Nondeterministic Chaos, and the Two-fold Singularity in Piecewise Smooth Flows

TLDR: The local dynamics of all possible two-folds in three dimensions are derived, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics.

Abstract: A vector field is piecewise smooth if its value jumps across a hypersurface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique in neither forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but nonperiodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behavior is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behavior can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.

Figures

Table 7.1

Figure 11. A magnification of Figure 9, making a linear coordinate change in the x1-x2 plane and stretching the positive x0 axis by a factor of 20. In (i) the orbit can now be seen, crossing repeatedly through the switching manifold and surrounding the focus cycle (depicted in red). In (ii), the orbit’s crossing points are seen to form a butterfly shape around the two (red) crossing points of the focus cycle.

Figure 3. Dynamics near a Teixeira singularity: (i) orbits outside the switching manifold curve around the folds and cross the manifold in CR1 and CR2, (ii) phase portraits of the sliding dynamics. At the origin the sliding vector field fs is set-valued, with elements pointing into the escaping region ES when V +V − > 1 and V +, V − < 0. In this case a one-parameter family of orbits intersects the origin along a unique direction (an eigenvector of f̃s from (3.4)). In all other cases the set-valued fs has elements which point into the sliding region SL, and only a single orbit that intersects the singularity.

Figure 12. Simulation of system (7.1) with V + = −4.99 and V − = −1/5, with a pseudoequilibrium in SL. (i) A single orbit originating near ES, with initial conditions (10−20,−10−6,−10−6), winds around the singularity and reaches SL, and then is repelled from the singularity toward a pseudoequilibrium (red dot at edge of SL). (ii) A linear coordinate change in the x1-x2 plane, and a stretching of the positive x0 axis by a factor of 20, resolves the crossing dynamics of the orbit.

Figure 5. The complete catalogue of local dynamics around a Teixeira singularity when p = V +V − − 1 " 0 is obtained by composing the possible phase portraits of sliding (s1), (s2), and crossing (c1), (c2s,u), (c3s,u). (s1), (s2) show the two possible phase portraits in the sliding and escaping regions SL and ES as the bifurcation parameter p changes sign. (c1), (c2s,u), (c3s,u) depict the intersections of orbits with the crossing regions CR1 and CR2 as p changes sign. (These are derived from the unfoldings in [12].) The crossing maps in (c2s) and (c3s) have a stable fixed point of node and focus type, respectively, while the dual cases in (c2u) and (c3u) have unstable fixed points. Altogether, composing cases (s1), (s2) with cases (c1), (c2s,u), (c3s,u), ten qualitatively different portraits are obtained.

Figure 6. Composite of the sliding portrait (s1) and the crossing portrait (c3u) in Figure 5. For p < 0 all orbits reach the sliding region, and a crossing orbit is illustrated. For p > 0 this system has an invariant set near the singularity, generated by the forward evolution of the hatched region in ES. The limit cycle (fixed point of the map φ) is shown.

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Citation for published version:
Colombo, A & Jeffrey, MR 2011, 'Nondeterministic chaos, and the two-fold singularity in piecewise smooth
flows', SIAM Journal on Applied Dynamical Systems, vol. 10, no. 2, pp. 423-451.
https://doi.org/10.1137/100801846
DOI:
10.1137/100801846
Publication date:
2011
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SIAM J. APPLIED DYNAMICAL SYSTEMS
c
!
2011 Society for Industrial and Applied Mathematics
Vol. 10, No. 2, pp. 423–451
Nondeterministic Chaos, and the Two-fold Singularity in Piecewise Smooth
Flows
∗
Alessandro Colombo
†
and Mike R. Jeffrey
‡
Abstract. A vector field is piecewise smooth if its value jumps across a hypersurface, and a two-fold singularity
is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in
piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible
two-folds in three dimensions, including nonlinear effects around certain bifurcat ions, finding that
they admit a flow ex hibiting chaotic but nondeterministic dynamics. In cases where the flow passes
through the two-fold, upon reaching the singularity it is unique in neither forward nor backward
time, meaning the causal link between inward and outward dynamics is severed. In one scenario
this occurs recurrently. The resultin g flow makes repeated, but nonperiodic, excursions from the
singularity, whose path and amplitude is not determined by previous excursions. We show that this
behavior is robust an d has many of the properties associated with chaos. Local geometry reveals
that the ch aotic behavior can be eliminated by varying a single parameter: the angular jump of the
vector field across the two-fold.
Key words. two-fold, sliding, Filippov, nondeterminism, chaos, bifurcation
AMS subject c lassifications. 34C23, 37G10, 37G35
DOI. 10.1137/100801846
1. Introduction. Piecewise smooth vector fields have appeared through out the history of
dynamical systems as models of mechanical and electronic devices (e.g., [1]) and, more recently,
have seen growing use in fields such as ecology, econ om ics , and neuroscience. Their spreading
use has naturally been accompanied by interest in their generic mathematical and dynamical
properties, which have been the s ubject of a number of recent books (e.g., [10,18,22,32]). Their
dynamics were formalized by Filippov [14], using differential inclusions (set-valued differential
equations; see [3]) to overcome the problem of indefiniteness of the vector field on the s urfaces
of discontinuity.
Although two-dimensional piec e w i s e smooth systems are now rather well understood (see,
for example, [14, 21]), a general understanding of dy namics in three or more dimensions is
crucially obstructed by the appearan ce of the so-called two-fold singular ity [30]. The two-fold
is a simple topological singularity that is generic in piecewise smo oth systems with three or
more dimensions. This implies that it may well be commonplace in systems of a piecewise
smooth nature. Contrarily, two-folds are neither well know n nor well understood, with regard
to either the theory of their dynamics or the frequency of their appearance in physical systems.
∗
Received by the editors July 13, 2010; accepted for publication (in revised form) by H. Kokubu March 1, 2011;
published electronically May 26, 2011.
http://www.siam.org/journals/siads/10-2/80184.html
†
Department of Electronics and Information, Via Ponzio 34/5, 20133 Milano, Italy (alessandro.colombo@
polimi.it).
‡
Engineering Mathematics D epartment, Queen’s Building, University of Bristol, BS8 1TR, Bristol UK (mike.
jeffrey@bris.ac.uk). The work of this author was supported by the EPSRC grant Making It Real.
423

