovery and Data Mining_. Springer, 2021, pp. 164–175.
  * [26] S. N. Aakur, S. Narayanan, V. Indla, A. Bagavathi, V. Laguduva Ramnath, and A. Ramachandran, “Mg-net: leveraging pseudo-imaging for multi-modal metagenome analysis,” in _Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part V 24_. Springer, 2021, pp. 592–602.
  * [27] J. M. Joyce, “Kullback-leibler divergence,” in _International encyclopedia of statistical science_. Springer, 2011, pp. 720–722.
  * [28] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton, “A simple framework for contrastive learning of visual representations,” in _International conference on machine learning_. PMLR, 2020, pp. 1597–1607.
  * [29] W. Ge, “Deep metric learning with hierarchical triplet loss,” in _Proceedings of the European Conference on Computer Vision (ECCV)_ , 2018, pp. 269–285.
  * [30] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2016, pp. 770–778.
  * [31] B. Liao, Y. Chen, Z. Wang, C. D. Smith, and J. Liu, “Comparative study on 1.5 t-3t mri conversion through deep neural network models,” in _2022 21st IEEE International Conference on Machine Learning and Applications (ICMLA)_. IEEE, 2022, pp. 963–968.
  * [32] Q. Zhu, P. Li, and Q. Li, “Attention retractable frequency fusion transformer for image super resolution,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2023, pp. 1756–1763.
  * [33] S. N. Aakur, V. Indla, V. Indla, S. Narayanan, A. Bagavathi, V. L. Ramnath, and A. Ramachandran, “Metagenome2vec: Building contextualized representations for scalable metagenome analysis,” in _2021 International Conference on Data Mining Workshops (ICDMW)_. IEEE, 2021, pp. 500–507.


# Versal deformations of
vector field singularities

Mauricio GARAY and Duco van Straten

###### Abstract.

When a singular point of a vector field passes through resonance, a formal
invariant cone appears. In the seventies, Pyartli proved that for
$(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a
family of invariant cylinders. In his thesis, Stolovitch established a new
type of normal form and proved that for a simple resonance and under
arithmetic conditions the cone is (the germ of) an analytic variety. In this
paper, we prove a versal deformation theorem for analytic vector fields with
an isolated singularity over Cantor sets. Our result implies that, under
arithmetic conditions, the resonant cone is the degeneration of a set of
invariant manifolds like in Pyartli’s example. For the multi-Hopf bifurcation,
that is for the $(-1,1)^{d}$-resonance, this implies the existence of
vanishing tori carrying quasi-periodic motions generalising previous results
of Chenciner and Li.

###### Contents

  1. 1 Introduction
  2. 2 Divergent and convergent normal forms
    1. 2.1 The Poincaré-Dulac normal form
    2. 2.2 Resonances
    3. 2.3 Positivity conditions
    4. 2.4 Examples and counter-examples
    5. 2.5 The resonant cone
    6. 2.6 The Stolovitch normal form
  3. 3 The formal versal deformation
    1. 3.1 Bruno variables
    2. 3.2 The detuning deformation
    3. 3.3 Formal normal form theorem
    4. 3.4 Relation to the Poincaré-Dulac normal form
    5. 3.5 The versality property
    6. 3.6 The Bruno-Stolovitch ideal
    7. 3.7 Piartly’s example
    8. 3.8 The multi-dimensional Hopf bifurcation
  4. 4 Versal deformations and invariant varieties
    1. 4.1 Bruno sequences
    2. 4.2 Functional analytic setting (part I)
    3. 4.3 The abstract versal deformation theorem
    4. 4.4 Functional analytic setting (part II)
    5. 4.5 Versal deformations over a Cantor set

##  1\. Introduction

In the formal neighbourhood of an analytic planar vector field one may find an
invariant algebroid curve, classically called a separatrix. The study of these
curves goes back to Briot-Bouquet, Dulac, Picard, Poincaré and was finally
solved by Camacho and Sad, who proved their existence in full generality [8,
15, 17, 36, 38].

These early investigations on separatrices were intimately connected to the
concept of normal forms and deformations near equilibrium; the idea being that
if a vector field can be reduced to a simple form by a change of variables,
then it would be easy to find the separatrices. But due to the appearance of
small denominators, the issue of convergence of the required transformation
turned out to be very non-trivial. In fact, the Poincaré-Siegel divergence
theorems suggest that there is no analytic normal forms or versal deformations
over a smooth base [37, 41].

In the work of Kolmogorov, Siegel, Bruno and Rüssmann profound results were
obtained with regard to convergence questions outside the Poincaré domain [14,
25, 40, 42]. As a rule, the normalising transformations diverge and only in
cases of integrability one may expect convergence of the normalising
transformations. The dichotomy theorems of Ilyashenko and Perez-Marco show
that these integrable cases are exceptional [23, 34, 35]. For excellent
overviews on these topics we refer the reader to [24, 29, 46].

So in the light of KAM theory, the best one can hope for is the construction
of a convergent versal deformations/normal forms over a Cantor set of positive
measure. It is the purpose of this paper to state and proves such a result.

Unlike the case of Hamiltonian dynamics and KAM theory, the Cantor set we
construct is not invariant and integral curves may not be all contained inside
the set. One may ask: what is the use a vector field over such a set? The
answer is quite simple: although the Cantor set is not invariant, it may very
well contain an interesting subset of persistant invariant varieties. In this
way we are led back to the very origins of normal forms alluded above.

Hyperbolic vector fields being linearisable in the $C^{\infty}$-category,
divergence and small denominators are invisible in the smooth category [43]
(incidentally this is not the case in the Gevrey category see [47]). This
underlines the relevance looking at vector fields in the complex domain, as
was traditionally done in the early works on the subject.

The simplest example is given by the $(1,-1)$ resonance, leading to the so-
called Hopf bifurcation:

$\left\\{\begin{matrix}\dot{x}=(-\alpha+\mu_{1}+\mu_{2}xy)x\\\
\dot{y}=(\alpha+\mu_{1}+\mu_{2}xy)y\end{matrix}\right.$

Classically one takes $\alpha$ is purely imaginary and restricts the flow to
the real subspace defined by $x=\bar{y}=z$. For $\mu=0$, the cone $xy=0$ is an
analytic invariant variety that we call the resonant cone. In the real case,
the curves $xy=\varepsilon$, that is the circles $|z|^{2}=\varepsilon$, are
limit cycles for $\varepsilon>0$ stable under perturbations [1, 22, 28]. In
the complex domain, these limit cycles vanishing at the singular point are
real parts of the invariant cylinders $xy=\varepsilon$ that degenerate into
the resonant cone [2, 3]. So we find here the classical situation of the
vanishing cycles at an $A_{1}$ singularity : a horizontal family of 1-cycles
generating the homology of the cylinder, which vanishes at the singular point
(see for instance [5, 6]). Pyartli proved the stability under perturbation of
the analytic cylinders degenerating into a cone [39].

It appears that Bruno thought Arnold had conjectured the existence of
invariant families in higher dimensions for simple resonances and disproved
this conjecture [11]. Incidentally, Bruno’s result was misquoted by Arnold in
his famous book on differential equation, where he attributes to Bruno and
Pyartli the proof of the existence of such families in dimension 3! (see [4,
Chapter 6, §36, E]). What Arnold conjectured or at least hoped was the
analyticity of the resonant cone for a single resonance and he wrote: However,
we can hope that these small perturbations do not destroy the invariant
manifold $M_{0}$.

Apparently unaware of Arnold’s problem, Écalle asserted that the conjecture
holds, but only under an arithmetic condition on the eigenvalues of the vector
field [18]. Analyticity of the resonant cone was finally proved in 1994 by
Stolovitch in his thesis. Stolovitch showed also the necessity of an
arithmetic condition [44]. Finally he extended the case of simple resonances
to that of so-called positive resonances, but apparently made a (minor)
mistake in defining positivity (see 2.3).

