Inclusion of higher-order terms in the border-collision normal form: persistence of chaos and applications to power converters.

While most engineering systems are designed to operate outside chaotic parameter regimes, in some situations the presence of chaos is advantageous. Examples include optical resonators for which additional energy can be stored when photons follow chaotic trajectories as without periodic motion they become trapped for longer times [1]. In mechanical energy harvesters the presence of chaos allows high-energy modes of operation to be stabilised with only a small control force [2]. Also, power converters, when run chaotically, have the advantage of increased electromagnetic compatibility due to broad spectral characteristics [3]. Regardless of whether or not chaos is desired, it is extremely helpful to understand when and why it occurs.

Power converters, and many other engineering systems, function by switching between different modes of operation. There is a growing understanding of the creation of chaos in such systems through use of the border-collision normal form (BCNF). This is a piecewise-linear (or more precisely, piecewise-affine) map that models the dynamics near parameter values at which a fixed point intersects a boundary between two modes of operation [4, 5]. The piecewise-linear nature of the BCNF makes it possible to prove results about chaotic attractors and their persistence, a phenomenon called robust chaos [6, 7, 8].

The BCNF is obtained using coordinate transformations to make the lowest order terms of the map as simple as possible. The map is then truncated, ignoring all terms that are nonlinear with respect to variables and parameters. As such, it is by no means clear that results for chaotic attractors in the BCNF carry over to the full nonlinear models originally being considered. In this paper we prove a persistence result showing that, under certain conditions, the existence of a chaotic attractor in the BCNF implies the existence of a chaotic attractor in the corresponding full model regardless of the nature of the nonlinear terms that have been neglected to form the BCNF. This shows that chaotic attractors created in border-collision bifurcations typically persist for an open interval of parameters beyond the bifurcation and justifies our use of the BCNF for determining when chaotic attractors are created.

In earlier work [9] we described a computational method for determining when the two-dimensional BCNF is chaotic by establishing the existence of a trapping region in phase space and a contracting-invariant expanding cone in tangent space. The trapping region guarantees the existence of an attractor, while the cone ensures it has a positive Lyapunov exponent. In this paper we use the robustness of these objects to demonstrate persistence with respect to higher-order terms. The most difficult technical issue we have to overcome is in showing that the higher-order terms do not cause orbits of the map to accumulate new symbolic itineraries that cannot be handled by the cone.

The remainder of the paper is arranged as follows. In section 2 we describe the two-dimensional BCNF and its relation to nonlinear models. We then state our persistence result in section 3. In section 4 we provide a detailed description of the geometric structure of the phase space of the BCNF that we use in section 5 to prove the persistence result. The result, together with computational methods of [9], are then applied to a model of power converters with pulse-width modulated control, section 6. We believe this verifies, for the first time, the long-standing belief from numerical simulations that chaos occurs robustly in these types of systems. Finally section 7 contains concluding remarks.

The study of border-collision bifurcations has a long history dating back to at least Feigin and coworkers in the context of relay control [10, 11]. The two-dimensional BCNF was introduced by Nusse and Yorke in [4] motivated by observations of anomalous behaviour in piecewise-linear economics models [12]. The BCNF has since been shown to display a remarkably rich array of dynamics, such as robust chaos [6], multi-stability [13], multi-dimensional attractors [14], and resonance regions with sausage-string structures [15, 16]. The BCNF is relevant for describing a wide-range of physical phenomena, another example being mechanical systems with stick-slip friction [17]; see [18] for a recent review.

Let $y\mapsto f(y;\mu)$ be a continuous, piecewise-smooth map with variable $y=(y_{1},y_{2})\in\mathbb{R}^{2}$ and parameter $\mu\in\mathbb{R}$. We are interested in the dynamics local to a border-collision bifurcation where a fixed point of $f$ collides with a switching manifold as $\mu$ is varied. Assuming the bifurcation occurs at single smooth switching manifold, only two pieces of the map are relevant to the local dynamics. By choosing coordinates so that the switching manifold is $y_{1}=0$, we can assume $f$ has the form

$$f(y;\mu)=\begin{cases}f_{L}(y;\mu),&y_{1}\leq 0,\\ f_{R}(y;\mu),&y_{1}\geq 0,\end{cases}$$

(2.1)

where $f_{L}$ and $f_{R}$ are $C^{1}$.

