[1]Tosin P. Adewumi

# Inner For-Loop for Speeding Up Blockchain Mining

Marcus Liwicki  Luleå University of Technology, E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, the authors propose to increase the efficiency of blockchain
mining by using a population-based approach. Blockchain relies on solving
difficult mathematical problems as proof-of-work within a network before
blocks are added to the chain. Brute force approach, advocated by some as the
fastest algorithm for solving partial hash collisions and implemented in
Bitcoin blockchain, implies exhaustive, sequential search. It involves
incrementing the nonce (number) of the header by one, then taking a double
SHA-256 hash at each instance and comparing it with a target value to
ascertain if lower than that target. It excessively consumes both time and
power. In this paper, the authors, therefore, suggest using an inner for-loop
for the population-based approach. Comparison shows that it's a slightly
faster approach than brute force, with an average speed advantage of about
1.67% or 3,420 iterations per second and 73% of the time performing better.
Also, we observed that the more the total particles deployed, the better the
performance until a pivotal point. Furthermore, a recommendation on taming the
excessive use of power by networks, like Bitcoin's, by using penalty by
consensus is suggested.

##  1 Introduction

Blockchain, as a digital distributed ledger, implies it’s an electronic,
decentralized permanent storage of transactions [1]. Its first successful
implementation was as the backbone of a peer to peer network for Bitcoin by
Nakamoto [1]. It has no requirement for a central server. Instead, it's
designed to operate within a network on a consensus basis [2]. Blockchain is
one of the latest and, arguably, the most promising technology for virtual
money, given the other previous attempts that have failed [2]. It has
potential usage in many fields and industries beyond money-related instances,
for example, in smart contracts, smart properties, legal documents, health
records, marriage certificates and a number of others [3].

The research question being addressed here is 'How can a population-based
algorithm be applied to speed up blockchain mining, if at all it can?' This is
because it has been established in other literature that optimization
algorithms or population-based searches work well for certain problems and
poorly for others [4, 5]. Lehre and Witt [6] showed that standard or basic
particle swarm optimization (PSO) with one or two particles stagnates even on
one-dimensional sphere. Population-based approaches use multiple solutions at
the same time [7], such as with genetic algorithms [8, 7] or PSO [9]. Brute
force is the widespread implementation, possibly the only approach, to mining
blockchain [2, 1]. This research uses the method of comparative study to
critically analyze results from several tests of the implemented population-
based algorithm and sequential brute force algorithm. The population-based
algorithm is a simple case of inner for-loop (IFL) program construct, i.e., a
nested for-loop within another loop. Brute force approach usually uses one
loop to iterate over all values and is the most intuitive approach to solving
a problem but has very low efficiency in some cases [10].

The main contribution of this research is the implementation of the inner for-
loop algorithm, demonstrated by Adewumi [11], to mine blockchain faster. The
relevance of this to power/energy consumption of the network machines will be
mentioned briefly in the discussion section of this paper. In addition, a
recommended total number of particles for the population is determined. The
search algorithm may be relevant and applicable to other search scenarios that
are reliant on brute force, for example, cryptography. The scope of this study
will include inner for-loop construct, blockchain network and SHA-256 hash.
This research does not address application-specific integrated circuit-
resistance (ASIC-resistance). The sections that follow include a brief
literature review, methodology, results, discussion and conclusion.

##  2 Literature Review

According to Narayanan et al [2], the concept of blockchain can be traced back
years ago to Haber and Stornetta (1991), who proposed secure timestamping for
digital documents. An anonymous individual or group, using the name Satoshi
Nakamoto, published 'Bitcoin: A Peer-to-Peer Electronic Cash System' in 2008
[1, 3]. Nakamoto [1] set out to solve the trust issue with regards to
electronic transactions through the creation of blockchain, the decentralized,
technological foundation of Bitcoin cryptocurrency, consequently eliminating
third-party financial institutions. In the network, a new transaction is
broadcast to all miners (nodes or machines) and each collects it and begins to
find the mathematical solution (called proof-of-work), by hashing values so
that the result begins with a number of zero bits. This is such that it falls
below the periodically reviewed target set by the network system. The proof-
of-work (PoW) is verifiable by executing a single hash. [1]

More specifically, sequential, brute force search is carried out by
incrementing the nonce (number) of a block header by one, then taking a double
SHA-256 hash at each instance and comparing it with a system-determined target
value to ascertain if lower than that target [12, 1]. Each header is made up
of a previous hash, merkle root, version, timestamp, bits (indicating the
difficulty level or target) and a nonce [1, 2]. The node that finds the
solution creates a new block based on it and broadcasts the new block to all
other nodes, who then verify if the proof-of-work is correct and the value has
not been spent. The nodes then add the new block to their chain
chronologically and prepare to create a new block by using the hash of the
accepted block as the previous block [1]. This difficult proof-of-work, which
consumes time and excessive power, has led to the production of special-
purpose processors called application-specific integrated circuits (ASICs) for
blockchain mining [2]. However, Bitcoin's network works by adjusting the
difficulty of the mathematical PoW so that it takes an average of ten minutes
to solve one [2]. Hence, the difficulty in its network has been growing
steadily over the years because of the powerful ASICs added to, or used to
replace older nodes. [2].

Adam Back's [12] hashcash proof-of-work model was used in the implementation
of a distributed timestamp on a peer-to-peer basis in Bitcoin [1]. The longest
chain in the network, always considered the correct one, will have the
greatest proof-of-work invested and represents the majority decision. If the
PoW is generated too fast, the difficulty increases and this is usually done
every two weeks, based on the average ten minutes the network expects to find
a solution each time [1, 2]. An attacker will have to do the proof-of-work of
a past chain and all previous chains to the point of the genesis block to be
able to modify a past block, which is near unrealistic. It has been asserted
that the fastest algorithm for computing partial hash collisions, upon which
hashcash cost-function is based, is brute force [12].

The nonce, timestamp and bits fields are 32 bits long. In generating a valid
nonce that solves the mathematical header problem, after all possible values
have been exhausted in the 32-bit nonce and no valid hash found, the value in
the coinbase can be modified, thereby triggering changes in the merkle tree,
and the process of iterating through possible values of nonce repeated. Other
fields that can be modified in the header to repeat the iteration through the
nonce when no valid hash is found are the timestamp and merkle root [2]. The
timestamp is registered in epoch time, that is, the time in seconds from 1970
UTC. Mining difficulty changes every 2,016 blocks, as a step function, which
translates to about every two weeks.

Secure Hash Algorithm-256 (SHA-256) [13] is a keyless, cryptographic hash
function with fixed output of 256 bits created by the National Institute of
Standards and Technology (NIST) in 2002 alongside two other versions, based on
previous standards [14]. Hash functions take in input string of any size and
efficiently compute output string of fixed size called digest [13, 2].
Collision, which implies two different inputs having the same hashed output,
is actually possible in hash functions, especially considering the birthday
paradox in probability [2]. This is, however, not practical to compute because
of the astronomically long time it will take computers to do so [2, 14]. There
are possible attacks on cryptographic hash functions, like pre-image attack or
collision attack, however, there hasn’t been any reported successful attack on
SHA-256 [14]. The ever-present risk of possible attacks with previous
cryptographic functions led NIST to organize competitions that led to the
creation of the SHA-3 family of functions [15]. Compared to SHA-3, however,
SHA-256 is still more popular. This is because support for SHA-1 has been
discontinued, SHA-3 is relatively new and SHA-256 has been the secure platform
for internet and network communication [13, 15].

In order to address the huge advantage ASICs have over regular computers, some
researchers have attempted new PoW algorithms that introduce ASIC-resistance,
like Scrypt [16], Ethash [17, 18] and Equihash [19]. These algorithms
generally, though unsuccessfully over time [19, 16], try to place restrictions
on the resources of ASICs by being memory-hard in some cases. Bitmain, an ASIC
manufacturer, overturned the objective of such algorithms by producing ASICs
optimized over such algorithms [16]. Equihash family of proofs of work also
uses parts of the algorithm employed in Bitcoin, particularly a fixed-time
difficulty filter requiring a nonce and leading zeros in hash. This is besides
additional, modified Wagner's algorithm. Equihash is not claimed to perform
faster than brute force, with regards to Bitcoin. Indeed, Biryukov and
Khovratovich, who introduced Equihash, pointed out that the best algorithms
for NP problems 'run in exponential time' [19], as do Wagner's algorithm and
brute force.

##  3 Methodology

This research adopted a comparative study method in order to evaluate brute
force algorithm and the newly implemented inner for-loop algorithm, using
various number of total particles in order to ascertain the preferred total
for the system. Although power analysis to determine the sample size to use
from the blockchain population in order to compare two means showed that a
higher sample size is desirable for greater power for small effect size, 120
was tested. The basic data were obtained by secondary means from the website
that keeps such records (www.blockchain.com). These include values of block
header fields of certain blocks in the Bitcoin blockchain. Simple random
sampling was used to pick all blocks by generating the blocks to be picked
using a simple Python program that randomly picked from 1 to 557,132 blocks
available as at January 2019 [20].

The programs were written in Python (version 3.7) programming language, using
an HP desktop having Intel Core i5-2500 at 3.30Ghz clock speed, 8GB RAM and
64bit Windows 10 operating system. Python, apparently, was more suitable as a
programming language because of its int datatype’s unrestricted size [21],
which is required when dealing with the huge numbers blockchain operates with.
For IFL, the maximum iteration value for the outer loop is determined by
capping it as the result of dividing the highest possible value of 2 to power
32 (since this is the highest possible value for the nonce field) by the
particle total. If this maximum iteration value is left simply as 2 to power
32 as with the brute force program, then the efficiency of the program will be
lost in cases when a hash solution or valid nonce is not found and adjustment
needs to be made to the header before restarting the search process. The code
is provided in the appendix for scrutiny and easy replication of this study.

Since memory access is slower for variables than having constants, the result
of the earlier cap can be, and was imputed, as a constant value for the outer
while loop for each particle total. The number of hash attempts for brute
force cases are the same for the IFL approach. It was ensured no other
unrelated application was open and running while each program executed. These
precautions minimized threats to internal validity so that things were
measured right. Parallel programming or multiple threads were not employed in
the codes, neither was machine-dependent optimization used.

The IFL program was run for each of the solved Bitcoin blockchain headers
using several particle totals of 2, 6, 20, 100, 200 and 1000. Due to the
extensive length of time running each test from nonce iteration of 0 to the
end will take, each was given a cap of 30,000,000 iterations and the execution
time recorded. Two runs were carried out per particle total per block and
average time calculated. The iterations per second was then calculated from
that, which was used to obtain time to valid nonce for each block. This
resulted in a total of 1,680 runs, including for brute force. Three blocks had
nonce within the 30,000,000 iterations and were correctly returned during the
tests.

##  4 Results

After 1,680 experimental runs, results were obtained and tabulated. It was
observed that the 200-Particle total is the best of the approaches tried. It
had the lowest time minimization though the speed advantage over brute force
was not very much. The 100-Particle IFL case is also better than brute force,
though lesser than its 200-Particle counterpart. The average speed improvement
of IFL (with 200 particles) over BF is 1.67% (about 3,420 iterations per
second) and 73% of the test instances show IFL performing better. Performance
from 2-Particle total starts out poorly, but as the number of particles
increases, the speed improves and time minimization gets better until after
200 particles. Table 1 gives a summary of the comparison of the key indicators
of the various approaches and this is expressed pictorially in Figure 1, while
Figure 2 shows how often brute force and 200-IFL performed better than each
other. Our null hypothesis that there is no difference between the means of
brute force and 200-Particle IFL is tested with statistical analysis of two
sample t-test. It shows that the computed t had a value of 5.953 against the
critical value of 2.576 (given alpha is 0.01), hence, we reject the null
hypothesis as there is a significant difference between the two means.

Tab. 1: Comparison of Approaches Approach | Mean iterations/s | Ratio to BF
---|---|---
Brute Force (BF) | 204,970 | 1.0000
2-Particle IFL | 192,198 | 0.9377
6-Particle IFL | 200,060 | 0.9760
20-Particle IFL | 203,708 | 0.9938
100-Particle IFL | 205,074 | 1.0005
200-Particle IFL | 208,393 | 1.0167
1000-Particle IFL | 204,932 | 0.9998
Fig. 1: Speed Ratio Comparison of Approaches Fig. 2: Better Performance Chart

##  5 Discussion

The results suggest the new algorithm utilized the machine-independent
optimization noted with IFL [11]. Further work might show that this advantage
may also be noticeable on a multiplied scale when parallel programming and
multiple threads are employed.

