A synopsis of the non-invertible two-dimensional border-collision normal form with applications to power converters.

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A synopsis of the non-invertible, two-dimensional,

border-collision normal form with applications to power

converters.

H.O. Fatoyinbo†and D.J.W. Simpson‡

†EpiCentre, School of Veterinary Science

‡School of Mathematical and Computational Sciences

Massey University

Palmerston North, 4410

New Zealand

October 27, 2022

Abstract

The border-collision normal form is a canonical form for two-dimensional, continuous

maps comprised of two aﬃne pieces. In this paper we provide a guide to the dynamics

of this family of maps in the non-invertible case where the two pieces fold onto the

same half-plane. We identify parameter regimes for the occurrence of key bifurcation

structures, such as period-incrementing, period-adding, and robust chaos. We then

apply the results to a classic model of a boost converter for adjusting the voltage of

direct current. It is known that for one combination of circuit parameters the model

exhibits a border-collision bifurcation that mimics supercritical period-doubling and is

non-invertible due to the switching mechanism of the converter. We ﬁnd that over a

wide range of parameter values, even though the dynamics created in border-collision

bifurcations is in general extremely diverse, the bifurcation in the boost converter can

only mimic period-doubling, although it can be subcritical.

1 Introduction

Periodic and non-periodic oscillations in systems of ordinary diﬀerential equations are usually

analysed by constructing a return map. In classical settings, where the diﬀerential equations

E-mail addresses: H.O. Fatoyinbo (h.fatoyinbo@massey.ac.nz), D.J.W. Simpson (d.j.w.simpson@

massey.ac.nz)

arXiv:2210.14445v1 [nlin.CD] 26 Oct 2022

induce a unique smooth ﬂow, such maps are typically smooth and invertible, at least locally

[1]. However, for piecewise-smooth diﬀerential equations and hybrid systems, return maps

are commonly piecewise-smooth [2].

The phase space of a piecewise-smooth map is characterised by the presence of one or more

switching manifolds where the map is nonsmooth. This nonsmoothness causes the dynamics

to change in a fundamental way at border-collision bifurcations where a ﬁxed point collides

with a switching manifold as parameters are varied. Under quite general conditions, these

dynamics are captured by a piecewise-linear family known as the border-collision normal

form [3, 4, 5]. In two dimensions this family can be written as

x

y7→ 









"τLx+y+µ

−δLx#, x ≤0,

"τRx+y+µ

−δRx#, x ≥0,

(1)

where the line x= 0 is the switching manifold, the parameter µ∈Rcontrols the border-

collision bifurcation, and τL, δL, τR, δR∈Rare additional parameters.

The normal form (1) is well-studied as it arises in diverse applications, and, as an extension

of the Lozi map [6], serves as a minimal model for chaotic and highly nonlinear dynamics.

The dynamics of (1) is remarkably rich — it exhibits chaos robustly [7, 8], can have any

number of coexisting attractors [9], of which all could be chaotic [10, 11]. Most studies of (1)

have focussed on parameter regimes where (1) is invertible, i.e. δLδR>0. For the invertible,

dissipative case, a review is provided by [12]; other works that characterise parameter space

in some detail include [13, 14, 15, 16].

However, it is perhaps under-appreciated that return maps of piecewise-smooth dynam-

ical systems are often, not only piecewise-smooth, but also non-invertible. This is because

a switch to a diﬀerent mode of operation can readily cause the ﬂow to fold back onto it-

self, as illustrated below for power converters. Some analysis of (1) has been done in the

non-invertible case. For example period-adding is illustrated numerically in [16], global bifur-

cations of a chaotic attractor in an equivalent map are described in [17], and two-dimensional

attractors are identiﬁed in [18]. In this paper we provide an overview of the dynamics of (1)

when it is non-invertible, speciﬁcally with δLδR<0. For the special case δLδR= 0, where

the long-term dynamics are essentially one-dimensional, refer to [19, 20].