Suppose the border-collision bifurcation occurs at $y=(0,0)={\bf 0}$ when $\mu=0$. By continuity, ${\bf 0}$ is a fixed point of both $f_{L}$ and $f_{R}$ when $\mu=0$:

$$f_{L}({\bf 0};0)=f_{R}({\bf 0};0)={\bf 0}.$$

(2.2)

From the map $f$ we can extract the four key values

$$\begin{split}\tau_{L}&={\rm trace}\big{(}{\rm D}f_{L}({\bf 0};0)\big{)},\\ \delta_{L}&={\rm det}\big{(}{\rm D}f_{L}({\bf 0};0)\big{)},\\ \tau_{R}&={\rm trace}\big{(}{\rm D}f_{R}({\bf 0};0)\big{)},\\ \delta_{R}&={\rm det}\big{(}{\rm D}f_{R}({\bf 0};0)\big{)},\end{split}$$

(2.3)

which determine the eigenvalues associated with ${\bf 0}$ for the two smooth components of (2.1). We can then use these values to form the piecewise-linear map

$$g(x)=\begin{cases}\begin{bmatrix}\tau_{L}&1\\ -\delta_{L}&0\end{bmatrix}x+\begin{bmatrix}1\\ 0\end{bmatrix},&x_{1}\leq 0,\\[12.80373pt] \begin{bmatrix}\tau_{R}&1\\ -\delta_{R}&0\end{bmatrix}x+\begin{bmatrix}1\\ 0\end{bmatrix},&x_{1}\geq 0.\end{cases}$$

(2.4)

This is the two-dimensional BCNF, except often $\mu$-dependence is retained in the constant term. In the remainder of this section we explain how (2.4) can be used to describe the local dynamics of (2.1) near the border-collision bifurcation.

We first write the components of (2.1) as

$$\begin{split}f_{L}(y;\mu)&=\begin{bmatrix}a^{L}_{11}&a_{12}\\ a^{L}_{21}&a_{22}\end{bmatrix}y+\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}\mu+\mathpzc{o}\mathopen{}\mathclose{{}\left(\|y\|+|\mu|}\right),\\ f_{R}(y;\mu)&=\begin{bmatrix}a^{R}_{11}&a_{12}\\ a^{R}_{21}&a_{22}\end{bmatrix}y+\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}\mu+\mathpzc{o}\mathopen{}\mathclose{{}\left(\|y\|+|\mu|}\right),\end{split}$$

(2.5)

where the two expressions share several coefficients due to the assumed continuity of (2.1) on $y_{1}=0$. By dropping the higher-order terms we obtain the piecewise-linear map

$$h(y;\mu)=\begin{cases}\begin{bmatrix}a^{L}_{11}&a_{12}\\ a^{L}_{21}&a_{22}\end{bmatrix}y+\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}\mu,&y_{1}\leq 0,\\[12.80373pt] \begin{bmatrix}a^{R}_{11}&a_{12}\\ a^{R}_{21}&a_{22}\end{bmatrix}y+\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}\mu,&y_{1}\geq 0.\end{cases}$$

(2.6)

This approximates (2.1) in the following sense: for any $\varepsilon>0$ there exists $\delta>0$ such that if $\|y\|+|\mu|<\delta$ then $\big{\|}f(y;\mu)-h(y;\mu)\big{\|}<\varepsilon\mathopen{}\mathclose{{}\left(\|y\|+|\mu|}\right)$.

The piecewise-linear map (2.6) satisfies the identity $h(\alpha y;\alpha\mu)=\alpha h(y;\mu)$ for any $\alpha>0$. For this reason the magnitude of $\mu$ only affects the spatial scale of the dynamics of (2.6): if $\Lambda\subset\mathbb{R}^{2}$ is an invariant set of $h$ for some $\mu_{0}$, then $\alpha\Lambda$ is an invariant set of $h$ with $\mu=\alpha\mu_{0}$ for all $\alpha>0$. In view of this scaling property, we can convert (2.6) to (2.4) for any $\mu>0$. This is achieved via the change of variables

$$x=\frac{1}{\gamma\mu}\begin{bmatrix}1&0\\ -a_{22}&a_{12}\end{bmatrix}y+\frac{1}{\gamma}\begin{bmatrix}0\\ a_{22}b_{1}-a_{12}b_{2}\end{bmatrix},$$

(2.7)

where

$$\gamma=(1-a_{22})b_{1}+a_{12}b_{2}\,,$$

(2.8)

and is valid assuming

	$\displaystyle a_{12}$	$\displaystyle\neq 0,$		(2.9)
	$\displaystyle\gamma$	$\displaystyle>0.$		(2.10)

The condition $a_{12}\neq 0$ ensures (2.7) is invertible (if $a_{12}=0$ then (2.6) can be partly decoupled [18]). We require $\gamma\neq 0$ so that $\mu$ unfolds the border-collision bifurcation in a generic fashion, while $\gamma>0$ ensures the left and right components of (2.6) transform to their respective components in (2.4). The case $\gamma<0$ can be accommodated by simply redefining $\mu$ as $-\mu$.