Given the fact that Bitcoin network's self-adjusting average mining time of 10
minutes has not stopped enthusiasts from securing more powerful machines to
mine even faster for the reward it offers [2, 3], then utilizing a machine-
independent algorithm, such as IFL, can also give an advantage. This is with
the knowledge that the system will still adjust its difficulty in the long
run. Besides, it would save some energy if it takes lesser time to solve the
proof-of-work. It would, therefore, be desirable to have a faster approach.
One solution to the ever-growing demand for more power in a blockchain
network, such as Bitcoin's, is to introduce penalty by consensus into such a
network. This concept of penalty by consensus, independent of IFL, can be used
to disqualify nodes which go above a dynamic power threshold set by the
consensus of the network. A future work can compare this new concept to ASIC-
resistant algorithms mentioned earlier and energy measurements evaluated.

##  6 Conclusion

This research work used the method of comparative study to critically analyze
results from several scenarios of the newly implemented inner for-loop program
and the widespread brute force algorithm on blockchain mining. From the
results obtained and analysis carried out, we can observe that this
population-based algorithm (implemented as a simple inner for-loop) can
successfully be applied to speed up blockchain mining. Furthermore, we
observed that fewer particles gave poor result but the more particles were
deployed in a population, the better the general performance until a pivotal
point. A recommended total number of particles for the population was
determined as 200 out of the several tried. Further work can be carried out
with a larger sample size. Other particle totals beyond 1000 may be
experimented with to find out if more optimal solutions exist. The inner for-
loop algorithm may also be tested on parallel processors. Different
programming languages may also be implemented to compare with the python
implementation for further external validity. Another implication of this
research worth investigating is the security of current cryptography
algorithms popularly implemented in brute force.

## References

  * [1] Satoshi Nakamoto.  Bitcoin: A peer-to-peer electronic cash system.  2008\.
  * [2] Arvind Narayanan, Joseph Bonneau, Edward Felten, Andrew Miller, and Steven Goldfeder.  Bitcoin and cryptocurrency technologies: A comprehensive introduction.  Princeton University Press, 2016.
  * [3] Michael Crosby, Pradan Pattanayak, Sanjeev Verma, Vignesh Kalyanaraman, et al.  Blockchain technology: Beyond bitcoin.  Applied Innovation, 2(6-10):71, 2016.
  * [4] Maurice Clerc.  From theory to practice in particle swarm optimization.  In Handbook of Swarm Intelligence, pages 3–36. Springer, 2011.
  * [5] Matthew Settles.  An introduction to particle swarm optimization.  Department of Computer Science, University of Idaho, 2, 2005.
  * [6] Per Kristian Lehre and Carsten Witt.  Finite first hitting time versus stochastic convergence in particle swarm optimisation.  In Advances in Metaheuristics, pages 1–20. Springer, 2013.
  * [7] Zahra Beheshti and Siti Mariyam Shamsuddin.  A review of population-based meta-heuristic algorithm.  International Journal of Advances in Soft Computing and Its Applications, 5:1–35, 03 2013.
  * [8] John H Holland.  Genetic algorithms.  Scientific american, 267(1):66–73, 1992.
  * [9] Russell Eberhart and James Kennedy.  A new optimizer using particle swarm theory.  In MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pages 39–43. Ieee, 1995.
  * [10] HL Willis.  Power distribution planning reference book, new york: M, 2004.
  * [11] Tosin P Adewumi.  Inner loop program construct: A faster way for program execution.  Open Computer Science, 8(1):115–122, 2018.
  * [12] Adam Back et al.  Hashcash-a denial of service counter-measure.  2002\.
  * [13] FIPS PUB.  Secure hash standard (shs).  FIPS PUB, 180(4), 2012.
  * [14] Rajeev Sobti and G Geetha.  Cryptographic hash functions: a review.  International Journal of Computer Science Issues (IJCSI), 9(2):461, 2012.
  * [15] Morris J Dworkin.  Sha-3 standard: Permutation-based hash and extendable-output functions.  Technical report, 2015.
  * [16] Mayank Raikwar, Danilo Gligoroski, and Katina Kralevska.  Sok of used cryptography in blockchain.  arXiv preprint arXiv:1906.08609, 2019.
  * [17] Gavin Wood et al.  Ethereum: A secure decentralised generalised transaction ledger.  Ethereum project yellow paper, 151(2014):1–32, 2014.
  * [18] Ainur R Zamanov, Vladimir A Erokhin, and Pavel S Fedotov.  Asic-resistant hash functions.  In 2018 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), pages 394–396. IEEE, 2018\.
  * [19] Alex Biryukov and Dmitry Khovratovich.  Equihash: Asymmetric proof-of-work based on the generalized birthday problem.  Ledger, 2:1–30, 2017.
  * [20] Leonard J Kazmier.  Theory and Problems of Business Statistics.  McGraw-Hill, 2004.
  * [21] H Swaroop.  A Byte of Python.  Independent, 2013.

## Appendix

Listing 1: Inner For Loop Example with 200 Particles

⬇

import time

import math

from hashlib import sha256

from struct import pack

from binascii import hexlify

from codecs import decode

version = pack("<L", 0x01)

previous_header_hash = decode("000000000009852382d88a5442c35c60214

0bf08a6c40f9d9475326572032ecf", ’hex’)[::-1]

merkle_root = decode("34d96742a37b46acd0d52a19be1bc0a2c4

50b3c94595117ce82dd03b39930570", ’hex’)[::-1]

date = pack("<L", 1290624043)

nbits = pack("<L", 453610282)

nbits_calc = hexlify(nbits[::-1])

base = 256

exponent = int(nbits_calc[0:2], 16) - 3

significand = int(nbits_calc[2:8], 16)

target = significand * (base ** exponent)

def spin(num_particles):

maximumiter = math.ceil(30000000/num_particles)

i = 0

nonce = -1

while i < 150000:

for j in range(0, num_particles):

nonce += 1

header_hash = sha256(sha256(version + previous_header_hash + merkle_root +
date + nbits + pack("<L", nonce)).digest()).digest()

if int(hexlify(header_hash[::-1]), 16) < target:

print("Nonce␣and␣Header␣Hash", nonce, hexlify(header_hash[::-1]))

return

i+=1

return

if __name__ == "__main__":

starttime = time.time()

spin(num_particles=200)

print("Time␣", time.time() - starttime)


#  Robust chaos in orientation-reversing and non-invertible two-dimensional
piecewise-linear maps.

I. Ghosh  School of Mathematical and Computational Sciences
Massey University
Colombo Road, Palmerston North, 4410
New Zealand  R. McLachlan  School of Mathematical and Computational Sciences
Massey University
Colombo Road, Palmerston North, 4410
New Zealand  D. Simpson  School of Mathematical and Computational Sciences
Massey University
Colombo Road, Palmerston North, 4410
New Zealand

###### Abstract

This paper concerns the two-dimensional border-collision normal form — a four-
parameter family of piecewise-linear maps generalising the Lozi family and
relevant to diverse applications. The normal form was recently shown to
exhibit a chaotic attractor throughout an open region of parameter space. This
was achieved by constructing a trapping region in phase space and an invariant
expanding cone in tangent space, but only allowed parameter combinations for
which the normal form is invertible and orientation-preserving. This paper
generalises the construction to include the non-invertible and orientation-
reversing cases. This provides a more complete and unified picture of robust
chaos by revealing its presence to be disassociated from the global
topological properties of the map. We identify a region of parameter space in
which the map exhibits robust chaos, and show that part of the boundary of
this region consists of bifurcation points at which the chaotic attractor is
destroyed.

##  1 Introduction

Robust chaos refers to the persistence of a chaotic attractor throughout open
regions of the parameter space of a family of dynamical systems [13]. Such
robustness is essential for chaos-based cryptography [20] and can be helpful
to devices that have advantageous functional characteristics when operated in
a chaotic regime, examples include power converters [6], optical resonators
[24], and energy harvesters [23].

Robust chaos occurs in diverse settings [17, 46]. Its occurrence in piecewise-
linear maps was popularised by Banerjee et al. [4], and later Glendinning and
Simpson [15] proved the presence of robust chaos in their setting. They
provided explicit parameter values yielding robust chaos, compared to earlier
results giving implicit conditions in more abstract settings [30, 34, 45], but
only dealt with two-dimensional, orientation-preserving maps. Piecewise-linear
maps that are not orientation-preserving are arguably more physically relevant
than those that are orientation-preserving, as discussed below. The purpose of
this paper is to extend the results of [15] to the non-orientation-preserving
cases. This is also a necessary first step towards generalising the results to
higher dimensions.

Specifically, we study the family

$f_{\xi}(x,y)=\begin{cases}\big{(}\tau_{L}x+y+1,-\delta_{L}x\big{)},&x\leq
0,\\\ \big{(}\tau_{R}x+y+1,-\delta_{R}x\big{)},&x\geq 0,\end{cases}$ (1.1)

known as the two-dimensional border-collision normal form. For any fixed
parameter point
$\xi=(\tau_{L},\delta_{L},\tau_{R},\delta_{R})\in\mathbb{R}^{4}$, we are
interested in the behaviour of iterations $(x,y)\mapsto f_{\xi}(x,y)$ on
$\mathbb{R}^{2}$. The family (1.1) is continuous but non-differentiable on the
line $x=0$, termed the switching manifold, and is a normal form in the sense
that any two-dimensional, piecewise-linear, continuous map with a single
switching manifold and satisfying a genericity condition can be transformed to
(1.1) through a change of coordinates [40, Appendix A]. It was originally
formulated by Nusse and Yorke [28] with an additional parameter $\mu$ that
when varied through zero brings about a border-collision bifurcation [2]. The
simplicity of (1.1) belies the incredible complexity of its dynamics; it can
exhibit two-dimensional attractors [12, 14], large numbers of coexisting
attractors [31, 35, 39], and Arnold tongues with a unique sausage-string
geometry [38]. In the special case $\tau_{L}=-\tau_{R}$ and
$\delta_{L}=\delta_{R}$, (1.1) reduces to the Lozi family [25].

At any point with $x\neq 0$ the derivative of (1.1) has determinant

$\det\big{(}({\rm D}f_{\xi})(x,y)\big{)}=\begin{cases}\delta_{L},&x<0,\\\
\delta_{R},&x>0.\end{cases}$

Consequently we have the following cases.

  * •

If $\delta_{L}>0$ and $\delta_{R}>0$ then $f_{\xi}$ is orientation-preserving.
This is the most natural case to consider from an applied perspective because
Poincaré maps derived from time-reversible flows on $\mathbb{R}^{n}$ are
necessarily orientation-preserving [44].

  * •

If $\delta_{L}<0$ and $\delta_{R}<0$ then $f_{\xi}$ is orientation-reversing.
This case was considered in the seminal work of Misiurewicz [27] for the Lozi
map. Such maps can be embedded into higher dimensional orientation-preserving
maps and in this way help us understand high-dimensional chaos in physical
systems [19].

  * •

If $\delta_{L}$ and $\delta_{R}$ have opposite signs then $f_{\xi}$ is non-
invertible with the left and right half-planes mapping onto either the top or
the bottom half-plane. Such maps, except piecewise-smooth instead of
piecewise-linear, apply to a wide range of power converters [3, 9, 47]. For
instance, the boost converter described by Banerjee et al. [4] regulates
output voltage via a switch that is activated whenever the current reaches a
threshold value. This causes the flow (in phase space) to fold back over
itself resulting in a stroboscopic map that is piecewise-smooth (due to the
switch) and non-invertible (due to the folding) [5, 10].

  * •

If one of $\delta_{L}$ and $\delta_{R}$ is zero then $f_{\xi}$ is non-
invertible with one half-plane mapping onto a line [21]. Such maps arise as
leading-order approximations to grazing-sliding bifurcations in relay control
systems [7] and mechanical systems with stick-slip friction [8, 42]. This is
because trajectories of the differential equations become constrained to a
codimension-one discontinuity surface causing the range of one piece of the
Poincaré map to have one less dimension than the domain of the map [7]. This
also occurs for the Hickian trade cycle model of Puu et al. [32] and the
influenza outbreak model of Roberts et al. [33].