More recently Stolovitch proved a KAM theorem for vector fields, namely he
showed the existence of a positive measure set of invariant manifolds,
assuming a triviality condition of the Poincaré-Dulac normal form [45]. For
the case of the Hopf bifurcation, the triviality condition of Stolovitch means
that the Poincaré-Dulac normal form defines a symplectic vector field, so
rather than an isolated limit cycle we have a family of circles, the system is
integrable and there is no bifurcation at all. Similarly for the multi-
dimensional Hopf bifurcation, the condition of Stolovitch means that the
system is Hamiltonian. Therefore in this case, the theorem implies the
existence of a positive measure set of complex analytic invariant manifolds
carrying a complex quasi-periodic motion. In the real elliptic case, these are
the complexification of KAM tori vanishing at the singularity and the theorems
of Stolovitch imply in particular that these form a set of positive measure
around the singularity (see also [19]).

From the standpoint of a singularist, the KAM theorem of Stolovitch means that
there are strata in the versal deformation space on which invariant varieties
form a positive measure set. So it is natural to ask if such a family could
exist over the whole space. Such a result turns out to be a consequence of our
versal deformation theorem. If we modify Arnold’s conjecture in the sense that
the family of analytic varieties is not analytic but only continuous (and even
Whitney $C^{\infty}$) over a set of positive measure, then it holds true.

The invariant manifolds being toric manifolds, the result might be seen as a
complex variant of the theory of quasi-periodic motions in the dissipative
context at a singularity. The general theory of such quasi-periodic motions in
the dissipative context has been established in the pioneering works of Broer,
Bruno, Huitema, Sevryuk and Takens (see [9, 10, 12]). Applying our result to
the multiple Hopf bifurcation, we obtain the existence of quasi-periodic
motions vanishing at the singularity. This generalises a result obtained by Li
in dimension 2 and 3 [26] (see also [16] for the corresponding result for
discrete dynamical systems).

In this paper we use, as Stolovitch, positivity assumptions on the linear part
of our vector field, but in fact we are confident that the results can be
proven under less restrictive conditions. It is for instance plausible that
the methods for the classification on planar vector field due Malgrange,
Martinet-Ramis, Moussu-Cerveau can be used [27, 31, 32, 33].

From a technical point of view, the proof of our versal deformation theorem is
a direct application of the general theory of normal forms developed in [21].
Therefore, with regards to the details of the proof of convergence, we refer
to that paper on the theory of Banach space valued functors.

##  2\. Divergent and convergent normal forms

In the sixties, Bruno started the systematic investigation of differential
equations over rings of formal power series, and complemented it with
important convergence resuls. But the formal case in itself is already a rich
subject.

###  2.1. The Poincaré-Dulac normal form

We denote by $\widehat{{\mathcal{O}}}$ the local ring of formal power series
in $d$-variables $x_{1},x_{2},\ldots,x_{d}$:

$\widehat{{\mathcal{O}}}:=\mathbb{C}[[x_{1},x_{2},\ldots,x_{d}]],$

and by $\widehat{{\mathcal{X}}}$ the corresponding module of derivations

$\widehat{{\mathcal{X}}}:=Der(\widehat{{\mathcal{O}}})=\bigoplus_{i=1}^{d}\widehat{{\mathcal{O}}}\partial_{x_{i}},$

So a formal vector field $X\in\widehat{{\mathcal{X}}}$ is of the form

$X=\sum_{i=1}^{d}a_{i}(x)\partial_{x_{i}},\;\;\;a_{i}(x)\in\widehat{{\mathcal{O}}},$

and can be written as an expansion of homogeneous components of increasing
degree:

$X=X_{0}+X_{1}+X_{2}+\dots$

Here and in the sequel the dots stand for higher order terms in the Taylor
expansion at the origin.

The vector field is called singular if $X_{0}=0$. We say it is an isolated
singularity, if the origin is scheme-theoretically the only point for which
$X_{0}=0$, or equivalently if the ideal

$(a_{1},a_{2},\ldots,a_{n})\subset\widehat{{\mathcal{O}}}$

is ${\mathcal{M}}$-primary, where ${\mathcal{M}}$ denotes the maximal ideal of
$\widehat{{\mathcal{O}}}$. Such a vector field $X$ induces derivations $X_{k}$
of ${\mathcal{M}}/{\mathcal{M}}^{k+1}$ and as any linear map, can be
decomposed into semi-simple and nilpotent part:

$X_{k}=S_{k}+N_{k},\;\;\;[S_{k},N_{k}]=0,$

and by taking the limit $k\longrightarrow+\infty$, we obtain the abstract
Poincaré-Dulac normal form

$X=S+N,\;\;\;[S,N]=0.$

###  2.2. Resonances

From now on we assume that the linear part of the vector field is semi-simple:

$X_{1}=S_{1}=S,\ N_{1}=0.$

In appropriate coordinates the semi-simple part takes the form

$S=\sum_{i=1}^{d}\lambda_{i}x_{i}\partial_{x_{i}},$

and we call

$\Lambda:=(\lambda_{1},\lambda_{2},\ldots,\lambda_{d})$

the frequency vector of $X$. We denote by

${\widehat{{\mathcal{O}}}}^{S}:=\\{f\in\widehat{{\mathcal{O}}}\;\;|\;\;S(f)=0\\}\subset\widehat{{\mathcal{O}}}$

the ring of invariants of $S$. A monomial

$x^{R}:=x_{1}^{r_{1}}x_{2}^{r_{r}}\ldots
x_{d}^{r_{d}},\;\;\;r_{i}\in\mathbb{N}$

is invariant if and only if

$(\Lambda,R):=\sum_{i=1}^{d}\lambda_{i}r_{i}=0,$

and the ring $\widehat{{\mathcal{O}}}^{S}$ is generated by a finite number of
such resonant monomials

$x^{R_{1}},x^{R_{2}},\ldots,x^{R_{p}},\;R_{j}\in\mathbb{N}^{d},$

so that

$\widehat{{\mathcal{O}}}^{S}=\mathbb{C}[[x^{R_{1}},x^{R_{2}},\ldots,x^{R_{p}}]].$

The vectors $R$ for which $(\Lambda,R)=0$ are called resonances and all
resonances are non-negative linear combinations of the basic resonances
$R_{i}$, but this representation in general is not unique.

Similarly, we denote by

$\widehat{{\mathcal{X}}}^{S}:=\\{V\in\widehat{{\mathcal{X}}}\;\;|\;\;[S,V]=0\\}$

the resonant vector fields, which are invariant under the infinitesimal action
by $S$. A monomial field

$x^{K}\partial_{x_{i}}=x_{1}^{k_{1}}x_{2}^{k_{2}}\ldots
x_{d}^{k_{d}}\partial_{x_{i}}$

belongs to $\widehat{{\mathcal{X}}}^{S}$ if and only if

$(\Lambda,K)-k_{i}=0.$

Note that the fields $x_{i}\partial_{x_{i}},i=1,2,\ldots,n$, always belong to
$\widehat{{\mathcal{X}}}^{S}$. The resonant vector fields
$\widehat{{\mathcal{X}}}^{S}$ form a module over
$\widehat{{\mathcal{O}}}^{S}$, which in fact is finitely generated. So there
are finitely many resonant fields

$V_{1},V_{2},\ldots,V_{m},$

such that any $V\in\widehat{{\mathcal{X}}}^{S}$ can be expressed as

$V=\sum_{i=1}^{m}\phi_{i}(x^{R_{1}},X^{R_{2}},\ldots,x^{R_{p}})V_{i}$

This immediately leads to the following result:

######  Theorem ([13, 17]).

Let $X$ be a formal vector field then there exists an automorphism $\varphi\in
Aut(\widehat{{\mathcal{O}}})$ and power series
$\phi_{1},\dots,\phi_{m}\in\mathbb{C}[[u]]:=\mathbb{C}[[u_{1},\dots,u_{p}]]$
such that $\varphi(X)$ is a resonant vector field i.e.:

$\varphi(X)=S+\sum_{i=1}^{m}\phi_{i}(x^{R_{1}},\dots,x^{R_{p}})V_{i}$

In the statement of the theorem and in the sequel, the derivation $\varphi(X)$
is defined by the commutative diagram

$\textstyle{\widehat{{\mathcal{O}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X}$$\scriptstyle{\varphi}$$\textstyle{\widehat{{\mathcal{O}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\widehat{{\mathcal{O}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi(X)}$$\textstyle{\widehat{{\mathcal{O}}}}$

Note that in differential geometry, a derivation $X$ is interpreted as a
vector field and the notation $\varphi_{*}X$ is used. As $\varphi(X)$ cannot
be understood in another way, we adopt this simplified notation which is more
adapted to our computations.