The transformation (2.7) performs a similarity transform to the Jacobian matrices of the components of the map, thus it preserves their traces and determinants. For this reason the values (2.3) were used to construct (2.4) — notice how $\tau_{L}$, $\delta_{L}$, $\tau_{R}$, and $\delta_{R}$ are the traces and determinants of the Jacobian matrices of the two components of (2.4).

In summary, for any map of the form (2.1) that has a border-collision bifurcation at $\mu=0$, we can use the values (2.3) to form the BCNF (2.4). Then, if the conditions (2.9)–(2.10) are satisfied, the BCNF is affinely conjugate to the piecewise-linear approximation to (2.1) for any $\mu>0$. Intuitively this approximation should be reasonable for sufficiently small values of $\mu$. Our persistence result in the next section gives conditions under which the approximation can indeed be justified.

Our main result, Theorem 3.11 below, links the existence of chaotic attractors of the piecewise-linear BCNF (2.4) to attractors of the nonlinear map (2.1) for small $\mu>0$. To state the result we need some preliminary definitions and conditions.

Let

$\displaystyle g_{L}(x)$

$\displaystyle=A_{L}x+\begin{bmatrix}1\\ 0\end{bmatrix},$

$\displaystyle g_{R}(x)$

$\displaystyle=A_{R}x+\begin{bmatrix}1\\ 0\end{bmatrix},$

denote the left and right components of the BCNF, $g$, where

$\displaystyle A_{L}$

$\displaystyle=\begin{bmatrix}\tau_{L}&1\\ -\delta_{L}&0\end{bmatrix},$

$\displaystyle A_{R}$

$\displaystyle=\begin{bmatrix}\tau_{R}&1\\ -\delta_{R}&0\end{bmatrix}.$

Theorem 3.11 requires the assumptions

	$\displaystyle\delta_{L}$	$\displaystyle>0,$		(3.1)
	$\displaystyle\delta_{R}$	$\displaystyle>\frac{\tau_{R}^{2}}{4},$		(3.2)
	$\displaystyle g^{-i}({\bf 0})_{2}$	$\displaystyle<0,\quad\text{for all $i\geq 1$},$		(3.3)

where the subscript in (3.3) indicates we are looking at the second component of the vector $g^{-i}({\bf 0})$. Conditions (3.1) and (3.2) imply that $g$ is a homeomorphism, so $g^{-1}$ exists and (3.3) is well-defined. Condition (3.3) ensures the backwards orbit of the origin does not enter the closed upper half-plane $\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{2}\geq 0}\right\}$. In fact, the backwards orbit is constrained to the fourth quadrant of $\mathbb{R}^{2}$ so is governed purely by $g_{R}$ and simply converges to the unique fixed point of $g_{R}$ (a repelling focus), see Fig. 1. Together conditions (3.1)–(3.3) allow us to divide phase space by the number of iterations required to cross the switching manifold

$$\Sigma=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}=0}\right\},$$

and this is achieved in section 4. Having a precise understanding of this division is critical to our proof of Theorem 3.11 presented in section 5.

To motivate the following definitions, consider the forward orbit of a point $x\in\mathbb{R}^{2}$ under $g$. Suppose it maps under $g_{L}$ $p$ times, then under $g_{R}$ $q$ times. That is, $g^{p+q}(x)=g_{R}^{q}\mathopen{}\mathclose{{}\left(g_{L}^{p}(x)}\right)$, hence ${\rm D}g^{p+q}(x)=A_{R}^{q}A_{L}^{p}$, assuming no iterates lie on $\Sigma$ where $g$ is non-differentiable. For orbits that go back and forth across $\Sigma$ without landing on $\Sigma$, which includes almost all orbits in the chaotic attractors we wish to analyse, derivatives of $g^{n}(x)$ for large $n$ can be expressed as products of matrices of the form $A_{R}^{q}A_{L}^{p}$. Consequently we can estimate Lyapunov exponents based on bounds for the values of $p$ and $q$.

Let

	$\displaystyle\Pi_{L}$	$\displaystyle=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}\leq 0}\right\},$
	$\displaystyle\Pi_{R}$	$\displaystyle=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}\geq 0}\right\},$

denote the closed left and right half-planes.

Now define the regions

	$\displaystyle\Phi_{L}$	$\displaystyle=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}<0,\,x_{2}\leq 0}\right\},$		(3.4)
	$\displaystyle\Phi_{R}$	$\displaystyle=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}>0,\,x_{2}\geq 0}\right\}.$		(3.5)

The following result shows that when an orbit crosses $\Sigma$ from right to left, it must arrive at a point in $\Phi_{L}$. Moreover, $\Phi_{L}$ is exactly the set of all such points. The set $\Phi_{R}$ admits the same characterisation for orbits crossing $\Sigma$ from left to right. The result follows immediately from the observation that $g^{-1}(x)_{1}$ and $x_{2}$ have opposite signs.