Let

$\Phi=\mathopen{}\mathclose{{}\left\\{\xi\in\mathbb{R}^{4}\,\middle|\,\tau_{L}>|\delta_{L}+1|,\,\tau_{R}<|\delta_{R}+1|}\right\\}$
(1.2)

denote the set of all $\xi$ for which $f_{\xi}$ has two saddle fixed points
(Lemma 2.1). Banerjee et al. [4] considered $\xi\in\Phi$ in the orientation-
preserving case and showed that on one side of a homoclinic bifurcation the
stable manifold of one fixed point has transverse intersections with the
unstable manifolds of both fixed points and argued that this implies the
existence of a chaotic attractor. Their conclusion was verified rigorously by
Glendinning and Simpson [15] using the methodology of Misiurewicz [27]. First,
a forward invariant region $\Omega\subset\mathbb{R}^{2}$ was identified by
using one of the unstable manifolds, and this was perturbed into a trapping
region that necessarily contains a topological attractor. Second, an invariant
expanding cone $\Psi\subset T\mathbb{R}^{2}$ (see §4) was constructed by using
eigenvectors of the two pieces of ${\rm D}f_{\xi}$. The existence of this
object implies that nearby forward orbits diverge exponentially and that the
attractor is chaotic in the sense of having a positive Lyapunov exponent. In
general this approach works brilliantly for piecewise-linear maps because the
invariance and expansion properties can be verified by simple, explicit
computations. For more complicated constructions the verification is best done
on a computer [16]. The construction is robust to nonlinear perturbations to
the pieces of the map and has been used to show that chaos persists for
intervals of parameter values beyond border-collision bifurcations [41].
Recently this approach has also been applied to piecewise-smooth maps with a
square-root singularity that arise for mechanical systems with hard impacts
[26]. Further techniques can reveal more properties of the attractor, such as
sensitive dependence on initial conditions [11], continuity with respect to
$\xi$ [1, 22], and the presence of an SRB measure [13].

In this paper, we extend the construction of [15] to the orientation-reversing
and non-invertible cases. We obtain a subset $\Phi_{\rm trap}\subset\Phi$ in
which $f_{\xi}$ has a trapping region and another subset $\Phi_{\rm
cone}\subset\Phi$ in which $f_{\xi}$ has an invariant expanding cone. It
follows that $f_{\xi}$ has a chaotic attractor for all $\xi\in\Phi_{\rm
trap}\cap\Phi_{\rm cone}$. This is an open subset of parameter space, hence
the chaos is robust with respect to the family $f_{\xi}$. We expect it is also
robust to nonlinear perturbations to the pieces of the map, as in [41].
Boundaries of $\Phi_{\rm trap}\cap\Phi_{\rm cone}$ are where some aspect of
the construction fails. As shown below, three of these boundaries correspond
to bifurcations where the chaotic attractor is destroyed. Beyond the other
boundaries the chaotic attractor appears to persist and we believe robust
chaos could be verified on a larger subset of parameter space by using a more
complicated construction, e.g. [40], but our aim here is not to optimise the
subset, only to obtain a reasonably large subset for both negative and
positive values of $\delta_{L}$ and $\delta_{R}$ by using a construction that
is both natural and simple.

The remainder of this paper is organised as follows. We start in §2 by
calculating the stable and unstable manifolds of the fixed points. Constraints
on the geometry of the manifolds as they are extended outwards from the fixed
points give rise to our definition of $\Phi_{\rm trap}$. Here we also define
$\Phi_{\rm cone}$ and state our main result (Theorem 2.2) for the existence of
a chaotic attractor. Being four-dimensional the sets $\Phi_{\rm trap}$ and
$\Phi_{\rm cone}$ are difficult to visualise; we show a range of two-
dimensional cross-sections to give some impressions of their size and shape.

In §3 we use the stable manifold of one of the fixed points to form a region
$\Omega\subset\mathbb{R}^{2}$ and show it is forward invariant under $f_{\xi}$
for any $\xi\in\Phi_{\rm trap}$. We then show how $\Omega$ can be perturbed
into a trapping region for any $\xi\in\Phi_{\rm trap}$. In §4 we identify a
cone that is both invariant and expanding for any $\xi\in\Phi_{\rm cone}$ and
prove Theorem 2.2. In §5 we further study the extent to which $\Phi_{\rm
trap}$ and $\Phi_{\rm cone}$ cover parameter space. We show a typical cross-
section of $\Phi_{\rm trap}\cap\Phi_{\rm cone}$ and overlay numerical
simulations whereby the presence robust chaos is estimated from forward
orbits. This helps clarify which boundaries of $\Phi_{\rm trap}$ and
$\Phi_{\rm cone}$ correspond to bifurcations and how much of the true robust
chaos region is detected by the straight-forward constructions. Final remarks
are provided in §6.

##  2 Sufficient conditions for a chaotic attractor

For any $\xi\in\Phi$ the map $f_{\xi}$ has fixed points

$\displaystyle X$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{-1}{\tau_{R}-\delta_{R}-1},\frac{\delta_{R}}{\tau_{R}-\delta_{R}-1}}\right),$
(2.1) $\displaystyle Y$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{-1}{\tau_{L}-\delta_{L}-1},\frac{\delta_{L}}{\tau_{L}-\delta_{L}-1}}\right),$
(2.2)

where $X$ is in the right half-plane and $Y$ is in the left half-plane. The
stability multipliers associated with $X$ and $Y$ are the eigenvalues of the
Jacobian matrices

$\displaystyle A_{L}$ $\displaystyle=\begin{bmatrix}\tau_{L}&1\\\
-\delta_{L}&0\end{bmatrix},$ $\displaystyle A_{R}$
$\displaystyle=\begin{bmatrix}\tau_{R}&1\\\ -\delta_{R}&0\end{bmatrix}.$ (2.3)

Since $\xi\in\Phi$, $A_{L}$ has eigenvalues $\lambda_{L}^{s}\in(-1,1)$ and
$\lambda_{L}^{u}>1$, while $A_{R}$ has eigenvalues $\lambda_{R}^{s}\in(-1,1)$
and $\lambda_{R}^{u}<-1$. This implies $X$ and $Y$ are saddles. In fact,
$\Phi$ is the set of all parameter combinations for which $f_{\xi}$ has two
saddle fixed points:

######  Lemma 2.1.

The map $f_{\xi}$ has two saddle fixed points if and only if $\xi\in\Phi$.

###### Proof.

Let $f^{L}$ and $f^{R}$ denote the left and right pieces of $f_{\xi}$,
respectively. If $\delta_{L}=\tau_{L}-1$ then $f^{L}$ has no fixed points,
while if $\delta_{L}\neq\tau_{L}-1$ then $f^{L}$ has the unique fixed point
$Y$, given by (2.2). Similarly if $\delta_{R}=\tau_{R}-1$ then $f^{R}$ has no
fixed points, while if $\delta_{R}\neq\tau_{R}-1$ then $f^{R}$ has the unique
fixed point $X$, given by (2.1).

If $\xi\in\Phi$ then $X$ and $Y$ are saddle fixed points of $f_{\xi}$, as
noted above. Conversely, suppose $f_{\xi}$ has two saddle fixed points. By the
above remarks, these must be $X$ and $Y$. Since they are fixed points of
$f_{\xi}$, $Y$ is in the left half-plane, so $\tau_{L}>\delta_{L}+1$ by (2.2),
and $X$ is in the right half-plane, so $\tau_{R}<\delta_{R}+1$ by (2.1). Since
they are saddles, also $\tau_{L}>-(\delta_{L}+1)$ and
$\tau_{R}<-(\delta_{R}+1)$, hence $\xi\in\Phi$. ∎

###  2.1 The stable and unstable manifolds of the fixed points

Since $X$ and $Y$ are saddles they have one-dimensional stable and unstable
manifolds. Their stable manifolds $W^{s}(X)$ and $W^{s}(Y)$ have kinks at
points where they meet the switching manifold $x=0$ and on preimages of these
points, while their unstable manifolds $W^{u}(X)$ and $W^{u}(Y)$ have kinks at
points where they meet $y=0$ (the image of the switching manifold) and on
images of these points. As each manifold (stable or unstable) emanates from
the fixed point, it coincides with the line through the fixed point with
direction given by the corresponding eigenvector of $A_{L}$ or $A_{R}$. We
will use a subscript $0$ to denote the part of the manifold that coincides
with this line. These are indicated in Fig. 1 for four different combinations
of the parameter values.

|
---|---
(a) $\delta_{L}>0,\delta_{R}>0$ | (b) $\delta_{L}>0,\delta_{R}<0$
|
(c) $\delta_{L}<0,\delta_{R}>0$ | (d) $\delta_{L}<0,\delta_{R}<0$
Figure 1:  Phase portraits of $f_{\xi}$ (1.1) for four different parameter
combinations $\xi=(\tau_{L},\delta_{L},\tau_{R},\delta_{R})\in\Phi_{\rm
trap}\cap\Phi_{\rm cone}$; (a) $\xi=(2.1,0.06,-1.7,0.18)$; (b)
$\xi=(2.1,0.4,-1.7,-0.55)$; (c) $\xi=(2.2,-0.3,-1.7,0.1)$; (d)
$\xi=(1.8,-0.75,-1.6,-0.4)$. Each plot shows the initial line segments of the
stable and unstable manifolds emanating from the fixed points $X$ and $Y$, as
well as an additional segment of the stable manifold of $Y$. We also show some
key points of intersection ($S$, $C$, $D$, $T$, and $Z$) of these segments
with the coordinate axes, see (2.4), (2.7), (2.9), (2.10), and (3.2)
respectively. To illustrate the chaotic attractor the black dots show iterates
of a typical forward orbit after transient dynamics has decayed.

Next, we describe some points where the stable and unstable manifolds
intersect $x=0$ and $y=0$ as these are central to our construction in §3. For
all $\xi\in\Phi$, $W^{s}_{0}(Y)$ has an endpoint on $x=0$, call it $S$. Since
$W^{s}_{0}(Y)$ has slope $-\lambda_{L}^{u}$ (due to the companion matrix form
(2.3)), from the above formula for $Y$ we obtain

$S=\mathopen{}\mathclose{{}\left(0,\frac{-\lambda_{L}^{u}}{\lambda_{L}^{u}-1}}\right).$
(2.4)

Notice $S_{2}<-1$ (here and throughout the paper for any $P\in\mathbb{R}^{2}$
we write $P_{1}$ and $P_{2}$, respectively, for its $x$ and $y$ components).
The manifold $W^{s}(Y)$ continues into the right half-plane but now with slope
$\frac{\phi_{1}(\xi)}{\lambda_{L}^{u}}$, where

$\phi_{1}(\xi)=\delta_{R}-\tau_{R}\lambda_{L}^{u}\,.$ (2.5)

Our trapping region construction requires that this linear segment, call it
$W_{1}^{s}(Y)$, intersects $y=0$. Certainly this is only possible if
$\phi_{1}(\xi)>0$. This inequality turns out to be sufficient to ensure
$W_{1}^{s}(Y)$ intersects $y=0$ except in the case $\delta_{L},\delta_{R}<0$
(Fig. 1-d). In this case $W_{1}^{s}(Y)$ is the line segment connecting $S$ and
$f_{\xi}^{-2}(S)$. So in this case for $W_{1}^{s}(Y)$ to intersect $y=0$ we
need $f_{\xi}^{-2}(S)$ to lie above $y=0$. A straightforward calculation
reveals that the $y$-component of $f_{\xi}^{-2}(S)$ is
$\frac{\phi_{2}(\xi)}{\lambda_{L}^{s}(\lambda_{L}^{u}-1)\delta_{R}}$, where

$\phi_{2}(\xi)=\delta_{R}(\lambda_{L}^{s}+1)-\lambda_{L}^{u}(\tau_{R}+(\delta_{R}+\tau_{R})\lambda_{L}^{s}).$
(2.6)

Thus we require $\phi_{2}(\xi)>0$ (because with $\delta_{L},\delta_{R}<0$ we
have $\lambda_{L}^{s}(\lambda_{L}^{u}-1)\delta_{R}>0$). In any case, if
$W_{1}^{s}(Y)$ intersects $y=0$, it does so at the point

$C=\mathopen{}\mathclose{{}\left(1+\frac{\phi_{3}(\xi)}{(\lambda_{L}^{u}-1)\phi_{1}(\xi)},0}\right),$
(2.7)

where

$\phi_{3}(\xi)=\delta_{R}-(\delta_{R}+\tau_{R}-(\tau_{R}+1)\lambda_{L}^{u})\lambda_{L}^{u}.$
(2.8)

Our construction also requires $C_{1}>1$, that is $\phi_{3}(\xi)>0$.

Now we consider the two unstable manifolds. For all $\xi\in\Phi$,
$W^{u}_{0}(Y)$ has an endpoint on $y=0$ at

$D=\mathopen{}\mathclose{{}\left(\frac{1}{1-\lambda_{L}^{s}},0}\right),$ (2.9)

and $W^{u}_{0}(X)$ has an endpoint on $y=0$ at

$T=\mathopen{}\mathclose{{}\left(\frac{1}{1-\lambda_{R}^{s}},0}\right).$
(2.10)

These points are indicated in Fig. 1.