###  2.3. Positivity conditions

We make the following assumptions on the eigenvalue $\Lambda$ of $S$:
P1) there are no algebraic relations between a minimal set of generators for
the ring of invariants. In other words, we assume that

$\widehat{{\mathcal{O}}}^{S}=\mathbb{C}[[u_{1},u_{2},\ldots,u_{p}]]$

is a power series ring.
P2) the module of $\widehat{{\mathcal{X}}}^{S}$ is freely generated by the
$x_{i}\partial_{x_{i}}$’s over the ring $\widehat{{\mathcal{O}}}^{S}$.

The following easy reformulation show why one can think of P1) and P2) as
positivity conditions.

######  Lemma.

The conditions P1) and P2) are respectively equivalent to the existence of
integral vectors $R_{1},\dots,R_{p}\in\mathbb{N}^{d}$ with $(L,R_{j})=0$, such
that:
1) Each vector $J\in\mathbb{N}^{d}$ with $(\Lambda,J)=0$ can be uniquely
written as

$J=\sum_{j=1}^{p}n_{j}R_{j},\;\;\;n_{j}\in\mathbb{N}.$

2) Each vector $K\in\mathbb{N}^{d}$ with $(\Lambda,K)=k_{i}$ for some $i$, can
be uniquely written as

$K=\sum_{j=1}^{d}m_{j}R_{j}+E_{j},\;\;\;m_{j}\in\mathbb{N},$

where $E_{1}=(1,0,\dots,0),\dots,E_{d}=(0,\dots,0,1)$ denotes the standard
basis vectors.

###  2.4. Examples and counter-examples

Let us consider a pair of opposite eigenvalues

$S=\lambda x\partial_{x}-\lambda y\partial_{y}$

The ring of invariants is generated by $xy$, the resonant fields are generated
by $x\partial_{x}$ and $y\partial_{y}$. Thus it satisfies the positivity
conditions. The formal normal form of a vector field with this linear part is

$(\lambda+g(xy))x\partial_{x}-(\lambda+h(xy))y\partial_{y}$

This vector field is Hamiltonian for the symplectic structure $dx\wedge dy$
precisely when $g=h$. A direct generalisation arises when eigenvalues form
opposite pairs

$S=\sum_{i=1}^{d}\lambda_{i}x_{i}\partial_{x_{i}}-\lambda_{i}y_{i}\partial_{y_{i}}$

and are generic otherwise, in the sense that

$\dim_{\mathbb{Q}}\sum_{i=1}^{d}\mathbb{Q}\lambda_{i}=d.$

The ring of invariants is generated by $x_{i}y_{i}$, $i=1,2,\ldots,d$, the
resonant fields are the $x_{i}\partial_{x_{i}}$, $y_{i}\partial_{y_{i}}$, so
that it is a case where the the positivity conditions hold. So the formal
normal form of a vector field with this linear part is

$S=\sum_{i=1}^{d}(\lambda_{i}+g_{i}(x_{1}y_{1},\ldots,x_{d}y_{d}))x_{i}\partial_{x_{i}}-(\lambda_{i}+h_{i}(x_{1}y_{2},\ldots,x_{d}y_{d}))y_{i}\partial_{y_{i}}$

Again the field is Hamiltonian for the symplectic structure

$\omega=\sum_{i=1}^{n}dx_{i}\wedge dy_{i}$

precisely when $g_{i}=h_{i}$.

A very different case is the $(1,1,-2)$-resonance:

$S=x\partial_{x}+y\partial_{y}-2z\partial_{z}$

The ring of invariants is generated by

$u:=x^{2}z,\;\;v:=y^{2}z,\;\;w:=xyz.$

Among these there is a relation

$uv-w^{2}=0,$

so the map

$\mathbb{C}^{3}\longrightarrow\mathbb{C}^{3},\
(x,y,z)\mapsto(x^{2}z,y^{2}z,xyz)$

is not surjective, but lands on the quadratic cone defined by the above
equation. The first positivity condition is not satisfied. Also, the module of
resonant vector fields is generated by

$x\partial_{x},\;\;y\partial_{y},\;\;z\partial_{z},\;\;x\partial_{y},\;\;y\partial_{x}$

The second positivity condition is not satisfied either and the relations

$vx\partial_{x}=wy\partial_{x},\;\;\;uy\partial_{y}=wx\partial_{y}$

show ${\mathcal{X}}^{S}$ is not a free module over the invariant ring
$\widehat{{\mathcal{O}}}^{S}$, showing that P2) does not hold either.

In [44, Section 4], Stolovitch seems to suggests that P1) implies P2). But if
we consider the case $\Lambda=(\alpha+\beta,\alpha,\beta)$, where $\alpha$ and
$\beta$ are $\mathbb{Q}$-independent and positive, there are no non-trivial
invariants, $\widehat{{\mathcal{O}}}^{S}=\mathbb{C}$, but there is a non-
trivial resonant vector field, namely $yz\partial_{x}$. So P1) is satisfied,
but not P2). By adding extra variables one obtains examples with
$\widehat{{\mathcal{O}}}^{S}\neq\mathbb{C}$.

###  2.5. The resonant cone

If $S$ satisfies the positivity assumptions, then any vector field in Poincaré
normal form is of the form

$X=S+N=\sum_{i=1}^{d}(\lambda_{i}+\phi_{i}(x^{R_{1}},x^{R_{2}},\ldots
x^{R_{p}}))x_{i}\partial_{x_{i}}$

In particular, such vector fields are of the form

$X=S+T,\;\;\;T=\sum_{i}^{d}a_{i}(x)x_{i}\partial_{x_{i}},\;\;\;a_{i}(x)\in\widehat{{\mathcal{O}}}$

that we will call formal logarithmic vector fields. Such a vector field have a
special property.

######  Definition.

The resonant ideal is the ideal $I$ generated by the invariants of positive
degree.

$I=(x^{R_{1}},x^{R_{2}},\ldots,x^{R_{p}})\subset\widehat{{\mathcal{O}}}$

The resonant cone is the formal scheme defined by the resonant ideal.

######  Proposition.

A logarithmic vector field

$X=S+T,\;\;\;T=\sum_{i}^{d}a_{i}(x)x_{i}\partial_{x_{i}}$

preserves the the resonant cone.

###### Proof.

$(S+T)x^{R_{j}}=\left((\Lambda,R_{j})+\sum_{i=1}^{d}a_{i}(x)r_{i,j}\right)x^{R_{j}}\in(x^{R_{1}},\dots,x^{R_{p}}).$

∎

######  Corollary.

If $S$ satisfies the positivity assumption, then any vector field

$X=S+\dots$

admits an invariant ideal isomorphic to the resonant ideal.

###### Proof.

By positivity, the normal form of the vector field is logarithmic, so the
result from the above proposition. ∎

###  2.6. The Stolovitch normal form

Like for the Poincaré-Siegel theorems, the transformation bringing a vector
field to Poincaré Dulac normal form is, as a general rule, divergent [14, 23,
34]. Therefore it is important to have a less restrictive but convergent
normal form.

We denote

${\mathcal{O}}:=\mathbb{C}\\{x_{1},x_{2},\ldots,x_{d}\\},\;\;\;{\mathcal{X}}:=Der({\mathcal{O}})=\oplus_{i=1}^{d}{\mathcal{O}}\partial_{x_{i}}$

the local ring of convergent power series and the module of analytic vector
fields.