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
424 ALESSANDRO COLOMBO AND MIK E R. JEFFREY
Figure 1. Dynamics at a switching manifold in a three-dimensional piecewise sm ooth system. T he vector
field switches between f
+
and f
−
. An orbit meeting the manifold may either: (i) cross through it, (ii) reach
it in finite time and then follow the sliding vector field f
s
, or (iii) escape it in finite time, though it may slide
along the manifold for some time before escaping.
The purpose of this paper is to present, in an organic and cons istent framework, all existing
results regarding the local dynamics near the two-fold. This also includes some novel results
about particular forms of the two-fold that reveal its role in the sudden onset of periodic orb its
and recu rrent nondeter ministic dynamics.
The two-fold was already well defined in [14]. In a piecewise smooth vector field, disconti-
nuities are assumed to occur across a hypersurface called the switching manifold. Since it is a
hypersurface, we can speak of the manifold as locally having two sides, and gen erically there
may exist p oints where the vector field is quadratically tangent to one side of the manifold
or the other. We call such a tangency a fold, because in the projection along the flow the
switch ing manifold has a simple fold. This assumes the system to be at least two-dimensional.
In high er dimensions there may generically exist points where two folds intersect transversely,
so that the vector field is tangent to both sides of the manifold, and this simple object is a
two-fold. A two-fold is an important organizing center because it brings together all of the
basic forms of dynamics possible in a piecewise smooth system. Filippov [14] described three
basic forms of dynamics that would occur at a switching manifold: crossing, sliding, and es-
caping, depending on the orientation of the vector field either side of the switching manifold ,
as illustrated in Figure 1. Crossing, shown in Figure 1(i), occurs where the component of the
vector field normal to the switch ing manifold has the same direction on both sides. In the two
other cases the normal comp onent of the vector field switches direction, so that the vector
field is either directed towards the switching manifold, giving sliding as in (ii), or is directed
away from the manifold , giving escaping as in (iii).
At a fold (see Figure 2) the vector field on one side of the switching manifold changes its
normal direction, formin g a boundary between crossing regions and sliding or escaping regions.
At a two-fold, the vector fields either side of the manifold both change their normal direction,
meaning that regions of all three dynamical beh aviors—crossing, sliding, an d es caping—meet,
and their boundaries intersect to form the singularity.
Escaping dynamics (see Figure 1(iii)) is typically neglected on the b asis that it simply
constitutes a tim e-reversal of sliding, and that escaping regions cannot be reached by a sys-
tem in forward time, making consequences of forward time nonuniqueness in these regions
irrelevant [10, 11, 27]. This assumption is incorrect at a two-fold, which can channel sliding
dynamics into the escaping region. This gives whole families of orbits robust access to regions