###  2.2 Homoclinic and heteroclinic bifurcations

In the orientation-preserving case ($\delta_{L},\delta_{R}>0$), Banerjee et
al. [4] noticed that as parameters are varied a chaotic attractor can be
destroyed when the points $C$ and $D$ collide. This type of bifurcation is a
homoclinic corner [37] where the kinks (corners) of the unstable manifold of
$Y$ lie on the stable manifold of $Y$ (and vice-versa). This is analogous to a
first homoclinic tangency [29] for smooth maps which are well understood as a
mechanism for the destruction of an attractor [18]. From the above formulas
for $C$ and $D$ we obtain

$\displaystyle
C_{1}-D_{1}=\frac{\phi_{4}(\xi)}{(\lambda_{L}^{u}-1)(1-\lambda_{L}^{s})\phi_{1}(\xi)},$
(2.11)

where

$\displaystyle\phi_{4}(\xi)=\delta_{R}-(\tau_{R}+\delta_{L}+\delta_{R}-(1+\tau_{R})\lambda_{L}^{u})\lambda_{L}^{u}.$
(2.12)

So the condition $\phi_{4}(\xi)>0$ (equivalent to equation (5) of [4]) ensures
that $C$ lies to the right of $D$ as in Fig. 1-a.

In the orientation-reversing case ($\delta_{L},\delta_{R}<0$) a chaotic
attractor can be destroyed when the points $C$ and $T$ collide. Here kinks of
the unstable manifold of $X$ lie on the stable manifold of $Y$ (and vice-
versa). We have

$\displaystyle
C_{1}-T_{1}=\frac{\phi_{5}(\xi)}{(\lambda_{L}^{u}-1)(1-\lambda_{R}^{s})\phi_{1}(\xi)},$
(2.13)

where

$\displaystyle\phi_{5}(\xi)=\delta_{R}-(\delta_{R}+\tau_{R}-(1+\lambda_{R}^{u})\lambda_{L}^{u})\lambda_{L}^{u}.$
(2.14)

The condition $\phi_{5}(\xi)>0$ ensures that $C$ lies to the right of $T$ as
in Fig. 1-d. In the special case of the Lozi map, $\phi_{5}(\xi)>0$ simplifies
(significantly) to equation (3) of Misiurewicz [27].

In view of the above discussion we define

$\displaystyle\Phi_{\rm
trap}=\mathopen{}\mathclose{{}\left\\{\xi\in\Phi\,\middle|\,\phi_{i}(\xi)>0,i=1,\ldots,5}\right\\}.$
(2.15)

This set is difficult to visualise because parameter space is four-
dimensional. Fig. 2 shows four different cross-sections of $\Phi_{\rm trap}$
obtained by fixing $\tau_{L}$ and $\tau_{R}$. Broadly speaking the size of the
cross-section decreases as the values of $\tau_{L}$ and $|\tau_{R}|$ increase.
Notice the topology of the cross-sections is different for different values of
$\tau_{L}$ and $\tau_{R}$. For instance in Fig. 2-a the boundary of the cross-
section is formed by $\phi_{1}(\xi)=0$, $\phi_{2}(\xi)=0$, and the boundary of
$\Phi$, whereas in Fig. 2-d the boundary is formed by $\phi_{1}(\xi)=0$,
$\phi_{3}(\xi)=0$, $\phi_{4}(\xi)=0$, $\phi_{5}(\xi)=0$, and the boundary of
$\Phi$. The figure also includes curves on which $f_{\xi}(C)=Y$ and
$f_{\xi}(C)=Z$, where $Z=f_{\xi}(S)$ is the intersection of $W_{0}^{s}(Y)$
with $y=0$. Together with the $\delta_{L}$ and $\delta_{R}$ axes, these curves
divide the cross-sections into six parts corresponding to six cases for the
vertices of the region $\Omega$ that we construct in §3.

|
---|---
(a) $\tau_{L}=1.6,\tau_{R}=-1.3$ | (b) $\tau_{L}=1.6,\tau_{R}=-1.5$
|
(c) $\tau_{L}=1.6,\tau_{R}=-1.7$ | (d) $\tau_{L}=1.6,\tau_{R}=-2.1$
Figure 2:  Four different two-dimensional cross sections of $\Phi_{\rm trap}$,
defined by (2.15). In each plot the cross-section is divided into six parts
(A)–(F) where $\Omega$ has certain vertices, see §3.

###  2.3 Sufficient conditions for robust chaos

Our construction of an invariant expanding cone $\Psi_{K}$ requires similar
constraints on the parameter values to those established above for the
trapping region. To this end we define

$\displaystyle\theta_{1}(\xi)$
$\displaystyle=\mathopen{}\mathclose{{}\left(\delta_{L}+\delta_{R}-\tau_{L}\tau_{R}}\right)^{2}-4\delta_{L}\delta_{R}\,,$
(2.16) $\displaystyle\theta_{2}(\xi)$
$\displaystyle=\tau_{L}^{2}+\delta_{L}^{2}-1+2\tau_{L}\,{\rm
min}\mathopen{}\mathclose{{}\left(0,-\tfrac{\delta_{R}}{\tau_{R}},q_{L},\tilde{a}}\right),$
(2.17) $\displaystyle\theta_{3}(\xi)$
$\displaystyle=\tau_{R}^{2}+\delta_{R}^{2}-1+2\tau_{R}\,{\rm
max}\mathopen{}\mathclose{{}\left(0,-\tfrac{\delta_{L}}{\tau_{L}},q_{R},\tilde{b}}\right),$
(2.18)

where

$\displaystyle q_{L}$
$\displaystyle=-\tfrac{\tau_{L}}{2}\mathopen{}\mathclose{{}\left(1-\sqrt{1-\tfrac{4\delta_{L}}{\tau_{L}^{2}}}}\right),$
$\displaystyle q_{R}$
$\displaystyle=-\tfrac{\tau_{R}}{2}\mathopen{}\mathclose{{}\left(1-\sqrt{1-\tfrac{4\delta_{R}}{\tau_{R}^{2}}}}\right),$

and

$\displaystyle\tilde{a}$
$\displaystyle=\frac{\delta_{L}-\delta_{R}-\tau_{L}\tau_{R}-\sqrt{\theta_{1}(\xi)}}{2\tau_{R}},$
$\displaystyle\tilde{b}$
$\displaystyle=\frac{\delta_{R}-\delta_{L}-\tau_{L}\tau_{R}-\sqrt{\theta_{1}(\xi)}}{2\tau_{L}},$

assuming $\theta_{1}(\xi)>0$. We then define

$\displaystyle\Phi_{\rm
cone}=\mathopen{}\mathclose{{}\left\\{\xi\in\Phi\,\middle|\,\theta_{i}(\xi)>0,i=1,\ldots,3}\right\\}.$
(2.19)

The condition $\theta_{1}(\xi)>0$ ensures $\theta_{2}(\xi)$ and
$\theta_{3}(\xi)$ are well-defined, and, as explained in §4, the conditions
$\theta_{2}(\xi)>0$ and $\theta_{3}(\xi)>0$ ensure that our cone $\Psi_{K}$ is
invariant and expanding.

Fig. 3 shows cross-sections of $\Phi_{\rm cone}$. Broadly speaking the size of
the cross-section increases as the values of $\tau_{L}$ and $|\tau_{R}|$
increase. Again the topology of the cross-sections is different for different
values of $\tau_{L}$ and $\tau_{R}$. Similar to Fig. 2 we have divided the
cross-sections into six parts corresponding to six cases for the boundary of
$\Psi_{K}$ (see §4). These correspond to different cases for the four
quantities in each of (2.17) and (2.18) that attain the minimum (respectively,
maximum) value.

|
---|---
(a) $\tau_{L}=1.6,\tau_{R}=-1.1$ | (b) $\tau_{L}=1.6,\tau_{R}=-1.5$
|
(c) $\tau_{L}=1.6,\tau_{R}=-2.2$ | (d) $\tau_{L}=1.6,\tau_{R}=-3$
Figure 3: Two-dimensional cross-sections of $\Phi_{\rm cone}$, defined by
(2.19). In each plot the cross-section has been divided into six parts
(I)–(VI) where the cone $\Psi_{K}$ has different boundaries, see Fig. 6.

Finally, we can state our main result.

######  Theorem 2.2.

For any $\xi\in\Phi_{\rm trap}\cap\Phi_{\rm cone}$ the normal form $f_{\xi}$
(1.1) has a topological attractor with a positive Lyapunov exponent.

This is proved at the end of §4. We have found that there are many
possibilities for the topology of cross-sections of $\Phi_{\rm
trap}\cap\Phi_{\rm cone}$ defining by fixing $\tau_{L}$ and $\tau_{R}$, and we
do not attempt to catagorise these in this paper. We provide one example in
§5, where we also identify critical values of $\tau_{L}$ and $\tau_{R}$ at
which the cross-sections of $\Phi_{\rm trap}$ and $\Phi_{\rm cone}$ vanish
entirely.

##  3 A forward invariant region and a trapping region

The following definition applies to any continuous map $f$ on the plane. Note
we write ${\rm int}(\cdot)$ for the interior of a set.

######  Definition 3.1.

A set $\Omega\subset\mathbb{R}^{2}$ is forward invariant if
$f(\Omega)\subset\Omega$. A compact set $\Omega\subset\mathbb{R}^{2}$ is a
trapping region if $f(\Omega)\subset{\rm int}(\Omega)$.

In this section we construct a triangle $\Omega$ and show that for any
$\xi\in\Phi_{\rm trap}$ this region is forward invariant under $f$
(Proposition 3.3). We then show there exists a perturbation of $\Omega$ that
is a trapping region for $f$ (Proposition 3.4). So that the non-invertible
cases can be accommodated, our $\Omega$ differs from the triangle constructed
by Glendinning and Simpson [15] for the orientation-preserving case, and by
Misiurewicz [27] for the orientation-reversing case. For clarity we now
suppress the $\xi$-dependency and write $f$ instead of $f_{\xi}$.

The linear segment of the stable manifold of $Y$ that contains $Y$, denoted
$W^{s}_{0}(Y)$, was shown in Fig. 1 for four different combinations of the
parameter values. In any case, this segment lies on the line
$y=-\lambda_{L}^{u}x+S_{2}$, where $S_{2}$ is the $y$-component of $S$, given
by (2.4). The point $S$ is the right-most point of $W^{s}_{0}(Y)$, and is easy
to see that in the other direction $W^{s}_{0}(Y)$ extends to infinity if
$\delta_{L}\geq 0$ and to the preimage of $S$ under the left piece of $f$
otherwise:

######  Lemma 3.1.

Suppose $\delta_{L}\in\mathbb{R}$ and $\tau_{L}>|\delta_{L}+1|$. Then

$W^{s}_{0}(Y)=\begin{cases}\mathopen{}\mathclose{{}\left\\{(x,y)\,\middle|\,-\frac{S_{2}}{\delta_{L}}\leq
x\leq 0,y=-\lambda_{L}^{u}x+S_{2}}\right\\},&\delta_{L}<0,\\\
\mathopen{}\mathclose{{}\left\\{(x,y)\,\middle|\,-\infty<x\leq
0,y=-\lambda_{L}^{u}x+S_{2}}\right\\},&\delta_{L}\geq 0,\end{cases}$ (3.1)

and $f(W^{s}_{0}(Y))\subset W^{s}_{0}(Y)$.

In particular $Z=f(S)\in W^{s}_{0}(Y)$. This point is the intersection of
$W^{s}_{0}(Y)$ with $y=0$ and given by

$\displaystyle Z$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{-1}{\lambda_{L}^{u}-1},0}\right).$
(3.2)

Now recall if $\phi_{1}(\xi)>0$ and $\phi_{2}(\xi)>0$ then $C\in W_{1}^{s}(Y)$
is given by (2.7).

######  Lemma 3.2.

Let $\xi\in\Phi$ with $\phi_{1}(\xi)>0$ and $\phi_{2}(\xi)>0$. Then

$Y,Z,f(Z),f(C),f^{2}(C)\in W^{s}_{0}(Y)\setminus\\{S\\}.$ (3.3)

###### Proof.

Certainly $Y,Z\in W^{s}_{0}(Y)\setminus\\{S\\}$ by construction. Also $f(Z)\in
W^{s}_{0}(Y)\setminus\\{S\\}$ because $W^{s}_{0}(Y)$ is forward invariant
(Lemma 3.1) and $Z=f(S)$ cannot be a preimage of $S$.