The following fundamental result is due to Stolovitch [44]:

######  Theorem.

Let $S$ be a linear vector field that satisfies the Bruno and positivity
conditions. Then for any analytic vector field

$X=S+\dots\in{\rm Der\,}({\mathcal{O}})$

there exists an analytic automorphism $\varphi\in{\rm Aut\,}({\mathcal{O}})$
such

$\varphi(X)=S\ {\rm mod\,}{\mathcal{X}}_{log}$

where

${\mathcal{X}}_{log}=\sum_{i=1}^{d}{\mathcal{O}}x_{i}\partial_{x_{i}}$

is the ${\mathcal{O}}$-module of logarithmic vector fields.

As remarked above, the formal version of this theorem trivially follows from
the Poincaré-Dulac normal form. The importance of the theorem lies in the
following corollary.

######  Corollary.

If $S$ satisfies the positivity conditions, then any analytic vector field
$X=S+\ldots$ admits an invariant analytic germ, isomorphic to the analytic
resonant cone.

Here by the phrase analytic resonant cone we mean of course the germ defined
by the ideal in ${\mathcal{O}}$ generated by the analytic resonant ideal
$(x^{R_{1}},x^{R_{2}},\ldots,x^{R_{p}})\subset{\mathcal{O}}$.

The above theorem of Stolovitch can be proven using our Lie iteration, whose
convergence can be shown in great generality (see also 4.3).

##  3\. The formal versal deformation

In this section, we consider a derivation with semi-simple linear part $S$. We
assume positivity conditions on $S$ and choose an integer basis
$R_{1},\dots,R_{j}$ of resonances.

###  3.1. Bruno variables

In [11], Bruno extended the ring $\widehat{{\mathcal{O}}}$ by adding variables
$u_{1},\dots,u_{p}$ corresponding to the $p$ monomial invariants
$x^{R_{i}},i=1,2,\ldots,p$. In the approach of Stolovitch in [45], this can be
seen as a generalisation the variables introduced by Moser in KAM-theory to
parametrise the invariant tori.

Geometrically, we introduce a space $\mathbb{C}^{p}$ with coordinates
$u_{1},\dots,u_{p}$ and define a map

$\mathbb{C}^{d}\longrightarrow\mathbb{C}^{p},x\mapsto(x^{R_{1}},x^{R_{2}},\ldots,x^{R_{p}}).$

The (completion at $0$ of the) fibre over $0$ is the resonant cone; note that
the semi-simple vector field $S$ preserves the fibres of $\varphi$. The graph
of this map is the space $\Sigma\subset\mathbb{C}^{n+p}$ defined by the
vanishing of the polynomials

$J_{1}:=u_{1}-x^{R_{1}},\;\;\;J_{2}:=u_{2}-x^{R_{2}},\ldots,\;\;\;J_{p}:=u_{p}-x^{R_{p}}.$

Any vector field $Y$ on $\mathbb{C}^{d}$ can be lifted to a vector field
$\widetilde{Y}$ on $\mathbb{C}^{d+p}$, tangent to $\Sigma$:

$Y\mapsto\widetilde{Y}:=Y+\sum_{j=1}^{p}Y(x^{R_{j}})\partial_{u_{j}},$

reflected algebraically by the statement

$\widetilde{Y}(J_{j})=0,\;\;j=1,2,\ldots,p.$

In our formal context we extend the power series ring
$\widehat{{\mathcal{O}}}$ by formal variables and work in

$\widehat{{\mathcal{O}}}[[u]]=\mathbb{C}[[x,u]]=\widehat{{\mathcal{O}}}[[u_{1},u_{2},\ldots,u_{p}]]=\mathbb{C}[[x_{1},\ldots,x_{d},u_{1},\ldots,u_{p}]]$

and rather than the manifold $\Sigma$, we consider the ideal

$J:=(J_{1},J_{2},\ldots,J_{p})\subset\widehat{{\mathcal{O}}}[[u]].$

Clearly, there is a ring-isomorphism

$\widehat{{\mathcal{O}}}[[u]]/J\longrightarrow\widehat{{\mathcal{O}}},\;\;\;x_{i}\mapsto
x_{i},\;\;\;u_{j}\mapsto x^{R_{j}}.$

We will use the lift $Y\mapsto\widetilde{Y}$ to identify
$\widehat{{\mathcal{X}}}$, $\widehat{{\mathcal{X}}}^{S}$ as subspaces of ${\rm
Der\,}(\widehat{{\mathcal{O}}}[[u]])$. and suppress the $\widetilde{}$ of the
notation.

######  Definition.

The ideal $J\subset\widehat{{\mathcal{O}}}[[u]]$ is called the graph ideal of
$S$.

We assign degree $1$ to the variables $x_{i}$ and

$|R_{i}|=\sum_{j=1}^{d}R_{i,j}$

to the variable $u_{j}$, so that the polynomials $I_{j}$ are homogeneous of
degree $|R_{j}|$. We filter the ring $\widehat{{\mathcal{O}}}[[u]]$, by the
order of a series, that is, the smallest degree of a monomial appearing in the
series.

This filtration of the ring induces a filtration on ${\rm
Der\,}(\widehat{{\mathcal{O}}}[[u]])$: a derivation is said to be of order $d$
if it maps terms of order $i$ to terms of order $i+d$, so that the order of
$S$ is $0$.

###  3.2. The detuning deformation

We now follow Martinet’s treatment of versal deformations and construct a ring
with extra parameters to describe a versal deformation. If we are given an
arbitrary deformation, given by a family with still other variables, we also
add these to the ring. The inducing maps are then obtained by solving
equations by the implicit function theorem [30]. The method is classical and
we used this technique also in [20].

Following Bruno, we work in the ring $\widehat{{\mathcal{O}}}[[u]]$ and with
lifted fields $\widehat{{\mathcal{X}}}\subset
Der(\widehat{{\mathcal{O}}}[[u]])$.

By a deformation of $S$ we mean a vector field depending on $l$ parameters
$\mu=(\mu_{1},\mu_{2},\ldots,\mu_{l})$ of the form

$X=S+T(\mu)\in\widehat{{\mathcal{X}}}[[\mu]],\;\;\;T(0)=0.$

Here

$\widehat{{\mathcal{X}}}[[\mu]]\subset Der(\widehat{{\mathcal{O}}}[[u,\mu]])$

consist of those vector fields of $\widehat{{\mathcal{X}}}[[u]]$ that contain
no $\partial_{\mu_{i}}$ (relative vector fields).

A special detuning deformation is obtained by introducing frequency variabes
$\phi_{1},\ldots,\phi_{d}$ and considering

$S_{v}=S+\sum_{i=1}^{d}\phi_{i}x_{i}\partial_{x_{i}}=\sum_{i=1}^{d}(\lambda_{i}+\phi_{i})x_{i}\partial_{x_{i}}\in\widehat{{\mathcal{X}}}[[\phi]]$

Under our running positivity assumptions we have

$\widehat{{\mathcal{X}}}^{S}=\oplus_{i=1}^{d}\widehat{{\mathcal{O}}}^{S}x_{i}\partial_{x_{i}},$

so the Poincaré-Dulac theorem suggests that the detuning deformation is versal
in a formal sense.

If we want to induce an arbitrary deformation $X=S+T(\mu)$ from the detuning
deformation, one first forms the sum of the two deformations, that is we
consider

$X_{v}=S_{v}+T(\mu)=S+\sum_{i=1}^{d}\phi_{i}x_{i}\partial_{x_{i}}+T(\mu),$

a deformation of $S$ with two sets of parameters $\phi$ and $\mu$. The
versality is then expressed by the existence of a certain automorphism

$\varphi\in Aut({\mathcal{O}}[[u,\phi,\mu]])$

that maps $X_{v}$, considered as a deformation of $S_{v}$, back to $S_{v}$. So
we will have to work with variables $u,\phi$ and $\mu$ and vector fields that
depend on these variables, but there will be no derivatives in the $\mu$ or
$\phi$ directions, and thus belong to

$\widehat{{\mathcal{X}}}[[\phi,\mu]]\subset{\rm
Der\,}_{\mathbb{C}[[\phi,\mu]]}(\widehat{{\mathcal{O}}}[[u,\phi,\mu]]).$

We also use the notation

$\widehat{{\mathcal{X}}}^{S}[[\phi,\mu]]\subset{\rm
Der\,}_{\mathbb{C}[[\phi,\mu]]}(\widehat{{\mathcal{O}}}[[u,\phi,\mu]])$

for the $\widehat{{\mathcal{O}}}[[\phi,\mu,u]]$-module generated by the
$(x^{R_{i}}-u_{i})x_{j}\partial_{x_{j}}$’s.