Now given a set $\Omega\subset\mathbb{R}^{2}$, suppose there exist numbers $1\leq p_{\rm min}\leq p_{\rm max}$ and $1\leq q_{\rm min}\leq q_{\rm max}$ such that

	$\displaystyle\chi_{L}(x)$	$\displaystyle\geq p_{\rm min}\,,\qquad\text{for all}~{}x\in\Omega\cap\Phi_{L}\,,$		(3.6)
	$\displaystyle\chi_{L}(x)$	$\displaystyle\leq p_{\rm max}\,,\qquad\text{for all}~{}x\in\Omega\cap\Pi_{L}\,,$		(3.7)
	$\displaystyle\chi_{R}(x)$	$\displaystyle\geq q_{\rm min}\,,\qquad\text{for all}~{}x\in\Omega\cap\Phi_{R}\,,$		(3.8)
	$\displaystyle\chi_{R}(x)$	$\displaystyle\leq q_{\rm max}\,,\qquad\text{for all}~{}x\in\Omega\cap\Pi_{R}\,.$		(3.9)

That is, any point in $\Omega\cap\Pi_{L}$ requires at most $p_{\rm max}$ iterations to cross $\Sigma$, and at least $p_{\rm min}$ iterations if its preimage lies in $\Pi_{R}$. The numbers $q_{\rm min}$ and $q_{\rm max}$ similarly bound the number of iterations required to cross $\Sigma$ from right to left.

For the given set $\Omega$, let

$${\bf M}_{\Omega}=\mathopen{}\mathclose{{}\left\{A_{R}^{q}A_{L}^{p}\,\big{|}\,p_{\rm min}\leq p\leq p_{\rm max},\,q_{\rm min}\leq q\leq q_{\rm max}}\right\}.$$

(3.10)

In Theorem 3.11, $\Omega$ is be assumed to be a trapping region for $g$, that is, $g(\Omega)\subset{\rm int}(\Omega)$, where ${\rm int}(\cdot)$ denotes interior. Also, to ensure a positive Lyapunov exponent, we assume ${\bf M}_{\Omega}$ has a contracting-invariant, expanding cone. This is defined as follows and illustrated in Fig. 2.

Finally we state the main result. We write $B_{r}(x)$ for the ball of radius $r>0$ centred at $x\in\mathbb{R}^{2}$.

Notice $\Lambda_{\mu}$ is chaotic in the sense of a positive Lyapunov exponent, as implied by (3.11).

In this section we describe the geometry of the regions of the left half-plane with different values of $\chi_{L}$ and regions of the right half-plane with different values of $\chi_{R}$ for the BCNF (2.4). The ways in which their images intersect can be used to establish conditions under which chaotic attractors exist [9]. The regions corresponding to values of $\chi_{L}$ were described in [19], so for these we simply state results without proof. The regions corresponding to values of $\chi_{R}$ can be described in a similar way; details of these calculations are provided in Appendix A.

First consider the sets $D_{p}$. As shown by Proposition 6.6 of [19], each $D_{p}$ is bounded by the lines $g_{L}^{-p}(\Sigma)$, $g_{L}^{-(p-1)}(\Sigma)$, and $\Sigma$, Fig. 3. If the line $g_{L}^{-p}(\Sigma)$ is not vertical, let $m_{p}$ denote its slope and $c_{p}$ denote its $x_{2}$-intercept, i.e.,

$$g_{L}^{-p}(\Sigma)=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{2}=m_{p}x_{1}+c_{p}}\right\}.$$

(4.3)

The next result is a simple consequence of results obtained in [19]. Recall $\Phi_{L}$ is the third quadrant (3.4).

For example with $(\tau_{L},\delta_{L})=(1.4,1)$, as in Fig. 3, we have $p^{*}=3$. Fig. 4 shows how the value of $p^{*}$, and hence the values of $p$ for which $D_{p}\cap\Phi_{L}\neq\varnothing$, depends on the values of $\tau_{L}$ and $\delta_{L}$. Between curves where $m_{i-1}=0$ and $m_{i}=0$, we have $D_{p}\cap\Phi_{L}\neq\varnothing$ if and only if $p\in\{1,2,\ldots i+1\}$. As $p\to\infty$ these curves accumulate on $\delta_{L}=\frac{\tau_{L}^{2}}{4}$ where $A_{L}$ has repeated eigenvalues. With $\delta_{L}>\frac{\tau_{L}^{2}}{4}$, $p^{*}=\infty$ and every $D_{p}\cap\Phi_{L}$ is non-empty.