Since $C_{1}>0$ we have
$f(C)=\mathopen{}\mathclose{{}\left(\tau_{R}C_{1}+1,-\delta_{R}C_{1}}\right)$,
and it is a simple exercise to use the formula (2.9) to show that $f(C)$ lies
on the line $y=-\lambda_{L}^{u}x+S_{2}$. Also $f(C)$ lies to the left of $S$
because
$f(C)_{1}=\frac{(\tau_{R}+\delta_{R})(\lambda_{L}^{u}-1)+\tau_{R}}{(\lambda_{L}^{u}-1)\phi_{1}(\xi)}$
is negative by inspection, and in the case $\delta_{L}<0$ the point $f(C)$
lies to the right of the left-most point of $W_{0}^{s}(Y)$ because
$f(C)_{1}+\frac{S_{2}}{\delta_{L}}=\frac{\phi_{2}(\xi)}{-\lambda_{L}^{s}(\lambda_{L}^{u}-1)\phi_{1}(\xi)}$
is positive. Thus $f(C)$ belongs to $W_{0}^{s}(Y)$ and is not an endpoint of
$W_{0}^{s}(Y)$, thus $f(C)\in W^{s}_{0}(Y)\setminus\\{S\\}$ and $f^{2}(C)\in
W^{s}_{0}(Y)\setminus\\{S\\}$ using again Lemma 3.1. ∎

Given $\xi\in\Phi$ with $\phi_{1}(\xi)>0$ and $\phi_{2}(\xi)>0$, let $Q$ and
$R$ be the upper-most and lower-most points of $\\{Y,Z,f(Z),f(C),f^{2}(C)\\}$,
respectively. Then let $\Omega$ be the compact filled triangle with vertices
$C$, $Q$, and $R$ (except $\Omega$ is a line segment in the special case
$\delta_{L}=\delta_{R}=0$). In other words, $\Omega$ is the convex hull of
$Y$, $Z$, $f(Z)$, $f(C)$, $f^{2}(C)$, and $C$.

There are six cases for the points that form the vertices of $\Omega$. These
are shown in Fig. 4 and correspond to the six parts of $\Phi_{\rm trap}$
indicated in Fig. 2. Fig. 4 also shows the set $f(\Omega)$. Notice in each
case $f(\Omega)$ has vertices at the images of the points $P$ and $V$ where
the boundary of $\Omega$ intersects $x=0$.

|
---|---
(A) $\delta_{L}=0.3,\delta_{R}=0.4$ | (B) $\delta_{L}=0.4,\delta_{R}=-0.4$
|
(C) $\delta_{L}=0.4,\delta_{R}=-0.6$ | (D) $\delta_{L}=-0.3,\delta_{R}=-0.4$
|
(E) $\delta_{L}=-0.5,\delta_{R}=0.1$ | (F) $\delta_{L}=-0.1,\delta_{R}=0.4$
Figure 4:  The triangle $\Omega$ and its image under $f$ with $\tau_{L}=1.6$,
$\tau_{R}=-1.5$, and six different values of $\delta_{L}$ and $\delta_{R}$.
These correspond to parts (A)–(F) of Fig. 2-b for different cases for the two
vertices of $\Omega$ in the left half-plane, denoted $Q$ and $R$ (the third
vertex is always $C$). In each case $f(\Omega)\subset\Omega$ by Proposition
3.3 because the parameter combinations belong to $\Omega_{\rm trap}$.

######  Proposition 3.3.

Let $\xi\in\Phi_{\rm trap}$. Then $f(\Omega)\subset\Omega$.

###### Proof.

The proof is long so we break it into three steps.

Step 1 — Characterise $f(\Omega)$.
The vertices $Q$ and $R$ lie in the left half-plane, while $C$ lies in the
right half-plane. Let $P$ denote the intersection of $\overline{QC}$ (the line
segment from $Q$ to $C$) with $x=0$, and $V$ denote the intersection of
$\overline{RC}$ with $x=0$, see Fig. 4. From the formula (2.7) for $C$, the
$y$-components of these points are given in terms of $Q$ and $R$ by

$\displaystyle P_{2}$
$\displaystyle=\frac{{\lambda_{L}^{u}}^{3}Q_{2}}{{\lambda_{L}^{u}}^{3}+[\lambda_{L}^{u}-Q_{2}(1-\lambda_{L}^{u})]\phi_{1}(\xi)},$
(3.4) $\displaystyle V_{2}$
$\displaystyle=\frac{{\lambda_{L}^{u}}^{3}R_{2}}{{\lambda_{L}^{u}}^{3}+[\lambda_{L}^{u}-R_{2}(1-\lambda_{L}^{u})]\phi_{1}(\xi)}.$
(3.5)

So $\Omega$ is the union of the quadrilateral $\Omega_{L}$ in the left half-
plane with vertices $P$, $Q$, $R$, and $V$, and the triangle $\Omega_{R}$ in
the right half-plane with vertices $C$, $P$, and $V$. Thus
$f(\Omega)=f(\Omega_{L})\cup f(\Omega_{R})$, where $f(\Omega_{L})$ and
$f(\Omega_{R})$ are polygons because each piece of $f$ is affine. Thus since
$\Omega$ is convex, to prove $f(\Omega)\subset\Omega$ it suffices to show that
the vertices of $f(\Omega_{L})$ and $f(\Omega_{R})$ belong to $\Omega$. These
vertices are the points $f(C)$, $f(P)$, $f(Q)$, $f(R)$, and $f(V)$.

Step 2 — Show $C$, $Q$, and $R$ map to $\Omega$.
Certainly $f(C)\in\Omega$ by the definition of $\Omega$. We now show
$f(Q),f(R)\in\overline{QR}$ (the left edge of $\Omega$). If $\delta_{L}>0$
then $\lambda_{L}^{s}>0$, so $f(Q)\in\overline{QY}$ and $f(R)\in\overline{RY}$
so certainly $f(Q),f(R)\in\overline{QR}$. Also if $\delta_{L}=0$, then
$f(Q)=f(R)=Y=Z\in\overline{QR}$. Finally if $\delta_{L}<0$, then $f(Q)$ and
$f(R)$ lie below $y=0$, so lie below $Z$, and hence below $Q$. In this case
$Q$ is either $f(C)$ or $Z$ (because $Y$, $f(Z)$, and $f^{2}(C)$ lie below
$y=0$), thus $f(Q)$ lies on or above $R$, by the definition of $R$. Also
$\lambda_{L}^{s}<0$, thus $f(R)$ lies above $f(Q)$, and hence above $R$. Thus
in any case $f(Q),f(R)\in\overline{QR}$.

Step 3 — Show $P$ and $V$ map to $\Omega$.
The points $f(P)$ and $f(V)$ lie on $y=0$, specifically

$\displaystyle f(P)$ $\displaystyle=\begin{bmatrix}P_{2}+1\\\ 0\end{bmatrix},$
$\displaystyle f(V)$ $\displaystyle=\begin{bmatrix}V_{2}+1\\\ 0\end{bmatrix},$
(3.6)

where $P_{2}$ and $V_{2}$ are given by (3.4) and (3.5). Also $V$ lies above
$S$, thus $f(V)$ lies to the right of $Z=f(S)$. Hence it remains for us to
show that $f(P)$ lies at or to the left of $C$, that is $C_{1}-(P_{2}+1)\geq
0$. To do this we consider the various possibilities for $Q$ in turn. There
are three cases: $Q$ is either $Y$, $Z$, or $f(C)$. This is because $f(Z)$
cannot lie above $Y$, while $f^{2}(C)$ cannot lie above $Y$ if $\delta_{L}>0$
and cannot lie above $Z$ if $\delta_{L}\leq 0$.

Case 1: With $Q=Z$ we have $P_{2}=0$. Thus $C_{1}-(P_{2}+1)>0=C_{1}-1>0$
because $\phi_{3}(\xi)>0$.

Case 2: With $Q=Y$, by substituting $Y_{2}$, given by (2.2), in place of
$Q_{2}$ in (3.4) we obtain

$C_{1}-(P_{2}+1)=\mathopen{}\mathclose{{}\left(1+\frac{\lambda_{L}^{s}{\lambda_{L}^{u}}^{2}}{{\lambda_{L}^{u}}^{2}(1-\lambda_{L}^{s})+\phi_{1}(\xi)}}\right)\frac{\phi_{4}(\xi)}{\phi_{1}(\xi)(\lambda_{L}^{u}-1)}.$

This case requires $\delta_{L}\geq 0$, thus $\lambda_{L}^{s}\geq 0$. Also
$C_{1}-(P_{2}+1)>0$ because $\phi_{1}(\xi)>0$ and $\phi_{4}(\xi)>0$.

Case 3: With $Q=f(C)$ we similarly obtain

$C_{1}-(P_{2}+1)=\mathopen{}\mathclose{{}\left(1+\frac{\lambda_{R}^{s}{\lambda_{L}^{u}}^{2}}{\lambda_{L}^{u}(\lambda_{L}^{u}-\delta_{R})\phi_{1}(\xi)}}\right)\frac{\phi_{5}(\xi)}{\phi_{1}(\xi)(\lambda_{L}^{u}-1)}.$

This case requires $\delta_{R}\leq 0$ (so that $f(C)_{2}\geq 0$), thus
$\lambda_{R}^{s}\geq 0$. Also $\phi_{1}(\xi)>0$ and $\phi_{5}(\xi)>0$, thus
$C_{1}-(P_{2}+1)>0$. ∎

######  Proposition 3.4.

Let $\xi\in\Phi_{\rm trap}$. Given $\varepsilon>0$ define

$\displaystyle C_{\varepsilon}$ $\displaystyle=C-(\varepsilon,0),$
$\displaystyle Q_{\varepsilon}$ $\displaystyle=Q+\varepsilon^{2}(C-R),$
$\displaystyle R_{\varepsilon}$ $\displaystyle=R+\varepsilon^{2}(C-Q),$

and let $\Omega_{\varepsilon}$ be the compact filled triangle with vertices
$C_{\varepsilon}$, $Q_{\varepsilon}$, and $R_{\varepsilon}$. Then
$f(\Omega_{\varepsilon})\subset{\rm int}(\Omega_{\varepsilon})$, for all
sufficiently small $\varepsilon>0$.

The triangle $\Omega_{\varepsilon}$ is shown in Fig. 5 for one combination of
parameter values. It has been defined so that its left edge lies to the right
of $W_{0}^{s}(Y)$ and is parallel to $W^{s}_{0}(Y)$. Due to the saddle nature
of $Y$, this edge shifts further to the right when iterated under $f$. Also
the left edge is an order $\varepsilon^{2}$ distance from $W_{0}^{s}(Y)$ to
ensure $f(C_{\varepsilon})$ lies to the right of the image of this edge.

Figure 5:  The trapping region $\Omega_{\varepsilon}$ and its image under $f$.
This figure was produced using $\xi=\\{1.6,0.3,-1.5,0.4\\}$, as in Fig. 4-A,
and $\varepsilon=0.3$ which is small enough that
$f(\Omega_{\varepsilon})\subset{\rm int}(\Omega_{\varepsilon})$ in accordance
with Proposition 3.4.

###### Proof.

Step 1 — Characterise $f(\Omega_{\varepsilon})$.
Assume $\varepsilon>0$ is sufficiently small that $Q_{\varepsilon}$ and
$R_{\varepsilon}$ lie to the left of $x=0$ and $C_{\varepsilon}$ lies to the
right of $x=0$. Then the line segment
$\overline{C_{\varepsilon}Q_{\varepsilon}}$ intersects $x=0$ at a unique point
$P_{\varepsilon}$, as does $\overline{C_{\varepsilon}R_{\varepsilon}}$ at a
point $V_{\varepsilon}$. Similar to the previous proof, it remains for us to
show that $C_{\varepsilon}$, $P_{\varepsilon}$, $Q_{\varepsilon}$,
$R_{\varepsilon}$, and $V_{\varepsilon}$ map to the interior of
$\Omega_{\varepsilon}$.

Step 2 — Show $C_{\varepsilon}$ maps to the interior of
$\Omega_{\varepsilon}$.
Let $O=(0,0)$ denote the origin and $I=(1,0)$ be its image under $f$. Also let
$\ell=\overline{CO}$. This line segment maps under the right piece of $f$ to
the line segment from $f(C_{\varepsilon})$ to $I$. Since
$C_{\varepsilon}\in\ell$ is an order $\varepsilon$ distance from $C$, its
image $f(C_{\varepsilon})\in f(\ell)$ is an order $\varepsilon$ distance from
$f(C)$, which belongs to $\overline{QR}$. Since
$\overline{Q_{\varepsilon}R_{\varepsilon}}$ is an order $\varepsilon^{2}$
distance from $\overline{QR}$, $f(C_{\varepsilon})$ must lie to the right of
$\overline{Q_{\varepsilon}R_{\varepsilon}}$ for sufficiently small
$\varepsilon>0$.