We assign degree $2$ to the $\mu$ variables, in this way making a deformation
of a linear vector field is obtained by adding terms of higher order.

###  3.3. Formal normal form theorem

######  Theorem.

Let $S=\sum\lambda_{i}x_{i}\partial_{x_{i}}$ be a linear vector field
satisfying the positivity assumptions and let

$S_{v}=\sum_{i=1}^{d}(\lambda_{i}+\phi_{i})x_{i}\partial_{x_{i}}\in\widehat{{\mathcal{X}}}[[\phi]]$

be its detuning deformation. Then for any perturbation of the form

$X_{v}:=S_{v}+\dots\in\widehat{{\mathcal{X}}}[[\phi,\mu]],$

where the dots stand for higher order terms; there exists an automorphism
$\varphi$ of the algebra $\widehat{{\mathcal{O}}}[[\phi,\mu,u]]$ which has the
following properties:

  1. 1)

$\varphi(X_{v})=S_{v}\quad{\rm mod\,}\
\widehat{{\mathcal{X}}}^{S}[[\phi,\mu]]$.

  2. 2)

$\varphi(u_{j})=u_{j}$ for $j=1,\dots,k$.

  3. 3)

$\varphi(\mu_{j})=\mu_{j}$ for $j=1,\dots,l$.

  4. 4)

$\varphi(\phi_{j})=\phi_{j}+\dots\in\mathbb{C}[[\phi,\mu,u]].$

###### Proof.

The proof is done by applying the Lie iteration. To save ink we set:

$R:=\widehat{{\mathcal{O}}}[[u,\phi,\mu]],{\rm
Der\,}:=\widehat{{\mathcal{X}}}[[\phi,\mu]]\subset{\rm Der\,}(R)$

The infinitesimal action on the versal deformation is given by map

${\rm Der\,}\longrightarrow{\rm Der\,},\ X\mapsto[X,S_{v}].$

It is $\mathbb{C}[[u,\phi,\mu]]$-linear and diagonal in the monomial basis:

$[x^{K}\partial_{x_{i}},S_{v}]=\lambda_{K,i}x^{K}\partial_{x_{i}},\;\;\;\lambda_{K,i}:=\left(\lambda_{i}+\phi_{i}-(\Lambda+\phi,K)\right).$

We define a $\mathbb{C}[[u,\phi,\mu]]$-linear map $L$ by putting for non-
resonant vector $K$ by

$\displaystyle L(x^{K}\partial_{x_{i}})$
$\displaystyle=\lambda^{-1}_{K,i}x^{K}\partial_{x_{i}}$ $\displaystyle
L(x_{i}\partial_{x_{i}})$ $\displaystyle=\partial_{\phi_{i}}$

As the module of resonant vector fields is freely generated by vector fields
of the form $x_{i}\partial_{x_{i}}$ the map $L$ is well-defined. Note that

$\displaystyle[u_{j}\partial_{\phi_{i}},S_{v}]$
$\displaystyle=u_{j}x_{i}\partial_{x_{i}}$
$\displaystyle=x^{R_{j}}x_{i}\partial_{x_{i}}+(u_{j}-x^{R_{j}})x_{i}\partial_{x_{i}}$
$\displaystyle=x^{R_{j}}x_{i}\partial_{x_{i}}\quad{\rm mod\,}J$

where $J$ is the graph ideal. Using $L$, for $V=S_{v}+T$, we define the map:

$j_{V}:m\mapsto Lm-L(Lm(T))=L(m-Lm(T)).$

This map satisfies

$[j_{V}(Y),S_{v}+T]=Y\ {\rm mod\,}J$

We define

$X_{0}=X_{v},\ S_{0}=0,\ A_{0}=S_{v},\ v_{0}=[j_{S_{v}}(X_{0})]_{1}^{2},$

where

$[-]_{a}^{b}$

denotes the sum of terms of degree $\geq a$ and $<b$ in the Taylor series at
$0$. The following Lie iteration which brings a vector field $X$ back to its
normal form :

$\begin{array}[]{ ll }&\\\ X_{n+1}&=e^{-[-,v_{n}]}X_{n},\\\ &\\\
S_{n+1}&=[X_{n}-[A_{n},v_{n}]]_{2^{n}}^{2^{n+1}},\\\ &\\\
A_{n+1}&=A_{n}+S_{n+1},\\\ &\\\
v_{n+1}&=j_{A_{n}}([X_{n}]_{2^{n+1}}^{2^{n+2}})\\\ &\\\ \end{array}$

∎

###  3.4. Relation to the Poincaré-Dulac normal form

The theorem implies the classical Poincaré-Dulac theorem. In this case, our
perturbation does not contain the $\mu$ parameters.

We start a vector field

$X=S+X^{\prime},\;\;X\in\widehat{{\mathcal{X}}}$

where $X^{\prime}$ are terms of positive order and $S$ is semi-simple. We
consider the associated perturbation of our corresponding versal deformation:

$X_{v}=S_{v}+X^{\prime}$

So $X$ is obtained from $X_{v}$ by setting all $\phi_{i}=0$. The above Theorem
Theorem tells us that there exists an automorphism $\varphi\in
Aut(\widehat{{\mathcal{O}}}[[\phi,u]])$ such that

$\varphi(X_{v})=S_{v}\;\;{\rm mod\,}\widehat{{\mathcal{X}}}^{S}[[\phi]]$

This automorphism $\varphi\in Aut(R)$ maps $\phi_{j}$ to
$\varphi(\phi_{j})=\phi_{j}+\ldots\in\mathbb{C}[[\phi,u]]$. By the implicit
function theorem, we may solve the equations $\varphi(\phi_{j})=0$ by
$\phi_{j}=g_{j}(u)$ for certain series $g_{j}(u)$. So we obtain

$\displaystyle\varphi(X)$
$\displaystyle=\sum_{i=1}^{d}(\lambda_{i}+g_{i}(u))x_{i}\partial_{x_{i}}+\sum_{j=1}^{p}(g,R_{j})u_{j}\partial_{u_{j}}\;{\rm
mod\,}\widehat{{\mathcal{X}}}^{S}$

which is the usual Poincaré-Dulac normal form using Bruno variables.

###  3.5. The versality property

The result implies that the detuning deformation is formally versal. Indeed,
let

$X=S+T(\mu)\in{\rm Der\,}_{\mathbb{C}}(\widehat{{\mathcal{O}}}[[\mu]])$

be an arbitrary deformation of $S$, so

$X=\sum_{i=1}^{d}(\lambda_{i}x_{i}+T_{i}(\mu,x))\partial_{x_{i}},$

such that $T_{i}(\mu=0)=0$. We form the sum deformation:

$X_{v}=S_{v}+T(\mu).$

The theorem above states that $X_{v}$ is mapped via an automorphism
$\varphi\in{\rm Aut\,}(\widehat{{\mathcal{O}}}[[u,\phi,\mu]])$ to $S_{v}$.
Again the restriction of $X_{v}$ to $\phi=0$ is the initial deformation $X$.
Solving the equations $\varphi(\phi_{j})=0$ by $\phi_{j}=g_{j}(u,\mu)$, we get
that

$\varphi(\widetilde{X})=\sum_{i=1}^{d}(\lambda_{i}+g_{i}(u,\mu))x_{i}\partial_{x_{i}}+(g,R_{i})u_{i}\partial_{u_{i}}\
{\rm mod\,}{\mathcal{X}}^{S}[[\mu]].$

If we wish one may eliminate the $u$-variables using the ideal $J$ and get

$\varphi(X)=\sum_{i=1}^{d}(\lambda_{i}+g_{i}(x^{R},\mu))x_{i}\partial_{x_{i}}$

###  3.6. The Bruno-Stolovitch ideal

Consider a vector field in normal form

$Y=\sum_{i=1}^{d}(\lambda_{i}+g_{i}(u,\mu))x_{i}\partial_{x_{i}}+\sum_{j=1}^{p}(g,R_{j})u_{j}\partial_{u_{j}}\;{\rm
mod\,}{\mathcal{X}}^{S}[[\mu]]$

here $g=(g_{1},\ldots,g_{p})$, $R_{j}=(R_{j,1},\ldots,R_{j,p}$ and
$(g,R_{j})=\sum_{i=1}^{p}a_{i}R_{j,i}$. The graph ideal $J$ is by definition
$Y$-invariant, but it contains also a bigger invariant ideal $I_{\mu}$
obtained by adding the functions $(g,R_{i})$’s.