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
NONDETERMINISTI C CHAOS AND THE TWO-FOLD 425
Figure 2. Tangencies in a piecewise smooth system, showing: (i) a visible fold, (ii) an invisible fold; these
form the boundaries between sliding (shaded) and crossing (unshaded) (reverse arrows to replace sliding with
escaping). (iii) Folds associated with the upper and lower fields cross to form a two-fold, where both vector fields
are tangent to the switching manifold (in the case illustrated, both folds are invisible).
of phase space that are infinitely repelling. This counterintuitive dynamical behavior, noticed
in Filippov’s seminal work [14], seems to have been overlooked ever since, though a similar
effect was discovered in the framework of nonstandard analysis [4] as the so-called canard
phenomen on. Canards are now a popular topic in singular perturbation theory [13, 28], with
numerous applications, of which a few examples are in neuron modellin g [24], chemical dy-
namics [6, 26], gas pressure dynamics [5], and ecology [9]. Despite qualitative similarities in
these approaches, their connection to the two-fold is as poor ly understood as the two-fold
itself. These connections are not the subject of this paper, and we restrict our interest to
understanding the two-fold in the context of generic piecewise smooth dynamical systems.
The study of dynamics around a two-fold has been mainly limited to a lowest order
approximation in three dimensions [14,16,29,30]. Such local analysis r eveals h ow an initially
smooth flow far from a discontinuity can evolve toward a state where its forward evolution
is set-valued. In this paper we review these results and extend them by carrying out a
comprehensive analysis of the nonlinear behavior of two-folds in three dimens ions. In so
doing, we determine the invariant sets that are present near the two-fold and decode their
complex dynamics.
In section 2 we define the two-fold singularity and its three types. We discuss the first of
these, the invisib le two-fold, or Teixeira singularity, in detail in section 3; we analyze its sliding
and crossing dynamics separately in sections 3.1 and 3.2, using them to reconstruct the full
system in sections 3.3 and 3.4. We briefly discuss the other forms, the visible (short for visible-
visible) two-fold in section 4, an d the visible-invisible two-fold in section 5, with a remark
on their b ifurcations in section 6. In section 7 we numerically simulate some particularly
interesting dynamics predicted in section 3, with some closing remarks in section 8.
2. The three flavors of two-fold. Consid er a three-dimensional piecewise smooth system
of ordinary differential equations
(2.1)
˙
x = f
+
(x) when h(x) > 0,
˙
x = f
−
(x) when h(x) < 0,
where the dot den otes differentiation with respect to tim e t ∈ R and where h(x) is a regular
scalar function of the state vector x =(x
0
,x
1
,x
2
) ∈ R
3
. For simplicity we set h(x)=x
0
, since
any piecewise s mooth system, in a region where h(x) = 0 defines a manif old, can be put into
this form through the approp riate change of var iables [14, 30]. Then, x
0
= 0 is the switching
manifold. Follow ing Filippov’s defi nition [14], the solution of (2.1) at th e switching manifold

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
426 ALESSANDRO COLOMBO AND MIK E R. JEFFREY
includes all solutions of the differential inclusion
(2.2)
˙
x ∈ f := f
−
+ λ(f
+
− f
−
),
where λ = 0 when h(x) < 0, λ = 1 when h(x) > 0, and λ ∈ [0, 1] w hen h(x) = 0, so that
f is a set-valued convex combination of f
+
and f
−
where h(x) = 0. In practice, when the
components of f
+
and f
−
normal to the sw itching manifold have opposite direction, f admits
a solution that lies on the switching manifold and satisfies the system given by
(2.3)
˙
x = f
s
(x) when x
0
=0,
where the sliding vector field, f
s
, is defined as
(2.4) f
s
= f
−
+
L
f
− h
L
f
− h − L
f
+ h
(f
+
− f
−
).
The symbol L
f
denotes the Lie derivative along the flow of a field f , given by L
f
= f ·∇ =
˙
x·
d
dx
.
Let L
2
f
denote the second Lie derivative L
2
f
=(L
f
)
2
. The dynamics in a general piecewise
smooth system is then a composite of the dynamics of f
+
, f
−
, and f
s
. We make the following
distinctions.
Definition of orbits and flow. An orbit segment is any smooth p ath x = x(t) satisfying
(2.2), entirely contained in one of the regions {x : h(x) > 0}, {x : h(x) < 0}, or {x : h(x)=
0}. An orbit is any continuous path x(t) that satisfies (2.2), formed by concatenating orbit
segments. The flow of (2.2) through a point
ˆ
x at time t is given by all points x(t + τ ) with
x(τ )=
ˆ
x for some τ ∈ R, x(t) satisfying (2.2). In the following, by the flow we mean the flow
of (2.2) unless otherwise stated.
An im portant consequ ence of this definition is that the flow through a point
ˆ
x in a sliding
region is not unique, because
ˆ
x always belongs to a one-parameter family of orbits (unless it is
an equ ilibrium). For example, the six orbits sh own in Figure 1(ii) each overlap in the sliding
region, so through any point
ˆ
x on the overlap the flow is nonunique; the same applies to the
six orbits shown in Figure 1(iii).
Now let us assume that both f
+
and f
−
have quadratic contact with the switching man-
ifold at the origin, that is,
L
f
+ h(0) = 0 and L
2
f
+
h(0) $=0,(2.5a)
L
f
−
h(0) = 0 and L
2
f
−
h(0) $=0.(2.5b)
Let us also require that neither f
+
nor f
−
has equilibria near the origin,
(2.6) f
+
$= 0 and f
−
$=0,
and that the pair of curves given by L
f
+
h = 0 and L
f
−
h = 0 on h = 0 intersect tr ansversely
at th e origin,
(2.7) det
!
∇h(0), ∇ L
f
+
h(0), ∇L
f
−
h(0)
"
$=0.