In regards to Theorem 3.11, while the values of $p_{\rm min}$ and $p_{\rm max}$ depend on the particular trapping region $\Omega$ being considered, Lemma 4.1 tells us that the value of $p_{\rm min}$ could always be as low as $1$, while the value of $p_{\rm max}$ cannot be more than $p^{*}+1$.

Next we note that the boundaries of the $D_{p}$ intersect the $x_{1}$-axis transversally. This is the case because each $g_{L}^{-p}(\Sigma)$ is a line intersecting the $x_{2}$-axis at $(0,c_{p})$ with $c_{p}<0$, see Lemma 6.2 of [19]. These transversal intersections are used to argue persistence in section 5.

We now turn our attention to the sets $E_{q}$. To characterise these we assume $\tau_{R}$ and $\delta_{R}$ satisfy (3.2) and (3.3). Condition (3.2) implies $g_{R}$ has the unique fixed point

$$x^{R}=\frac{1}{\delta_{R}-\tau_{R}+1}\begin{bmatrix}1\\ -\delta_{R}\end{bmatrix},$$

(4.4)

which lies in the fourth quadrant. Condition (3.3) ensures that the set of $q$ for which $E_{q}\cap\Phi_{R}\neq\varnothing$ admits a relatively simple characterisation and that the boundaries of the $E_{q}$ intersect the $x_{1}$-axis transversally.

By an analogous argument to that for the sets $D_{p}$, the lines $g_{R}^{-q}(\Sigma)$, $g_{R}^{-(q-1)}(\Sigma)$, and $\Sigma$ form the boundaries of the $E_{q}$. However, their layout is more complicated than that of the $D_{p}$. Fig. 3 shows a typical example.

If the line $g_{R}^{-q}(\Sigma)$ is not vertical, let $n_{q}$ denote its slope and $d_{q}$ denote its $x_{2}$-intercept, i.e.,

$$g_{R}^{-q}(\Sigma)=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{2}=n_{q}x_{1}+d_{q}}\right\}.$$

(4.5)

If $g_{R}^{-q}(\Sigma)$ is vertical we write $n_{q}=\infty$ (in this case $g_{R}^{-q}(\Sigma)$ is the line $x_{1}=-\frac{d_{q-1}}{\delta_{R}}$, see Appendix A).

Lemma 4.3 is proved in Appendix A. The values of $q^{*}$ and $q^{**}$ are determined by the values of $\tau_{R}$ and $\delta_{R}$ as indicated in Fig. 5. As we move about the $(\tau_{R},\delta_{R})$-plane, the value of $q^{*}$ changes by one when we cross a curve where $n_{q}=0$, and the value of $q^{**}$ changes by one when we cross a curve where $d_{q}=0$. These curves accumulate on $\delta_{R}=\frac{\tau_{R}^{2}}{4}$ past which $A_{R}$ has real eigenvalues and condition (3.2) is no longer satisfied. In Fig. 5, condition (3.3) is satisfied above the piecewise-smooth black curve.

In regards to Theorem 3.11, Lemma 4.3 implies $q_{\rm min}\geq q^{*}$ and $q_{\rm max}\leq q^{**}$. Analogous to Lemma 4.2 we have the following result (proved in Appendix A).

The proof is divided into six steps.

Step 1 — Robustness of the cone.
Let $C$ be a contracting-invariant, expanding cone for ${\bf M}_{\Omega}$ of (3.10) and let $c>1$ be a corresponding expansion factor as in Definition 3.2. Here we show that $C$ is an invariant expanding cone for a collection of matrices $\hat{A}_{R}^{q}\hat{A}_{L}^{p}$ that are sufficiently close to some matrix $A_{R}^{q}A_{L}^{p}$ in ${\bf M}_{\Omega}$. Note, the expansion factor will be $\frac{c+1}{2}$ instead of $c$.

We first define a function $\mathcal{F}$ from the space of $2\times 2$ matrices to the space of subsets of $\mathbb{R}^{2}$ as follows: given a $2\times 2$ matrix $N$, let

$$\mathcal{F}(N)=\mathopen{}\mathclose{{}\left\{\frac{Nv}{\|v\|}\,\middle|\,v\in C\setminus\{{\bf 0}\}}\right\}.$$

By the definition of an contracting-invariant expanding cone, given $M\in{\bf M}_{\Omega}$ we have $\mathcal{F}(M)\subset{\rm int}(C)$ and $\mathcal{F}(M)\cap B_{c}({\bf 0})=\varnothing$. The function $\mathcal{F}$ is continuous, so if $N$ is sufficiently close to $M$ then $\mathcal{F}(N)\subset C$ and $\mathcal{F}(N)\cap B_{\frac{c+1}{2}}({\bf 0})=\varnothing$. That is, for all $v\in C$,

$$\begin{split}Nv&\in C,\\ \|Nv\|&\geq\frac{c+1}{2}\|v\|.\end{split}$$

(5.1)