Also, $f(\ell)$ lies inside the triangle $QRI$. Since $C_{\varepsilon}$ lies
to the right of $I$ (because $\phi_{3}(\xi)>0$ and assuming $\varepsilon$ is
sufficiently small), this triangle lies below
$\overline{Q_{\varepsilon}C_{\varepsilon}}$ and above
$\overline{R_{\varepsilon}C_{\varepsilon}}$. Thus $f(C_{\varepsilon})$ lies
below $\overline{Q_{\varepsilon}C_{\varepsilon}}$ and above
$\overline{R_{\varepsilon}C_{\varepsilon}}$. Thus $f(C_{\varepsilon})$ lies
inside all three edges of $\Omega_{\varepsilon}$, hence
$f(C_{\varepsilon})\in{\rm int}(\Omega_{\varepsilon})$.

Step 3 — Show $P_{\varepsilon}$ and $V_{\varepsilon}$ map to the interior of
$\Omega_{\varepsilon}$.
Let $A$ be any point on $\overline{CZ}$ with $A\neq C,Z$. Then there exists
sufficiently small $\varepsilon>0$ such that $A\in{\rm
int}(\Omega_{\varepsilon})$. We know $f(P)$ and $f(V)$ are located on
$\overline{CZ}\setminus\\{C,Z\\}$, because by construction $Z_{1}\leq
f(V)_{1}<f(P)_{1}<C_{1}$, so belong to ${\rm int}(\Omega_{\varepsilon})$ for
sufficiently small $\varepsilon>0$. The same is true for $f(P_{\varepsilon})$
and $f(V_{\varepsilon})$ because $P_{\varepsilon}\to P$ and
$V_{\varepsilon}\to V$ as $\varepsilon\to 0$.

Step 4 — Show $Q_{\varepsilon}$ and $R_{\varepsilon}$ map to the interior of
$\Omega_{\varepsilon}$.
For brevity we just show $f(Q_{\varepsilon})\in{\rm
int}(\Omega_{\varepsilon})$ ($f(R_{\varepsilon})\in{\rm
int}(\Omega_{\varepsilon})$ can be shown similarly). Let $d_{1}>0$ be the
distance that $\overline{Q_{\varepsilon}R_{\varepsilon}}$ lies to the right of
$\overline{QR}$, see Fig. 5. Then $f(Q_{\varepsilon})$ lies a distance
$\lambda_{L}^{u}d_{1}$ to the right of $\overline{QR}$, as $Y$ is a saddle
fixed point with stable direction $\overline{QR}$ and unstable eigenvalue
$\lambda_{L}^{u}$. Since $\lambda_{L}^{u}>1$, the point $f(Q_{\varepsilon})$
lies to the right of the line $\overline{Q_{\varepsilon}R_{\varepsilon}}$.

If $Q\neq Y$, then $f(Q)$ lies an order $1$ distance below $\overline{CQ}$,
thus $f(Q_{\varepsilon})$ lies below
$\overline{C_{\varepsilon}Q_{\varepsilon}}$ for sufficiently small
$\varepsilon>0$. Now consider the case $Q=Y$. As shown in Fig. 5 let $d_{2}$
be the vertical displacement from $f(Q_{\varepsilon})$ upwards to the line
$\overline{Q_{\varepsilon}C_{\varepsilon}}$ (we will show $d_{2}>0$). By
direct calculations
$d_{2}=\beta\varepsilon^{2}+O\mathopen{}\mathclose{{}\left(\varepsilon^{3}}\right)$
where

$\beta=\delta_{L}(C_{1}-R_{1})(C_{1}+(\tau_{L}-2)Y_{1})-R_{2}(C_{1}+(\delta_{L}-1)Y_{1}).$

Let $p=C_{1}-D_{1}$ and notice $p>0$ by (2.11) because $\phi_{4}(\xi)>0$.
Using also
$D_{1}=\mathopen{}\mathclose{{}\left(1-\lambda_{L}^{u}}\right)Y_{1}$ by (2.2)
and (2.9), we obtain

$\beta=\delta_{L}(C_{1}-R_{1})\mathopen{}\mathclose{{}\left(p-Y_{1}\mathopen{}\mathclose{{}\left(1-\lambda_{L}^{s}}\right)}\right)-R_{2}\mathopen{}\mathclose{{}\left(p-Y_{1}\lambda_{L}^{u}\mathopen{}\mathclose{{}\left(1-\lambda_{L}^{s}}\right)}\right),$

which is positive by inspection (e.g. $Y_{1}<0$). This shows that $d_{2}>0$
for sufficiently small values of $\varepsilon$, that is $f(Q_{\varepsilon})$
lies below the upper edge of $\Omega_{\varepsilon}$. By similar calculations
one can show that $f(Q_{\varepsilon})$ lies above the lower edge of
$\Omega_{\varepsilon}$, and therefore $f(Q_{\varepsilon})\in{\rm
int}(\Omega_{\varepsilon})$. ∎

##  4 Invariant expanding cones

In this section we first define cones and what it means for them to be
invariant and expanding under an arbitrary matrix $A$. We then focus on the
Jacobian matrices $A_{L}$ and $A_{R}$ of the normal form (1), because in order
for a cone to establish chaos in (1) it needs to be invariant and expanding
for both $A_{L}$ and $A_{R}$. We then explicitly construct such a cone for any
$\xi\in\Phi_{\rm cone}$ (Proposition 3), and use this to prove Theorem 2.2.

######  Definition 4.1.

A set $C\subset\mathbb{R}^{2}$ is a cone if $\alpha v\in C$ for all $v\in C$
and $\alpha\in\mathbb{R}$.

######  Definition 4.2.

Given $A\in\mathbb{R}^{2\times 2}$, a cone $C\subset\mathbb{R}^{2}$ is

  1. i)

invariant under $A$ if $Av\in C$ for all $v\in C$, and

  2. ii)

expanding under $A$ if there exists $c>1$ such that $\|Av\|\geq c\|v\|$ for
all $v\in C$.

In this paper we use the Euclidean norm $\|v\|=\sqrt{v_{1}^{2}+v_{2}^{2}}$ and
it suffices to consider cones of the form

$\Psi_{K}=\mathopen{}\mathclose{{}\left\\{\alpha\begin{bmatrix}1\\\
m\end{bmatrix}\,\middle|\,\alpha\in\mathbb{R},m\in K}\right\\},$ (4.1)

where $K$ is an interval. Since $v\mapsto Av$ is a linear map, to verify
invariance and expansion of a cone $\Psi_{K}$, it suffices to verify
properties (i) and (ii) for vectors of the form $v=\begin{bmatrix}1\\\
m\end{bmatrix}$:

######  Lemma 4.1.

If $Av\in\Psi_{K}$ for all $v=\begin{bmatrix}1\\\ m\end{bmatrix}$ with $m\in
K$, then $\Psi_{K}$ is invariant under $A$. If there exists $c>1$ such that
$\|Av\|\geq c\|v\|$ for all $v=\begin{bmatrix}1\\\ m\end{bmatrix}$ with $m\in
K$, then $\Psi_{K}$ is expanding under $A$.

Now we focus on the Jacobian matrices

$A_{J}=\begin{bmatrix}\tau_{J}&1\\\ -\delta_{J}&0\end{bmatrix},$

where $J\in\\{L,R\\}$, of the normal form (1.1). The slope of
$v=\begin{bmatrix}1\\\ m\end{bmatrix}$ is $m$ and the slope of $A_{J}v$ is

$\displaystyle G_{J}(m)=-\frac{\delta_{J}}{\tau_{J}+m},$ (4.2)

assuming $m\neq-\tau_{J}$. Notice

$\displaystyle\frac{dG_{J}(m)}{dm}=\frac{\delta_{J}}{(\tau_{J}+m)^{2}},$ (4.3)

thus $G_{J}(m)$ is increasing if $\delta_{J}>0$, decreasing if $\delta_{J}<0$,
and flat if $\delta_{J}=0$. In any case, $G_{J}(m)$ is monotone and so in
order to verify invariance under $A_{J}$ it suffices to consider the endpoints
of $K$:

######  Lemma 4.2.

Let $\tau_{J},\delta_{J}\in\mathbb{R}$ and $K=[a,b]$ be an interval with
$-\tau_{J}\notin K$. If $a\leq G_{J}(a)\leq b$ and $a\leq G_{J}(b)\leq b$ then
$\Psi_{K}$ is invariant under $A_{J}$.

###### Proof.

Since $-\tau_{J}\notin K$, by (4.2) and (4.3) $G_{J}(m)$ is continuous and
monotone on $K$. Thus for any $m\in K$, $G_{J}(m)$ is equal to or lies between
the values $G_{J}(a)$ and $G_{J}(b)$. Thus $a\leq G_{J}(m)\leq b$. That is,
the slope of $A_{J}\begin{bmatrix}1\\\ m\end{bmatrix}$ belongs to $K$, thus
$A_{J}\begin{bmatrix}1\\\ m\end{bmatrix}\in\Psi_{K}$. Hence $\Psi_{K}$ is
invariant under $A_{J}$ by Lemma 4.1. ∎

Next we introduce the function

$H_{J}(m)=\mathopen{}\mathclose{{}\left\|A_{J}\begin{bmatrix}1\\\
m\end{bmatrix}}\right\|^{2}-\mathopen{}\mathclose{{}\left\|\begin{bmatrix}1\\\
m\end{bmatrix}}\right\|^{2}=\tau_{J}^{2}+\delta_{J}^{2}-1+2\tau_{J}m.$ (4.4)

It is easy to show that if $H_{J}(m)>0$ for all $m$ in a compact interval $K$,
then $\Psi_{K}$ is expanding under $A_{J}$. Since $H_{J}(m)$ is a linear
function of $m$ it again suffices to consider the endpoints of $K$:

######  Lemma 4.3.

Let $\tau_{J},\delta_{J}\in\mathbb{R}$ and $K=[a,b]$ be an interval. If
$H_{J}(a)>0$ and $H_{J}(b)>0$ then $\Psi_{K}$ is expanding under $A_{J}$.

###### Proof.

Let $h=\min[H_{J}(a),H_{J}(b)]>0$. By (4.4), $H_{J}(m)\geq h$ for all $m\in
K$. Then for any $m\in K$ the vector $v=\begin{bmatrix}1\\\ m\end{bmatrix}$
satisfies

$\|A_{J}v\|^{2}=H_{J}(m)+\|v\|^{2}\geq
h+\|v\|^{2}=\mathopen{}\mathclose{{}\left(\tfrac{h}{\|v\|^{2}}+1}\right)\|v\|^{2}\geq\mathopen{}\mathclose{{}\left(\frac{h}{n}+1}\right)\|v\|^{2},$

where $n=\max_{m\in K}(1+m^{2})$. Thus $\Psi_{K}$ is expanding under $A_{J}$
(with expansion factor $c=\sqrt{\frac{h}{n}+1}>1$) by Lemma 4.1. ∎

To prove chaos in (1.1) we need to choose $K=[a,b]$ so that $\Psi_{K}$ is
invariant under $A_{L}$ and $A_{R}$. This favours the interval $K$ being
relatively large. However, we want $K$ to be as small as possible in order to
maximise the parameter region over which it is expanding under $A_{L}$ and
$A_{R}$. This balancing act motivates the following calculations that form the
basis of our definition of $K$ given below in Proposition 4.4.

For each $J\in\\{L,R\\}$, the fixed point equation $G_{J}(m)=m$ is quadratic
in $m$. If $\delta_{J}\neq 0$ and $\delta_{J}<\frac{\tau_{J}^{2}}{4}$, then
$G_{J}$ has exactly two fixed points

$\displaystyle q_{J}$
$\displaystyle=-\frac{\tau_{J}}{2}\mathopen{}\mathclose{{}\left(1-\sqrt{1-\frac{4\delta_{J}}{\tau_{J}^{2}}}}\right),$
$\displaystyle r_{J}$
$\displaystyle=-\frac{\tau_{J}}{2}\mathopen{}\mathclose{{}\left(1+\sqrt{1-\frac{4\delta_{J}}{\tau_{J}^{2}}}}\right).$
(4.5)

|
---|---
(I) $\delta_{L}=0.4$, $\delta_{R}=0.4$. | (II) $\delta_{L}=0.4$, $\delta_{R}=-0.2$.
|
(III) $\delta_{L}=0.4$, $\delta_{R}=-0.6$. | (IV) $\delta_{L}=-0.5$, $\delta_{R}=-0.5$.
|
(V) $\delta_{L}=-0.3$, $\delta_{R}=0.2$. | (VI) $\delta_{L}=-0.3$, $\delta_{R}=0.4$.
Figure 6:  Cobweb diagrams of the slope maps $G_{L}(m)$ and $G_{R}(m)$ (4.2)
with $\tau_{L}=1.6$, $\tau_{R}=-1.5$, and six different values of $\delta_{L}$
and $\delta_{R}$. These correspond to parts (I)–(VI) of Fig. 3-b. In each case
$K=[a,b]$, defined by (4.10) and (4.11), is forward invariant under $G_{L}$
and $G_{R}$ as shown. This implies $\Psi_{K}$ is invariant under $A_{L}$ and
$A_{R}$ (see Lemma 4.2). In each case $K$ contains the fixed points $q_{L}$
and $q_{R}$; in case (IV) $K$ is defined using one of the two period-two
solutions.