We begin in §2 by showing how the piecewise-linear normal form (1) applies to border-

collision bifurcations of arbitrary, two-dimensional, piecewise-smooth maps. We then study

(1) subject to δR<0< δL, the speciﬁc signs being chosen without loss of generality. The

dynamical complexity of (1) means we cannot hope to characterise all dynamics, and for this

reason our approach is to chart the essential features. With µ < 0 the dominate bifurcation

structures are period-adding and robust chaos, §3, while with µ > 0 the map exhibits period-

incrementing and robust chaos, §4.

In §5 we illustrate the results with the power converter model of Deane [21]. This model

exhibits a border-collision bifurcation, and by determining the part of the parameter space of

the normal form that this bifurcation corresponds to, we can use our results to characterise

the bifurcation. Although in general the dynamics created in border-collision bifurcations is

extremely diverse, it appears this bifurcation acts exclusively as a piecewise-smooth version

of period-doubling. Finally §6 provides concluding remarks.

2 Border-collision bifurcations and the normal form

A border-collision bifurcation occurs when a ﬁxed point of a piecewise-smooth map collides

with a switching manifold. Here we consider a two-dimensional map with variables u, v ∈

Rand parameter η∈R. We assume a border-collision bifurcation occurs at the origin

(u, v) = (0,0) when η= 0, and we wish to understand the dynamics in a neighbourhood of

(u, v;η) = (0,0; 0).

We assume the switching manifold is smooth, at least locally, so there exists a smooth

coordinate change that shifts the switching manifold to the line u= 0 [22]. Then, assuming

the map is continuous and piecewise-C2, locally it has the form

u

v7→ 









"aL

11u+a12v+b1η

21u+a22v+b2η#+O(|u|+|v|+|η|)2, u ≤0,

"aR

11u+a12v+b1η

21u+a22v+b2η#+O(|u|+|v|+|η|)2, u ≥0,

(2)

for some aL

11, aL

21, aR

11, aR

21, a12, a22, b1, b2∈R. Note that the vand ηcoeﬃcients of the two

pieces of (2) are the same. This is a consequence of the assumed continuity of (2) on u= 0.

Next we work to bring (2) into the normal form (1).

2.1 A derivation of the normal form

By construction the origin is a ﬁxed point of (2) when η= 0. We now consider ﬁxed points

of (2) for small η∈R. Let

c= (1 −a22)b1+a12b2.(3)

Then the left (u≤0) piece of (2) has the ﬁxed point

uL(η)

vL(η)=1

(1 −aL

11)(1 −a22)−a12aL

21 c

21b1+1−aL

11b2η+Oη2,(4)

assuming the denominator in (4) is non-zero. Similarly the right (u≥0) piece of (2) has the

ﬁxed point

uR(η)

vR(η)=1

(1 −aR

11)(1 −a22)−a12aR

21 c

21b1+1−aR

11b2η+Oη2,(5)

assuming its denominator is non-zero. In a suﬃciently small neighbourhood of (u, v;η) =

(0,0; 0), these are the only ﬁxed points of (2).

Notice we require c6= 0 for the ﬁxed points to move away from the switching manifold

at a rate that is asymptotically proportional to η. Thus c6= 0 is the transversality condition

[23, 24] that ensures ηunfolds the border-collision bifurcation in a generic fashion.