To be precise, there exists $\eta_{1}>0$ such that if $\|N-M\|<\eta_{1}$ for some $M\in{\bf M}_{\Omega}$ then (5.1) is satisfied for all $v\in C$. Furthermore, there exists $\eta_{2}>0$ such that $\|\hat{A}_{L}-A_{L}\|<\eta_{2}$ and $\|\hat{A}_{R}-A_{R}\|<\eta_{2}$ implies

$$\big{\|}\hat{A}_{R}^{q}\hat{A}_{L}^{p}-A_{R}^{q}A_{L}^{p}\big{\|}<\eta_{1},$$

(5.2)

for all $p_{\rm min}\leq p\leq p_{\rm max}$ and $q_{\rm min}\leq q\leq q_{\rm max}$, and that any such $\hat{A}_{L}$ and $\hat{A}_{R}$ are invertible (which is possible because $A_{L}$ and $A_{R}$ are invertible by (3.1) and (3.2)).

Step 2 — Bounds related to $g(\Omega)$.
By the definition of $p_{\rm min}$, $p_{\rm max}$, $q_{\rm min}$, and $q_{\rm max}$, see (3.6)–(3.9), we have $\Omega\cap\Phi_{L}\subset\bigcup_{p=p_{\rm min}}^{p_{\rm max}}D_{p}$ and $\Omega\cap\Phi_{R}\subset\bigcup_{q=q_{\rm min}}^{q_{\rm max}}E_{q}$. In this step we use the results of section 4 and the fact that $\Omega$ maps to its interior under $g$ to control the behaviour of points inside and near $g(\Omega)$.

Since $g(\Omega)\subset{\rm int}(\Omega)$ is compact there exists $\varepsilon_{1}>0$ such that

$$B_{\varepsilon_{1}}(g(\Omega))\subset\Omega,$$

(5.3)

as illustrated in Fig. 6. Now consider the set $U=B_{\varepsilon_{1}}\mathopen{}\mathclose{{}\left(g(\Omega)\cap\Phi_{L}}\right)\cap\Pi_{L}$. This set is contained in $\Omega\cap\Phi_{L}$ except has some points just above the negative $x_{1}$-axis. By Lemma 4.2, we can assume $\varepsilon_{1}$ is small enough that $U$ does not intersect any ‘other’ regions $D_{p}$, that is $U\subset\bigcup_{p=p_{\rm min}}^{p_{\rm max}}D_{p}$. Further, by shrinking this set by a small amount, the result will be bounded away from the other regions $D_{p}$. That is, there exists $\varepsilon_{2}>0$ such that for all $x\in B_{\frac{\varepsilon_{1}}{2}}\mathopen{}\mathclose{{}\left(g(\Omega)\cap\Phi_{L}}\right)\cap\Pi_{L}$:

1. i)

   $g_{L}^{p}(x)_{1}<-\varepsilon_{2}$ for all $p=1,\ldots,p_{\rm min}-1$, and

2. ii)

   if $g_{L}^{p}(x)_{1}\leq 0$ for all $p=p_{\rm min},\ldots,p_{\rm max}-1$, then $g_{L}^{p_{\rm max}}(x)_{1}>\varepsilon_{2}$.

Also, $\Omega\cap\Pi_{L}\subset\bigcup_{p=1}^{p_{\rm max}}D_{p}$ by (3.7). Thus by (5.3) we can assume $\varepsilon_{2}$ is small enough that for all $x\in B_{\frac{\varepsilon_{1}}{2}}(g(\Omega))$:

1. iii)

   if $g_{L}^{p}(x)_{1}\leq 0$ for all $p=1,\ldots,p_{\rm max}-1$, then $g_{L}^{p_{\rm max}}(x)_{1}>\varepsilon_{2}$.

In view of Lemma 4.4 we can assume $\varepsilon_{1}$ and $\varepsilon_{2}$ are small enough that analogous bounds also hold for $g_{R}^{q}(x)_{1}$.

Step 3 — Change of coordinates.
Here we apply the coordinate change (2.7) for converting $f$ to $g$ plus higher order terms. For small $\mu>0$ this coordinate change represents a spatial blow-up of phase space near the origin. Since the higher order terms are small near the origin we are able to control the higher order terms by assuming $\mu$ is sufficiently small. Here we also let $r>0$ be such that $\Omega\subset B_{r}({\bf 0})$.