######  Definition.

The Bruno-Stolovitch ideal of the vector field $Y$ is the ideal

$I_{\mu}=J+((g,R_{1}),\ldots,(g,R_{p}))\subset\widehat{{\mathcal{O}}}[[u,\mu]]$

######  Lemma.

The ideal $I_{\mu}$ is $Y$-invariant:

$f\in I_{\mu}\implies Y(f)\in I_{\mu}$

###### Proof.

A direct computations shows that

$Y(u_{i})=(g,R_{i})u_{i}\quad{\rm mod\,}J$

and therefore $Y(u_{i})\in I_{\mu}$. In vector notations

$Y(g)=(D_{u}g)Y(u)$

which can be interpreted as the image of the vector field $Y$ under the
differential $D_{u}g$ of the map $g$. Therefore

$Y((g,R_{i}))=((D_{u}g)Y(u),R_{i})\quad{\rm mod\,}J$

and the right hand side lies in $I_{\mu}$. This proves the lemma. ∎

This means that the ideal $I$ generated by the $x^{R_{j}}-u_{j}$’s and the
functions $(g,R_{j})$ is an invariant ideal of the vector field.

Assume for a moment that for a fixed value of $\mu$ the functions $g_{i}$’s
are analytic in a small neighbourhood of the origin. Then the vector field
$Y(\mu,-)$ is analytic and tangent to the Bruno-Stolovitch varieties:

$V_{\mu}:=\\{(x,u)\in\mathbb{C}^{d+p}:(g(u,\mu),R_{i})=0,\ u=x^{R}\\}.$

As Bruno showed in [11], as a general rule, one cannot hope to have even a
$\mu$-dependant analytic family of the varieties. However in the formal case,
the varieties $V_{\mu}$ are formal schemes and, as one would expect from
classical KAM theory, these form a Whitney $C^{\infty}$-family of analytic
varieties over a positive measure set.

###  3.7. Piartly’s example

The versal deformation

$Y=(\alpha+\mu+xy+\dots)x\partial_{x}+(-\alpha+\mu+xy+\dots)y\partial_{y}.$

is given by:

$S_{v}=(\alpha+\phi_{1})x\partial_{x}+(-\alpha+\phi_{2})y\partial_{y}$

and by versality of this deformation, there exists an automorphism $\varphi$
such that

$\varphi(\widetilde{Y})=(\alpha+g_{1}(u,\mu))x\partial_{x}+(-\alpha+g_{2}(u,\mu))y\partial_{y}+(g_{1}+g_{2})u\partial_{u}\
{\rm mod\,}M$

As the only resonant exponent is $(1,1)$, the Bruno-Stolovitch ideal is
generated by

$((1,1),g)=g_{1}(u,\mu)+g_{2}(u,\mu)$

and by the function $u-xy$ which generates the resonance ideal.

An explicit computations shows that we have the expansion

$g_{1}(u,\mu)+g_{2}(u,\mu)=2u+\mu+\dots$

The function

$g_{1}(xy,\mu)+g_{2}(xy,\mu)=2xy+\mu+\dots$

defining the Bruno ideal has an $A_{1}$-singularity.

If we assume this expansion to be analytic for some small value of of $\mu$
then the manifold $V_{\mu}$ is a cylinder for $\mu\neq 0$ and a cone for
$\mu=0$. Assuming $\alpha$ imaginary, we may restrict the field to the real
plane $\mathbb{R}^{2}\approx\mathbb{C}\subset\mathbb{C}^{2}$ defined by the
equation

$g_{1}(z\bar{z},\mu)+g_{2}(z\bar{z},\mu)=2|z|^{2}+\mu+\dots$

is diffeomorphic to a circle for $\mu<0$. We obtain the standard picture of
the family of vanishing cycles of the $A_{1}$-singularity (see [5, 7]).

###  3.8. The multi-dimensional Hopf bifurcation

It is easy to generalise the above picture to higher dimensions. We start with
a vector field of the form

$Y=\sum_{i=1}^{d}(\alpha_{i}+\mu_{i}+x_{i}y_{i}+\dots)x_{i}\partial_{x_{i}}+(-\alpha_{i}+\mu_{i}+x_{i}y_{i}+\dots)y_{i}\partial_{y_{i}}$

where the $\alpha_{i}$ are $\mathbb{Q}$-independent. We consider the versal
deformation:

$S_{v}=\sum_{i=1}^{d}(\alpha_{i}+\phi_{i})x_{i}\partial_{x_{i}}+\sum_{i=1}^{d}(-\alpha_{i}+\phi_{i+d})y_{i}\partial_{y_{i}}.$

The vector field $\widehat{Y}$ is isomorphic to

$Y=\sum_{i=1}^{d}(\alpha_{i}+g_{i}(u,\mu))x_{i}\partial_{x_{i}}+\sum_{i=1}^{d}(-\alpha_{i}+g_{i+d}(u,\mu))y_{i}\partial_{y_{i}}$

The resonance ideal is generated by the $u_{i}-x_{i}y_{i}$’s and the Bruno-
Stolovitch ideal is generated by these functions to which we add the functions
$g_{i}+g_{i+d}$. In this case, if the function turns out to be analytic form
some $\mu$-value and if we consider real structure $x_{=}\overline{y_{i}}$ as
for $d=1$, the real parts of the manifolds $V_{\mu}$, for $\mu_{i}<0$, are
tori and the vector field carries a quasi-periodic motion.

##  4\. Versal deformations and invariant varieties

Iteration above was defined on the level of formal power series. But one can
define a similar iteration defined for appropriate Banach spaces of analytic
functions. The convergence of the iteration procedure then is a direct
consequence of the general theory developed in [21] and that we will now
briefly recall. We will be very sketchy as full details were already given in
[21]. It is likely that one may produce complete and direct proofs along the
lines of Stolovitch [44, 45], but this would require a much longer line of
arguments and computations.

###  4.1. Bruno sequences

For convergence, our precedure requires Bruno-type conditions. To fix
notations, we briefly recall the standard facts.

######  Definition ([14]).

A strictly monotone positive sequence $a$ is called a Bruno sequence if the
infinite product

$\prod_{k=0}^{\infty}a_{k}^{1/2^{k}}$

converges to a strictly positive number or equivalently if

$\sum_{k\geq 0}\left|\frac{\log a_{k}}{2^{k}}\right|<+\infty.$

We denote respectively by ${\mathcal{B}}^{+}$ and ${\mathcal{B}}^{-}$ the set
of increasing and decreasing Bruno sequences.
Attached to a frequency vector $\Lambda\in\mathbb{C}^{d}$, we define the
sequence $\sigma(\Lambda)$ with terms

$\sigma(\Lambda)_{k}:=\min_{i}\\{|(\Lambda,J-E_{i})|\neq
0:J\in\mathbb{N}^{d}\setminus\\{0\\},\|J\|\leq 2^{k}\\}$

Here $E_{i}(=0,\dots,0,1,0,\dots,0)$ denotes the standard basis of the vector
space $\mathbb{C}^{d}$. We say that the vector $\Lambda$ is Bruno if the
sequence $\sigma(\Lambda)$ is a decreasing Bruno sequence.