In view of the switching condition in (2), uL, vLis a ﬁxed point of (2) only if uL≤0,

in which case we say it is admissible. Similarly uR, vRis an admissible ﬁxed point of (2)

if uR≥0. If uL<0, the stability of uL, vLis governed by the eigenvalues of the Jacobian

matrix of (2) evaluated at uL, vL. This matrix is aL

11 a12

21 a22+O(η). To leading order its

trace and determinant are

τL=aL

11 +a22 , δL=aL

11a22 −a12aL

21 .(6)

We similarly deﬁne

τR=aR

11 +a22 , δR=aR

11a22 −a12aR

21 ,(7)

for the right piece of (2). By removing the nonlinear terms from each piece of (2) and

applying the coordinate change

x=u, (8)

y=−a22u+a12v+ (a22b1−a12b2)η, (9)

µ=cη, (10)

we arrive at the normal form (1) with τL,δL,τR, and δRgiven by (6) and (7). This coordinate

change is invertible when c6= 0 (discussed above) and a12 6= 0 (otherwise (2) decouples into

two one-dimensional maps).

2.2 The utility of the normal form

The dynamics of the normal form (1) approximates the dynamics of (2) for small values

of u,v, and η. Presently there is little mathematical theory clarifying the validity of this

approximation, but in practice the normal form is useful for characterising the dynamics

created in border-collision bifurcations. For example, if the normal form exhibits a hyperbolic

periodic solution, this solution also exists for (2) [5]. This type of persistence result has

recently been extended to chaotic attractors [25].

Since the normal form is piecewise-linear, the structure of its dynamics is independent of

the magnitude of µ. If µ < 0 its value can be scaled to −1, while if µ > 0 its value can be

scaled to 1. We therefore study the normal form with µ=−1 to understand the dynamics

on one side of the border-collision bifurcation, and with µ= 1 to understand the dynamics

on the other side of the border-collision bifurcation.

With µ= 0 the origin is a ﬁxed point of the normal form. If it is asymptotically stable

then (2) has a local attractor on each side of the border-collision bifurcation [26]. However,

the normal form is non-diﬀerentiable at the origin, so its stability can be diﬃcult to ascertain

[27, 28]. For instance in the non-invertible case the origin can be asymptotically stable even

when both pieces of the map are area-expanding [29].

If δLδR<0, then, in view of the substitution (x, y, µ)7→ (−x, −y, −µ) that leaves the

normal form invariant other than switching ‘left’ and ‘right’, we can assume δL>0 and

δR<0. With δL>0 and δR<0 the normal form maps both left and right half-planes to

the upper half-plane (y≥0).

Figure 1: The lower plot is a bifurcation diagram of the normal form (1) with (11). As the

value of µis increased through 0, a stable period-3 solution changes to a stable period-4

solution. The upper plots are phase portraits with µ=±1 (red circle: unstable ﬁxed point;

red triangles: unstable period-2 solution; blue circles: stable period-3 and period-4 solutions;

green line: switching manifold).

2.3 An example of the dynamics of the normal form

Fig. 1 shows a bifurcation diagram and phase portraits of the normal form (1) with

(τL, δL, τR, δR) = (−1,2,−1,−0.2).(11)

Here δL>0 and δR<0 so all invariant sets lie in the upper half-plane (y≥0). For this

example the border-collision bifurcation brings about a transition from a stable period-3

solution to a stable period-4 solution. As with all bounded invariant sets of the normal form,

these solutions contract linearly to the origin as µ→0, and this is evident in the bifurcation

diagram. The period-3 solution for µ < 0 consists of one point in the left half-plane and two

points in the right half-plane. Its symbolic representation is therefore LRR (or any cyclic

permutation of this) and we refer to it as an LRR-cycle. Similarly the period-4 solution

for µ > 0 is an LRRR-cycle. Based on symbolic representations, we can characterise the

existence, admissibility, and stability of periodic solutions in a general manner [5, 30].

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Asynopsisofthenon-invertible,two-dimensional,border-collisionnormalformwithapplicationstopowerconverters.H.O.FatoyinboyandD.J.W.SimpsonzyEpiCentre,SchoolofVeterinarySciencezSchoolofMathematicalandComputationalSciencesMasseyUniversityPalmerstonNorth,4410NewZealandOctober27,2022AbstractTheborder-colli...

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A synopsis of the non-invertible two-dimensional border-collision normal form with applications to power converters.

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