We first decompose (2.7) into the spatial blow-up

$$x=\frac{\tilde{y}}{\gamma\mu},$$

(5.4)

and the bounded coordinate change

$$\tilde{y}=\phi_{\mu}(y)=\begin{bmatrix}y_{1}\\ -a_{22}y_{1}+a_{12}y_{2}+(a_{22}b_{1}-a_{12}b_{2})\mu\end{bmatrix}.$$

(5.5)

Notice (5.5) is invertible because $a_{12}\neq 0$, (2.9). The coordinate change (5.5) transforms $f$ to a map

$$\tilde{f}(\tilde{y};\mu)=\begin{cases}\tilde{f}_{L}(\tilde{y};\mu),&\tilde{y}_{1}\leq 0,\\ \tilde{f}_{R}(\tilde{y};\mu),&\tilde{y}_{1}\geq 0,\end{cases}$$

(5.6)

where

$$\begin{split}\tilde{f}_{L}(\tilde{y};\mu)&=A_{L}\tilde{y}+\begin{bmatrix}\gamma\mu\\ 0\end{bmatrix}+E_{L}(\tilde{y};\mu),\\ \tilde{f}_{R}(\tilde{y};\mu)&=A_{R}\tilde{y}+\begin{bmatrix}\gamma\mu\\ 0\end{bmatrix}+E_{R}(\tilde{y};\mu).\end{split}$$

(5.7)

The higher order terms $E_{L}$ and $E_{R}$ are $C^{1}$ and $\mathpzc{o}\mathopen{}\mathclose{{}\left(\|\tilde{y}\|+|\mu|}\right)$. Thus given any $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)>0$ such that

$$\begin{split}\frac{\mathopen{}\mathclose{{}\left\|E_{L}(\tilde{y};\mu)}\right\|}{\|\tilde{y}\|+\mu}&<\frac{\gamma\varepsilon}{\gamma r+1},\qquad\text{for all}~{}(\tilde{y};\mu)\in B_{\delta\gamma r}({\bf 0})\times(0,\delta),\\ \frac{\mathopen{}\mathclose{{}\left\|E_{R}(\tilde{y};\mu)}\right\|}{\|\tilde{y}\|+\mu}&<\frac{\gamma\varepsilon}{\gamma r+1},\qquad\text{for all}~{}(\tilde{y};\mu)\in B_{\delta\gamma r}({\bf 0})\times(0,\delta),\end{split}$$

(5.8)

and

$$\begin{split}\big{\|}{\rm D}\tilde{f}_{L}(\tilde{y};\mu)-A_{L}\big{\|}&<\eta_{2}\,,\qquad\text{for all}~{}(\tilde{y};\mu)\in B_{\delta\gamma r}({\bf 0})\times(0,\delta),\\ \big{\|}{\rm D}\tilde{f}_{R}(\tilde{y};\mu)-A_{R}\big{\|}&<\eta_{2}\,,\qquad\text{for all}~{}(\tilde{y};\mu)\in B_{\delta\gamma r}({\bf 0})\times(0,\delta).\end{split}$$

(5.9)

Then, since $\gamma>0$ (2.10), the spatial blow-up (5.4) converts (5.6) with $\mu>0$ to a map of the form

$$\tilde{g}(x;\mu)=\begin{cases}\tilde{g}_{L}(x;\mu),&x_{1}\leq 0,\\ \tilde{g}_{R}(x;\mu),&x_{1}\geq 0,\end{cases}$$

(5.10)

where $\tilde{g}_{L}(x;\mu)=\frac{1}{\gamma\mu}\tilde{f}_{L}(\gamma\mu x;\mu)$ and $\tilde{g}_{R}(x;\mu)=\frac{1}{\gamma\mu}\tilde{f}_{R}(\gamma\mu x;\mu)$. By (5.7)–(5.9) we have

$$\begin{split}\mathopen{}\mathclose{{}\left\|\tilde{g}_{L}(x;\mu)-g_{L}(x)}\right\|&<\varepsilon,\qquad\text{for all}~{}(x;\mu)\in B_{r}({\bf 0})\times(0,\delta),\\ \mathopen{}\mathclose{{}\left\|\tilde{g}_{R}(x;\mu)-g_{R}(x)}\right\|&<\varepsilon,\qquad\text{for all}~{}(x;\mu)\in B_{r}({\bf 0})\times(0,\delta),\end{split}$$

(5.11)

and

$$\begin{split}\mathopen{}\mathclose{{}\left\|{\rm D}\tilde{g}_{L}(x;\mu)-A_{L}}\right\|&<\eta_{2}\,,\qquad\text{for all}~{}(x;\mu)\in B_{r}({\bf 0})\times(0,\delta),\\ \mathopen{}\mathclose{{}\left\|{\rm D}\tilde{g}_{R}(x;\mu)-A_{R}}\right\|&<\eta_{2}\,,\qquad\text{for all}~{}(x;\mu)\in B_{r}({\bf 0})\times(0,\delta).\end{split}$$