###  4.2. Functional analytic setting (part I)

We fix the frequency vector $\Lambda$ of our linear vector field $S$. We will
replace the the ring $\widehat{{\mathcal{O}}}[[u]]$ of formal power series by
an appropriate system of Banach space of holomorphic functions. To describe
these, we first consider the space $\mathbb{C}^{d}\times\mathbb{C}^{p}$, with
coordinates $x$, $u$. We put

$D^{d}_{s}:=\\{x\in\mathbb{C}^{d}\;|\;|x_{i}|<s,\;i=1,2,\ldots,d\\}$
$D^{p}_{s}:=\\{u\in\mathbb{C}^{p}\;|\;|u_{j}|<s,\;j=1,2,\ldots,p\\}$

We define

$U=D^{d}\times D^{k}\longrightarrow\mathbb{N}\times\mathbb{R}_{>0}$

as a relative set with fibres independent on $n\in\mathbb{N}$:

$U_{n,s}=D^{d}_{s}\times D^{p}_{s}$

We then consider the Banach spaces of holomorphic functions and vector fields
on these sets (with continuous extension to the boundary),

$E_{n,t}:={\mathcal{O}}^{c}(U_{n.t}),\;\;\;F_{n,t}:={\mathcal{X}}^{c}(U_{n,t})$

for which there are corresponding restriction maps of norm $\leq 1$

$E_{n,t}\longrightarrow E_{n,s},\;\;s<t,\;\;\;\;E_{m,s}\longrightarrow
E_{n,s},\;\;n>m$

and similarly for ${\mathcal{X}}$. In the terminology of [21] such a system of
Banach spaces is called an Arnold space and write simply
$E={\mathcal{O}}^{c}(W)$, etc.

The set $U_{n,s}$ has the Huygens property for the constant sequence
$a_{n}=1$, by which we mean

$U_{n,s}+D_{a_{n}(t-s)}\subset U_{n,t}.$

This guarantees that the locality estimates are satisfied for differential
operators and when passing from one Arnold space to another.

In the iteration process, we will also have to choose an appropriate sequence

$s_{0}>s_{1}>s_{2}\ldots>s_{n}>s_{n+1}>\ldots>s_{\infty}>0$

and form the ’pull-back’, i.e. we form

$E_{n}:=E_{n,s_{n}},n=0,1,2,\ldots$

for which we still have restriction maps

$E_{0}\longrightarrow E_{1}\longrightarrow E_{2}\longrightarrow\ldots$

As sequence $(x_{n})$ with $x_{n}\in E_{n}$ is called convergent if it is a
Cauchy sequence in the sense that

$\forall\varepsilon>0,\ \exists N,\;n\geq N\implies|\iota
x_{n}-x_{n+p}|\leq\varepsilon$

where $\iota=\iota_{n+p,n}$ is the restriction map from $E_{n}$ to $E_{n+p}$.

###  4.3. The abstract versal deformation theorem

######  Theorem ([21]).

Let $\mathfrak{g}$ be an Arnold-Lie algebra, $X\in\mathfrak{g}$ and
$\mathfrak{h}\subset\mathfrak{g}$ a sub-Arnold-Lie algebra which is a direct
summand, that is, the projection on $\mathfrak{h}$ is a local operator whose
norm is an integrable Bruno sequence. Assume that

  1. 1)

there exists a a local operator $L\in{\mathcal{L}}(\mathfrak{g},{\bf M})$ and
a cutoff operator $\kappa\in{\mathcal{K}}(M,M)$ solving approximatively the
homological equation modulo $\mathfrak{h}$, in the sense that:

$[L(Y),X]=Y+\kappa(Y)\quad{\rm mod\,}\mathfrak{h}$

  2. 2)

the norm sequence $|L|,\ |\kappa|$ are integrable Bruno sequences.

Then there exists $r\in\mathbb{R}_{>0}$ and $k$ such that for any $Y\in
rt^{k}(B_{\mathfrak{g}})_{t}$ there exists
$v\in\Gamma(\mathbb{N},\mathfrak{g})$ satisfying

$e^{v}(X+rY)=\iota X\ {\rm mod\,}\mathfrak{h}$

We will not give all the details here but we will indicate which functional
space should be considered to deduce the theorem as in the next paragraph we
prove a stronger result with parameters for families of vector fields.

Appyling the abstract versal deformation theorem with
$\mathfrak{g}={\mathcal{X}}^{c}(U)$ and $\mathfrak{h}$ the
${\mathcal{O}}^{c}(U)$-module generated by $({\mathcal{X}}^{S})^{c}(U)$, we
deduce the Stolovitch normal form theorem 2.6.

###  4.4. Functional analytic setting (part II)

We will now formulate and sketch the proof of the above theorem with
additional parameters $\phi$ and $\mu$. We have to make only notational
changes in our set-up-

For the domain of $\phi$-variables, we have to throw away the appropriate
neighbourhoods of the resonant hyperplanes. We fix a decreasing sequence
$a=(a_{n})$ and $s_{0}>0$ and consider the set

$Z\longrightarrow\mathbb{N}\times]0,s_{0}]$

with fibres

$Z_{n,s}:=Z_{n,s}(\Lambda,a,s_{0}):=\\{\phi\in D_{s}^{d}:\forall k\leq
n,\sigma(\Lambda+\phi)_{k}\geq a_{k}(s_{0}-s)\\}$

This set has the Huygens property, by which we mean

$Z_{n,s}+D_{a^{*}_{n}(t-s)}\subset
Z_{n,t},\;\;\;a^{*}_{n}:=\frac{a_{n}}{2^{n}}.$

The addition of the $\mu$-variables is straightforward. We consider

$D^{l}_{s}:=\\{\mu\in\mathbb{C}^{l}\;|\;|\mu_{j}|<s,\;j=1,2,\ldots,l\\}$

and get a relative set

$W\longrightarrow\mathbb{N}\times]0,s_{0}]$

with fibre

$W_{n,s}:=D^{d}_{s}\times Z_{n,s}\times D^{p}_{s}\times
D^{l}_{s}\subset\mathbb{C}^{d}\times\mathbb{C}^{d}\times\mathbb{C}^{p}\times\mathbb{C}^{l},$

where $\iota=\iota_{n+p,n}$ is the restriction map from $E_{n}$ to $E_{n+p}$.

###  4.5. Versal deformations over a Cantor set

Applying the abstract versal deformation theorem with

$\mathfrak{g}={\mathcal{X}}^{c}(W),\mathfrak{h}=({\mathcal{X}}^{S})^{c}(W)$

and the map $L$ defined analoguous to the $L$ in 3.3, we get the following
result:

######  Theorem.

Let $\Lambda\in\mathbb{C}^{d}$ be a Bruno vector satisfying the positivity
conditions. Consider a derivation

$S_{v}=\sum_{i=1}^{d}(\lambda_{i}+\phi_{i})x_{i}\partial_{x_{i}}\in{\mathcal{X}}^{c}(W)_{0}$

of the algebra ${\mathcal{O}}^{c}(W)$. Then for any other derivation of the
form

$X_{v}:=S_{v}+\dots\in{\mathcal{X}}^{c}(W)_{0}$

there exists a morphism

$\varphi:{\mathcal{X}}^{c}(W)_{0}\longrightarrow{\mathcal{X}}^{c}(W)_{\infty}$

which has the following properties:

  1. 1)

$\varphi(X_{v})=S_{v}\quad{\rm mod\,}\ ({\mathcal{X}}^{c}(W))^{S}$.

  2. 2)

$\varphi(u_{j})=u_{j}$ for $j=1,\dots,k$.

  3. 3)

$\varphi(\mu_{j})=\mu_{j}$ for $j=1,\dots,l$.

  4. 4)

$\varphi(\phi_{j})=\phi_{j}+\dots\in{\mathcal{O}}^{c}(D^{d+p})$

As a corollary we get that the functions defining the Bruno-Stolovitch ideal
lie in ${\mathcal{O}}^{c}(W)_{\infty}$, in particular they define a continuous
family of invariant manifolds. Assuming non-degeneracy conditions on the
frequency (for instance that the map is a submersion), these are parametrised
by a positive measure set. For the particular case of the multi-dimensional
Hopf bifurcation, we get a family of tori carrying quasi-periodic motions
generalising the theorem of Chenciner-Li [16, 26]. Stolovitch KAM theorem [45]
can also be deduced from the convergence of the Lie iteration on this setting.