In order for $\Psi_{K}$ to be invariant under $A_{L}$ and $A_{R}$, we define
$K$ so that it contains $q_{L}$ and $q_{R}$, see Fig. 6. So the smallest
interval we consider is $K=[q_{L},q_{R}]$,. In the orientation-preserving case
($\delta_{L},\delta_{R}>0$), this interval indeed gives invariance, as shown
in [15]. In the orientation-reversing case ($\delta_{L},\delta_{R}<0$),
invariance requires that $K$ contains a period-two solution. The equation
$(G_{R}\circ G_{L})(m)=m$ is quadratic in $m$ with discriminant

$\theta_{1}(\xi)=\mathopen{}\mathclose{{}\left(\delta_{L}+\delta_{R}-\tau_{L}\tau_{R}}\right)^{2}-4\delta_{L}\delta_{R}\,,$

repeating (2.16). So if $\theta_{1}(\xi)>0$ there are two period-two
solutions, Fig. 6-(IV). Of these the inner-most solution is
$\\{\tilde{a},\tilde{b}\\}$, where

$\displaystyle\tilde{a}$
$\displaystyle=\frac{\delta_{L}-\delta_{R}-\tau_{L}\tau_{R}-\sqrt{\theta_{1}(\xi)}}{2\tau_{R}},$
$\displaystyle\tilde{b}$
$\displaystyle=\frac{\delta_{R}-\delta_{L}-\tau_{L}\tau_{R}-\sqrt{\theta_{1}(\xi)}}{2\tau_{L}},$
(4.6)

satisfying $G_{L}(\tilde{a})=\tilde{b}$ and $G_{R}(\tilde{b})=\tilde{a}$. So
in this case the smallest interval we can take is $K=[\tilde{a},\tilde{b}]$,
as used by Misiurewicz [27].

In the non-invertible cases the slope maps $G_{L}$ and $G_{R}$ are either both
non-negative or both non-positive, see again Fig. 6. Thus a simple and
effective choice for one endpoint of $K$ is $m=0$. In this case the smallest
interval leading to invariance uses also one of $q_{L}$, $q_{R}$, or the image
of $m=0$ under $G_{L}$ or $G_{R}$:

$\displaystyle G_{L}(0)$ $\displaystyle=-\frac{\delta_{L}}{\tau_{L}},$
$\displaystyle G_{R}(0)$ $\displaystyle=-\frac{\delta_{R}}{\tau_{R}}.$ (4.7)

Proposition 4.4 shows that all cases can be accommodated by simply defining
$a$ and $b$ as the minimum and maximum of all points suggested above. Recall
$\Phi_{\rm cone}$ was defined in §2 as the set of all $\xi\in\Phi$ for which
$\theta_{1}(\xi)$, $\theta_{2}(\xi)$, and $\theta_{3}(\xi)$ are positive. The
condition $\theta_{1}(\xi)>0$ ensures $\tilde{a}$ and $\tilde{b}$ are well-
defined, while, if $a$ and $b$ are given by (4.10) and (4.11),

$\displaystyle\theta_{2}(\xi)$
$\displaystyle=\tau_{L}^{2}+\delta_{L}^{2}-1+2\tau_{L}a=H_{L}(a),$ (4.8)
$\displaystyle\theta_{3}(\xi)$
$\displaystyle=\tau_{R}^{2}+\delta_{R}^{2}-1+2\tau_{R}b=H_{R}(b),$ (4.9)

and $\theta_{2}(\xi)>0$ and $\theta_{3}(\xi)>0$ ensure $\Psi_{K}$ is invariant
and expanding.

######  Proposition 4.4.

Let $\xi\in\Phi_{\rm cone}$ and $K=[a,b]$ where

$\displaystyle a$
$\displaystyle=\min\mathopen{}\mathclose{{}\left[0,-\tfrac{\delta_{R}}{\tau_{R}},q_{L},\tilde{a}}\right],$
(4.10) $\displaystyle b$
$\displaystyle=\max\mathopen{}\mathclose{{}\left[0,-\tfrac{\delta_{L}}{\tau_{L}},q_{R},\tilde{b}}\right].$
(4.11)

Then $\Psi_{K}$ is invariant and expanding under $A_{L}$ and $A_{R}$.

To prove Proposition 4.4 we first establish three lemmas. The first of these
provides bounds on the other fixed points, $r_{L}$ and $r_{R}$, of $G_{L}$ and
$G_{R}$.

######  Lemma 4.5.

Let $\xi\in\Phi$. Then $r_{L}<\frac{1-\delta_{L}^{2}-\tau_{L}^{2}}{2\tau_{L}}$
and $r_{R}>\frac{1-\delta_{R}^{2}-\tau_{R}^{2}}{2\tau_{R}}$.

###### Proof.

We have $\delta_{L}+1<\tau_{L}$, hence $(\delta_{L}+1)^{2}<\tau_{L}^{2}$, and
so $(\delta_{L}-1)^{2}<\tau_{L}^{2}-4\delta_{L}$. By multiplying the last two
inequalities together we obtain
$(\delta_{L}^{2}-1)^{2}<\tau_{L}^{2}(\tau_{L}^{2}-4\delta_{L})$, so
$\delta_{L}^{2}-1<\tau_{L}\sqrt{\tau_{L}^{2}-4\delta_{L}}$ which rearranges to
$r_{L}<\frac{1-\delta_{L}^{2}-\tau_{L}^{2}}{2\tau_{L}}$ using (4.5). The
result for $r_{R}$ follows similarly. ∎

######  Lemma 4.6.

With the assumptions of Proposition 4.4, $a=\tilde{a}$ if and only if
$\delta_{L}\leq 0$ and $\delta_{R}\leq 0$; similarly $b=\tilde{b}$ if and only
if $\delta_{L}\leq 0$ and $\delta_{R}\leq 0$.

###### Proof.

First suppose $\delta_{L}\leq 0$ and $\delta_{R}\leq 0$. Then
$\tau_{L}\tau_{R}\delta_{L}\geq 0$ (also $\theta_{1}(\xi)>0$ by assumption),
so we can use (4.6) to obtain

$2\tau_{R}(\tilde{a}+\tau_{L})=-\sqrt{\theta_{1}(\xi)}-\sqrt{\theta_{1}(\xi)+4\tau_{L}\tau_{R}\delta_{L}}<0.$

Thus $\tilde{a}>-\tau_{L}$ (because $\tau_{R}<0$). Also $\tilde{b}<-\tau_{R}$
by a similar argument.

Notice $\delta_{L}\leq 0$ implies $G_{L}(m)\geq 0$ for all $m>-\tau_{L}$, so
$q_{L}\geq 0$, $G_{L}(0)=-\frac{\delta_{L}}{\tau_{L}}\geq 0$, and
$G_{L}(\tilde{a})=\tilde{b}\geq 0$. Similarly $G_{R}(m)\leq 0$ for all
$m<-\tau_{R}$, so $q_{R}\leq 0$, $G_{R}(0)=-\frac{\delta_{R}}{\tau_{R}}\leq
0$, and $G_{R}(\tilde{b})=\tilde{a}\leq 0$. Also $G_{L}(m)$ is non-increasing,
so $\tilde{a}\leq 0$ implies $G_{L}(\tilde{a})\geq G_{L}(0)$. Thus
$\tilde{b}\geq-\frac{\delta_{L}}{\tau_{L}}\geq 0\geq q_{R}$, so $b=\tilde{b}$.
Similarly $\tilde{a}\leq-\frac{\delta_{R}}{\tau_{R}}\leq 0\leq q_{L}$, so
$a=\tilde{a}$.

Conversely suppose $a=\tilde{a}$. Then $\tilde{a}=G_{R}(\tilde{b})\leq 0$, so
$\delta_{R}\leq 0$. Thus $G_{R}(m)$ is non-increasing, so
$\tilde{a}=G_{R}(\tilde{b})\leq G_{R}(0)$ implies $\tilde{b}\geq 0$. Thus
$\tilde{b}=G_{L}(\tilde{a})\geq 0$, so $\delta_{L}\leq 0$, as required (also
$b=\tilde{b}$ implies $\delta_{L}\leq 0$ and $\delta_{R}\leq 0$ in a similar
fashion). ∎

######  Lemma 4.7.

With the assumptions of Proposition 4.4, $-\tau_{L}<a$ and $b<-\tau_{R}$.

###### Proof.

For brevity we just show $-\tau_{L}<a$ ($b<-\tau_{R}$ can be shown similarly).
Using $\theta_{2}(\xi)>0$ and Lemma 4.5 we obtain

$a>\frac{1-\delta_{L}^{2}-\tau_{L}^{2}}{2\tau_{L}}>r_{L}\,.$ (4.12)

Thus if $\delta_{L}\geq 0$ then

$r_{L}+\tau_{L}=\frac{\tau_{L}}{2}\mathopen{}\mathclose{{}\left(1-\sqrt{1-\frac{4\delta_{L}}{\tau_{L}^{2}}}}\right)\geq
0,$

and so $a>-\tau_{L}$. Now suppose $\delta_{L}<0$. If $\delta_{R}\geq 0$ then
$a=0>-\tau_{L}$, while if $\delta_{R}<0$ then $a=\tilde{a}$ (by Lemma 4.6) and
$-\tau_{L}<\tilde{a}$ as shown in the proof of Lemma 4.6. ∎

###### Proof of Proposition 4.4.

We first show $G_{L}(a)\geq a$ and $G_{L}(b)\geq a$. If $\delta_{L}\leq 0$
then $G_{L}(m)\geq 0$ for all $m\in K$ (using Lemma 4.7), so certainly
$G_{L}(a)\geq a$ and $G_{L}(b)\geq a$. Now suppose $\delta_{L}>0$. In this
case $G_{L}(m)\geq m$ for all $r_{L}\leq m\leq q_{L}$ (i.e. at and between the
fixed points of $G_{L}$). Observe $r_{L}<a\leq q_{L}$ by (4.12) and the
definition of $a$, thus $G_{L}(a)\geq a$. Also $G_{L}(m)$ is increasing thus
$G_{L}(b)>G_{L}(a)\geq a$.

Next we show $G_{L}(a)\leq b$ and $G_{L}(b)\leq b$. If $\delta_{L}\geq 0$ then
we have $G_{L}(m)\leq 0$ for all $m\in K$ (using Lemma 4.7), so certainly
$G_{L}(a)\leq b$ and $G_{L}(b)\leq b$. Now suppose $\delta_{L}<0$. If
$\delta_{R}\leq 0$ then Lemma 4.6 implies $a=\tilde{a}$ and
$b=\tilde{b}=G_{L}(\tilde{a})$, so $b=G_{L}(a)$, if $\delta_{R}>0$ then $a=0$
and $b\geq G_{L}(0)$, so $b\geq G_{L}(a)$. Also $G_{L}(m)$ is decreasing thus
$G_{L}(b)<G_{L}(a)\leq b$.

Now from Lemma 4.2 we can conclude that $\Phi_{K}$ is invariant under $A_{L}$.
Invariance under $A_{R}$ can be proved in a similar fashion.

Next we prove expansion. By (4.8), $\theta_{2}(\xi)>0$ implies $H_{L}(a)>0$.
Also

$H_{L}(b)=\tau_{L}^{2}+\delta_{L}^{2}-1+2\tau_{L}b=H_{L}(a)+2\tau_{L}(b-a)$
(4.13)

is positive because $\tau_{L}>0$ and $b\geq a$. Thus $\Psi_{K}$ is expanding
for $A_{L}$ by Lemma 4.3. By a similar argument $\Psi_{K}$ is also expanding
for $A_{R}$. ∎

###### Proof of Theorem 2.2.

Choose any $\xi\in\Phi_{\rm trap}\cap\Phi_{\rm cone}$. By Proposition 3.4
there exists a trapping region $\Omega_{\varepsilon}$ for $f$. Then
$\bigcap_{n\geq 0}f^{n}(\Omega_{\varepsilon})$ is an attracting set and
contains a topological attractor $\Lambda$. By Proposition 4.4 there exists a
non-empty cone $\Psi_{K}$ that is invariant and expanding for both $A_{L}$ and
$A_{R}$ with some expansion factor $c>1$.