(5.12)

Further assume $\varepsilon\leq\varepsilon_{1}$ so that, by (5.3), $\Omega$ is a trapping region for $\tilde{g}(x;\mu)$ with any $\mu\in(0,\delta)$. That $\Omega$ is a trapping region implies $\tilde{g}$ has a topological attractor $\Gamma_{\mu}\subset\Omega$. Then $\Lambda_{\mu}=\phi_{\mu}^{-1}\mathopen{}\mathclose{{}\left(\gamma\mu\Gamma_{\mu}}\right)$ is the corresponding attractor of $f$. Since $\gamma\mu\Gamma_{\mu}\subset B_{\gamma\mu r}({\bf 0})$ and $\phi_{\mu}^{-1}(\tilde{y})$ is a linear function of the pair $(\tilde{y};\mu)$, there exists $s>0$ such that $\Lambda_{\mu}\subset B_{\mu s}({\bf 0})$ for all $\mu\in(0,\delta)$.

Step 4 — Extend bounds on $g$ to the perturbed map $\tilde{g}$.
In Step 2 we obtained bounds on the number of iterations required for orbits of the BCNF $g$ to cross $\Sigma$. Here we show the same bounds hold for the perturbed map $\tilde{g}$.

We first extend the definitions of $\chi_{L}$ and $\chi_{R}$ to $\tilde{g}$. Given $x\in\mathbb{R}^{2}$, let $\chi_{L,\mu}(x)$ be the smallest $p\geq 1$ for which $\tilde{g}^{p}(x;\mu)\notin\Pi_{L}$ and let $\chi_{R,\mu}(x)$ be the smallest $q\geq 1$ for which $\tilde{g}^{q}(x;\mu)\notin\Pi_{R}$, if such $p$ and $q$ exist. In view of Lemma 3.1, we can similarly generalise $\Phi_{L}$ and $\Phi_{R}$ by defining

$$\begin{split}\Phi_{L,\mu}&=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}<0,\,\tilde{g}^{-1}(x;\mu)_{1}\geq 0}\right\},\\ \Phi_{R,\mu}&=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{2}\,\big{|}\,x_{1}>0,\,\tilde{g}^{-1}(x;\mu)_{1}\leq 0}\right\}.\end{split}$$

(5.13)

These sets are sketched in Fig. 6 and for sufficiently small $\mu<0$ they are within $\frac{\varepsilon_{1}}{2}$ of $\Phi_{L}$ and $\Phi_{R}$ in the bounded set $\Omega$. Next, take $0<\varepsilon<\frac{\varepsilon_{1}}{2}$ small enough so that, by (5.11), $\tilde{g}(\Omega;\mu)\cap\Phi_{L,\mu}\subset B_{\frac{\varepsilon_{1}}{2}}\mathopen{}\mathclose{{}\left(g(\Omega)\cap\Phi_{L}}\right)$ and $\tilde{g}(\Omega;\mu)\cap\Phi_{R,\mu}\subset B_{\frac{\varepsilon_{1}}{2}}\mathopen{}\mathclose{{}\left(g(\Omega)\cap\Phi_{R}}\right)$ for all $\mu\in(0,\delta)$. For small enough $\varepsilon>0$, (5.11) also implies

$$\begin{split}\mathopen{}\mathclose{{}\left\|\tilde{g}_{L}^{p}(x;\mu)-g_{L}^{p}(x)}\right\|&<\varepsilon_{2},\qquad\text{for all}~{}(x;\mu)\in\Omega\times(0,\delta)~{}\text{and all}~{}p=1,2,\ldots,p_{\rm max}\,,\\ \mathopen{}\mathclose{{}\left\|\tilde{g}_{R}^{q}(x;\mu)-g_{R}^{q}(x)}\right\|&<\varepsilon_{2},\qquad\text{for all}~{}(x;\mu)\in\Omega\times(0,\delta)~{}\text{and all}~{}q=1,2,\ldots,q_{\rm max}\,.\end{split}$$

(5.14)

Then by (i) and (ii) of Step 2, and analogous bounds on $g_{R}^{q}(x)_{1}$,

$$\begin{split}p_{\rm min}&\leq\chi_{L,\mu}(x)\leq p_{\rm max},\qquad\text{for all}~{}(x;\mu)\in\tilde{g}(\Omega;\mu)\cap\Phi_{L,\mu}\times(0,\delta),\\ q_{\rm min}&\leq\chi_{R,\mu}(x)\leq q_{\rm max},\qquad\text{for all}~{}(x;\mu)\in\tilde{g}(\Omega;\mu)\cap\Phi_{L,\mu}\times(0,\delta).\end{split}$$

(5.15)