## References

  * [1] A. Andronov.  Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues.  Comptes Rendus de l’Académie des Sciences de Paris, 189:559–561, 1929.
  * [2] V. I. Arnold.  Remarks on singularities of finite codimension in complex dynamical systems.  Functional Analysis and its Applications, 3(1):1–5, 1969.
  * [3] V.I. Arnold.  Lectures on bifurcations in versal families.  In V.I. Arnold Collected Works, pages 271–340. Springer, 1972.
  * [4] V.I. Arnold.  Chapitres supplémentaires de la théorie des équations différentielles ordinaires.  MIR, 1980.
  * [5] V.I. Arnold, A.N. Varchenko, and S. Goussein-Zade.  Singularity of differentiable mapping, Vol. II.  Nauka:Moscow, 1982.  English transl.: Birkhauser, 382p., Basel(1986).
  * [6] V.I. Arnold, V.A. Vassiliev, V.V. Goryunov, and O.V. Lyashko.  Singularity Theory I, Dynamical Systems VI.  Springer-Verlag, 1988.  English transl., 245p. (1993).
  * [7] V.I. Arnold, V.A. Vassiliev, V.V. Goryunov, and O.V. Lyashko.  Singularity Theory II, Dynamical Systems VIII.  VINITI, Moscow, 1989.  English transl.:Springer-Verlag, 235p., (1993).
  * [8] C. Briot and T. Bouquet.  Recherches sur les propriétés des fonctions définies par des équations différentielles.  J. École Polytech., 21:133–197, 1856.
  * [9] H.W. Broer, G.B. Huitema, and M.B. Sevryuk.  Families of quasi-periodic motions in dynamical systems depending on parameters.  In Nonlinear Dynamical Systems and Chaos, pages 171–211. Springer, 1996.
  * [10] H.W. Broer, G.B. Huitema, and M.B. Sevryuk.  Quasi-periodic motions in families of dynamical systems: order amidst chaos.  Springer, 2009.
  * [11] A. D. Bruno.  Normal form of differential equations with a small parameter.  Matematicheskie Zametki, 16(3):407–414, 1974.
  * [12] A. D. Bruno.  Integral analytic sets.  Dokl. Akad. Nauk SSSR, 220(6):1255–1258, 1975.  (in Russian).
  * [13] A.D. Bruno.  The normal form of differential equations.  Dokl. Akad. Nauk SSSR, 157(6):1276–1279, 1964.  (In Russian).
  * [14] A.D. Bruno.  Analytic form of differential equations I.  Trans. Moscow Math. Soc., 25:131–288, 1971.
  * [15] C. Camacho and P. Sad.  Invariant varieties through singularities of holomorphic vector fields.  Annals of mathematics, 115(3):579–595, 1982.
  * [16] A. Chenciner.  Bifurcations de points fixes elliptiques.  Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 61(1):67–127, 1985.
  * [17] H. Dulac.  Solutions d’un système d’équations différentielles dans le voisinage de valeurs singulières.  Bulletin de la Société mathématique de France, 40:324–383, 1912.
  * [18] J. Écalle.  Singularités non abordables par la géométrie.  Annales de l’institut Fourier, 42(1-2):73–164, 1992.
  * [19] M. Garay.  Degenerations of invariant Lagrangian manifolds.  Journal of Singularities, 8:50–67, 2014.
  * [20] M. Garay and D. van Straten.  Hamiltonian normal forms.  ArXiv:1909.06053, 2019.
  * [21] M. Garay and D. van Straten.  A category of Banach space functors.  ArXiv: 2010.02320, 2020.
  * [22] E. Hopf.  Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems.  Berichten der Mathematisch-Physischen Klasse der Sächsischen Akademie der Wissenschaften zu Leipzig, 94:1–22, 1942.
  * [23] Y. Ilyashenko.  Divergence of series reducing an analytic differential equation to linear normal form at a singular point.  Functional Analysis and its applications, 13(3):227–229, 1979.
  * [24] Y. Ilyashenko and S Yakovenko.  Lectures on Analytic Differential Equations, volume 86 of Graduate Studies in Mathematics.  American Mathematical Society, 2008.
  * [25] A.N. Kolmogorov.  On the conservation of quasi-periodic motions for a small perturbation of the Hamiltonian function.  Dokl. Akad. Nauk SSSR, 98:527–530, 1954.  (In Russian).
  * [26] X. Li.  On the persistence of quasi-periodic invariant tori for double Hopf bifurcation of vector fields.  Journal of Differential Equations, 260(10):7320–7357, 2016.
  * [27] Bernard Malgrange et al.  Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques.  Astérisque, 92(93):82, 1981.
  * [28] J.E. Marsden and M. McCracken.  The Hopf bifurcation and its applications, volume 19.  Springer Science & Business Media, 2012.
  * [29] J. Martinet.  Normalisation des champs de vecteurs holomorphes (d’après A.-D. Brjuno), volume 770 of Sem. Bourbaki, Lecture Notes in Mathematics.  Springer, 1981.  272 pp.
  * [30] J. Martinet.  Singularities of smooth functions and maps, volume 58 of Lecture Notes Series.  Cambridge University Press,, 1982.  272 pp.
  * [31] J. Martinet and J.-P. Ramis.  Problèmes de modules pour des équations différentielles non linéaires du premier ordre.  Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 55(1):63–164, 1982.
  * [32] J. Martinet and J.-P. Ramis.  Classification analytique des équations différentielles non linéaires résonnantes du premier ordre.  Annales scientifiques de l’École Normale Supérieure, 16(4):571–621, 1983.
  * [33] R. Moussu and D. Cerveau.  Groupes d’automorphismes de $(\mathbb{C},0)$ et équations différentielles $ydy+\cdots=0$.  Bulletin de la Société Mathématique de France, 116(4):459–488, 1988.
  * [34] R. Pérez-Marco.  Total Convergence or General Divergence in Small Divisors.  Communications in Mathematical Physics, 223(3):451–464, 2001.
  * [35] R. Pérez-Marco.  Convergence or generic divergence of the Birkhoff normal form.  Annals of Mathematics, 2:557–574, 2003.
  * [36] E. Picard.  Traité d’analyse, Tome 3. Gauthier-Villars, 3 edition, 1928.  660 pp.
  * [37] H. Poincaré.  Sur le problème des trois corps et les équations de la dynamique.  Acta mathematica, 13(1):3–270, 1890.
  * [38] H. Poincaré.  Sur les propriétés des fonctions définies par les équations aux différences partielles (Première thèse, 1879).  Oeuvres de Henri Poincaré, Tome I. Gauthiers-Villars, 1951.
  * [39] A.S. Pyartli.  Birth of complex invariant manifolds close to a singular point of a parametrically dependent vector field.  Functional Analysis and Its Applications, 6(4):339–340, 1972.
  * [40] H. Rüssmann.  Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung.  Math. Annalen, 154:285–306, 1964.
  * [41] C.L. Siegel.  On the integrals of canonical systems.  Annals of Mathematics, 42(3):806–822, 1941.
  * [42] C.L. Siegel.  Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung.  Nach. Akad. Wiss. Göttingen, math.-phys., pages 21–30, 1952.
  * [43] S. Sternberg.  On the structure of local homeomorphisms of Euclidean n-space, II.  American Journal of Mathematics, 80(3):623–631, 1958.
  * [44] L. Stolovitch.  Sur un théorème de Dulac.  Annales de l’institut Fourier, 44(5):1397–1433, 1994.
  * [45] L. Stolovitch.  A KAM phenomenon for singular holomorphic vector fields.  Publications Mathématiques de l’Institut des Hautes Études Scientifiques., 102:99–165, 2005.
  * [46] L. Stolovitch.  Progress in normal form theory.  Nonlinearity, 22(7):R77, 2009.
  * [47] L. Stolovitch.  Smooth Gevrey normal forms of vector fields near a fixed point.  Annales de l’Institut Fourier, 63(1):241–267, 2013.