Choose any $z\in\Lambda$ and let $v\in\Psi_{K}$ be non-zero. The Lyapunov
exponent $\lambda(z,v)$ for $z$ in the direction $v$ is the limiting rate of
separation of the forward orbits of $z$ and $z+\Delta v$ for arbitrarily small
$\Delta>0$ [43]. If the forward orbit of $z$ does not intersect the switching
manifold then the derivative of the $n^{\rm th}$ iterate of $z$ under $f$ is
well-defined for all $n\geq 1$ and

$\lambda(z,v)=\limsup_{n\to\infty}\frac{1}{n}\ln\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\|{\rm
D}f^{n}(x)v}\right\|}\right).$ (4.14)

Observe

${\rm D}f^{n}(x)={\rm
D}f\mathopen{}\mathclose{{}\left(f^{n-1}(x)}\right)\cdots{\rm
D}f\mathopen{}\mathclose{{}\left(f(x)}\right){\rm D}f(x),$

where each of the $n$ matrices on the right-hand side is either $A_{L}$ or
$A_{R}$. By the invariance and expansion of $\Psi_{K}$,
$\mathopen{}\mathclose{{}\left\|{\rm D}f^{n}(x)v}\right\|\geq c^{n}\|v\|$ for
all $n$, so $\lambda(z,v)\geq\ln(c)>0$. If instead the forward orbit of $z$
intersects the switching manifold, $\lambda(z,v)$ can similarly be evaluated
and bounded using one-sided directional derivatives because $f$ is piecewise-
linear, see [40] for details. ∎

##  5 Further remarks on the parameter regions $\Phi_{\rm trap}$ and
$\Phi_{\rm cone}$.

In §2 we described two-dimensional cross-sections of $\Phi_{\rm trap}$ and
$\Phi_{\rm cone}$ defined by fixing the values of $\tau_{L}>0$ and
$\tau_{R}<0$. For the most part larger values of $\tau_{L}$ and $|\tau_{R}|$
yield smaller cross-sections of $\Phi_{\rm trap}$ and larger cross-sections of
$\Phi_{\rm cone}$, see Fig. 2 and Fig. 3. This is because with larger values
of $\tau_{L}$ and $|\tau_{R}|$ the map is more strongly expanding, hence less
amenable for the existence of a trapping region but more amenable for the
existence of an invariant expanding cone.

Figure 7:  The shaded region shows where cross-sections of $\Phi_{\rm trap}$
and $\Phi_{\rm cone}$, defined by fixing the values of $\tau_{L}$ and
$\tau_{R}$, are both non-empty. As we exit the shaded region through the
lower-right boundaries the cross-section of $\Phi_{\rm trap}$ vanishes; as we
exit through the upper and left boundaries the cross-section of $\Phi_{\rm
cone}$ vanishes.

Fig. 7 shows critical curves in the $(\tau_{L},\tau_{R})$-plane where the
cross-sections vanish entirely. To explain this figure we treat the critical
curves one by one. First consider $(\tau_{L},\tau_{R})$ at a point just above
the critical curve $\tau_{R}=\frac{-\tau_{L}}{\tau_{L}-1}$. Here the
$\Phi_{\rm trap}$ cross-section has three vertices, $P^{(1)}$, $P^{(2)}$, and
$P^{(3)}$, as shown in Fig. 8-a. It is a simple exercise to show that as
parameters are varied each vertex reaches the origin
$(\delta_{L},\delta_{R})=(0,0)$ on the critical curve. For instance the upper
vertex is where $\phi_{3}(\xi)=0$ and $\phi_{4}(\xi)=0$ intersect at
$P^{(1)}=\mathopen{}\mathclose{{}\left(0,\frac{(\tau_{R}-(\tau_{R}+1)\tau_{L})\tau_{L}}{1-\tau_{L}}}\right)$
and solving $P^{(1)}_{2}=0$ gives $\tau_{R}=\frac{-\tau_{L}}{\tau_{L}-1}$.
Thus here the $\Phi_{\rm trap}$ cross-section contracts to a point and
vanishes.

With instead $(\tau_{L},\tau_{R})$ at a point just to the left of
$\tau_{R}=\frac{-1}{\tau_{L}-2}$, the $\Phi_{\rm trap}$ cross-section has
three vertices at different points $P^{(4)}$, $P^{(5)}$, and $P^{(6)}$, as
shown in Fig. 8-b. Explicit calculations reveal that each vertex reaches the
corner $(\delta_{L},\delta_{R})=(\tau_{L}+1,\tau_{R}-1)$ when
$\tau_{R}=\frac{-1}{\tau_{R}-2}$. For instance,
$P^{(4)}=\mathopen{}\mathclose{{}\left(\frac{(\tau_{L}\tau_{R}-\tau_{R}+1)(\tau_{R}-1)}{\tau_{R}^{2}},\tau_{R}-1}\right)$,
and solving $P^{(4)}_{1}=\tau_{L}+1$ gives $\tau_{R}=\frac{-1}{\tau_{R}-2}$.
Thus here the $\Phi_{\rm trap}$ cross-section again vanishes.

|
---|---
(a) $\tau_{L}=2.7,\tau_{R}=-1.5$ | (b) $\tau_{L}=2.4,\tau_{R}=-2.1$
|
(c) $\tau_{L}=0.98,\tau_{R}=-1.5$ | (d) $\tau_{L}=1.6,\tau_{R}=-0.98$
Figure 8: Further sample cross-sections of $\Phi_{\rm trap}$ (panels (a) and
(b)) and $\Phi_{\rm cone}$ (panels (c) and (d)). These use the parameter
points $(\tau_{L},\tau_{R})$ shown in Fig. 7.

With $\tau_{L}$ just less than $1$ and $\tau_{R}<-1$ the $\Phi_{\rm cone}$
cross-section appears as in Fig. 8-c. As parameters are varied the $\Phi_{\rm
cone}$ cross-section vanishes when the vertices
$P^{(7)}=\big{(}\tau_{L}(\tau_{R}+1),-(\tau_{R}+1)\big{)}$ and
$P^{(8)}=\big{(}-\sqrt{1-\tau_{L}^{2}},-(\tau_{R}+1)\big{)}$ coincide. Solving
$P^{(7)}_{1}=P^{(8)}_{1}$ yields the critical curve
$\tau_{R}=-\frac{\tau_{L}+\sqrt{1-\tau_{L}^{2}}}{\tau_{L}}$ of Fig. 7.
Similarly with $\tau_{L}>1$ and $\tau_{R}$ just greater than $-1$ the
$\Phi_{\rm cone}$ cross-section appears as in Fig. 8-d. The cross-section
vanishes when $P^{(9)}$ and $P^{(10)}$ coincide on
$\tau_{L}=-\frac{-\tau_{R}+\sqrt{1-\tau_{R}^{2}}}{\tau_{R}}$.

Figure 9: A two-parameter bifurcation diagram of (1) with $\tau_{L}=1.7$ and
$\tau_{R}=-1.9$. The boundary of $\Phi_{\rm trap}\cap\Phi_{\rm cone}$ is shown
in yellow. The coloured regions show the result of a numerical simulation
(white: no attractor; green: chaotic attractor; other colours: periodic
attractor). Some periods are indicated (e.g. in the large pink region the map
has a stable period-$2$ solution. In the blue strip at the top the fixed point
$X$ is stable.

The geometry and topology of cross-sections of $\Phi_{\rm trap}\cap\Phi_{\rm
cone}$ admit many possibilities and a complete analysis is beyond the scope of
this paper. Here we examine one example, Fig. 9. Here the cross-section of
$\Phi_{\rm trap}\cap\Phi_{\rm cone}$ is bounded by the following curves (going
anticlockwise): $\delta_{L}=\tau_{L}-1$, $\delta_{R}=-\tau_{R}-1$,
$\phi_{4}(\xi)=0$, $\phi_{3}(\xi)=0$, $\phi_{5}(\xi)=0$, and
$\theta_{2}(\xi)=0$ (which has a kink at $\delta_{L}=0$). The first two of
these curves are boundaries of our overall parameter region $\Phi$, the next
three curves are boundaries of $\Phi_{\rm trap}$, and the last curve is a
boundary of $\Phi_{\rm cone}$.

Fig. 9 also shows the result of a simple numerical simulation to investigate
the nature of the attractor. For each point in a $300\times 300$ equispaced
grid of $(\delta_{L},\delta_{R})$ values, we computed $10^{7}$ iterates of the
forward orbit using a random initial condition. Green points are where an
estimate of the maximal Lyapunov exponent was positive, white points are where
the orbit appeared to diverge (its norm exceeded $10^{4}$), and other points
are where there exists a stable periodic solution (determined by solving for
periodic solutions exactly).

Note the numerical simulation gives an imperfect picture. For example, we
believe no attractor exists immediately to the left of the heteroclinic
bifurcation $\phi_{5}(\xi)=0$, yet some green points are present here (with
small $\delta_{L}>0$ and $-3<\delta_{R}<-2$) because orbits often experience a
long transient before diverging and $10^{7}$ iterations are insufficient to
detect this.

Nevertheless, the numerics effectively highlights the fact that three of the
boundaries of $\Phi_{\rm trap}\cap\Phi_{\rm cone}$ are bifurcations where the
chaotic attractor is destroyed, so these boundaries cannot be improved upon.
As we cross $\delta_{R}=-\tau_{R}-1$ the fixed point $X$ becomes stable,
$\phi_{4}(\xi)=0$ is a homoclinic bifurcation where the attractor is destroyed
[4], and $\phi_{5}(\xi)=0$ is a heteroclinic bifurcation where the attractor
is destroyed. Elsewhere the attractor is destroyed in a variety of other
global bifurcations. Above $\phi_{3}(\xi)=0$ and to the right of
$\delta_{L}=\tau_{L}-1$ the trapping region construction of §3 fails. An
alternate construction that partially deals with this is described in [40].
Below $\theta_{2}(\xi)=0$ the cone $\Psi_{K}$ of §4 fails to be expanding. For
some parameter combinations below $\theta_{2}(\xi)=0$ it is possible to
construct an invariant expanding cone, and hence verify the presence of chaos,
by using an induced map [16].

##  6 Discussion

We have extended the constructive proof of [15] for robust chaos in the
border-collision normal form to the orientation-reversing and non-invertible
settings. Specifically we identified an open parameter region $\Phi_{\rm
trap}$, where the map has a trapping region, and an open parameter region
$\Phi_{\rm cone}$, where it has an invariant expanding cone, see Figs. 2 and
3. Throughout $\Phi_{\rm trap}\cap\Phi_{\rm cone}$ the map has an attractor
with a positive Lyapunov exponent, Theorem 2.2.

In Fig. 9 we considered a typical two-dimensional cross-section of parameter
space. Numerical results showed robust chaos throughout $\Phi_{\rm
trap}\cap\Phi_{\rm cone}$, corroborating Theorem 2.2 as expected. The robust
chaos terminates at three boundaries of $\Phi_{\rm trap}\cap\Phi_{\rm cone}$:
$\delta_{R}=-\tau_{R}-1$, $\phi_{4}(\xi)=0$, and $\phi_{5}(\xi)=0$, and
appears to persist beyond the other boundaries. We expect our construction
could be adapted to verify robust chaos beyond these boundaries, and already
this has been achieved in some cases [16, 40].

Notice in Fig. 9 the cross-section of $\Phi_{\rm trap}\cap\Phi_{\rm cone}$
includes a neighbourhood of $(\delta_{L},\delta_{R})$. This is the case for
many values of $\tau_{L}$ and $\tau_{R}$ and is a significant achievement of
this paper because it shows robust chaos is not lost as we cross from the
orientation-preserving setting to the orientation-reversing and non-invertible
settings. Thus the presence of robust chaos is not dependent on the global
topological properties of the map. Moreover, this provides a path for robust
chaos to be demonstrated in higher dimensional maps. The $n$-dimensional
border-collision normal form [36] can have two saddle fixed points: $X$ with
an eigenvalue $\lambda_{R}^{u}<-1$, and $Y$ with an eigenvalue
$\lambda_{L}^{u}>1$. If all other eigenvalues associated with $X$ and $Y$ are
sufficiently small in absolute value (which in two dimensions means
$(\delta_{L},\delta_{R})$ is sufficiently close to $(0,0)$), and with
appropriate constraints on the values of $\lambda_{R}^{u}$ and
$\lambda_{L}^{u}$, we believe the map must have a chaotic attractor, and this
is a tantalising avenue for future research.

##  7 Acknowledgments

This work was supported by Marsden Fund contract MAU1809, managed by Royal
Society Te Apārangi.

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