Frequently asked questions

Q1. What have the authors contributed in "Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows∗" ?

The authors show that this behavior is robust and has many of the properties associated with chaos. 

Q2. What does the Poincaré map (3.9) mean?

The changes of variables that place the generic Poincaré map (3.9) in the form (3.12) ensures that, for p sufficiently close to 0, the positive u1 axis lies strictly inside the domain D of the map φ. 

Q3. How many portraits are obtained from the Teixeira singularity?

The complete dynamics around the Teixeira singularity is obtained by stitching together any one of the portraits (s1), (s2) for the sliding dynamics, with any one of the portraits (c1)–(c3) for the crossing dynamics, for a grand total of ten qualitatively different phase portraits. 

Q4. What is the meaning of the term B2,0?

Due to the involution condition, the term B2,0 is identically null near p = 0, and this second order expansion is structurally unstable. 

Q5. What is the definition of sensitivity for a set-valued flow?

A set-valued flow ψ exhibits sensitive dependence on an invariant set X if there is a fixed r such that for each x ∈ X and any " > 0 there is a nearby y ∈ B!(x) ∩X such that the diameter of ψt(x) ∪ ψt(y) is greater than r for some t ≥ 0. 

Q6. What is the axis of the singular eigenvector?

In the (x1, x2) coordinates and at the bifurcation, the coordinate axis ξ1 points in the direction of the singular eigenvector (−1,−V +) of φ|p=0, while the axis ξ2 points in the x1 direction; hence the change of variables has turned the plane to align the singular eigenvector with the ξ1 direction, and in the new variables the ξ1 axis is strictly inside CR1 and CR2. 

Q7. What is the last crossing of the orbit?

After the last crossing (lower right corner of CR1), the orbit is seen to impact SL toward the top of the figure, evolve via an almost straight path toward the point (x1, x2) ! (0.00025, 0.00005), then to the singularity. 

Q8. What are the bifurcations that must occur at the transition between scenarios (c1) and?

The bifurcations that must occur at the transition between scenarios (c1) and (c3), for p near 0 suggest the existence of other structures, e.g., quasiperiodic orbits, that may emerge as p crosses 0. 

Q9. What is the preimage of the positive x2 axis?

This implies that D lies between the negative x2 axis and the preimage of the positive x2 axis under the map φ−; since φ− is an involution, the preimage is a curve given by φ−(x1 = 0, x2 > 0). 

Q10. What is the smallest order approximation to the Teixeira singularity?

The lowest order approximation to the Teixeira singularity analyzed in [16] revealed an interesting bifurcation, in the form of an invariant nonsmooth diabolo (an invariant double cone with a crease at the switching manifold) that self-annihilates through a loss of hyperbolicity. 

Q11. What is the definition of a structurally unstable scenario?

A structurally unstable scenario occurs when the image of the border of ES under φ (or multiple iterations of φ) is tangent to the border of SL. 

Q12. How can the authors compare the crossing dynamics to the sliding vector field f s?

It should be remarked that, although the authors derived the crossing dynamics from generic forms for the maps φ± in section 3.2, the crossing dynamics can be derived directly by integrating a local series expansion of the vector fields f±, allowing them to be compared directly to the sliding vector field f s. 

Q13. What is the meaning of the term nondeterministic chaos?

The term “nondeterministic chaos” has previously appeared in [8] in a somewhat different setting, though referring to a similar loss of uniqueness in which an infinity of orbits recurrently pass through a single point in finite time. 

Q14. What is the nature of the unfolding?

The unfolding reveals the bifurcation of the Teixeira singularity as a new route to the sudden appearance of periodic orbits and more complex invariant sets in piecewise smooth systems. 

Q15. What is the bifurcation diagram of the crossing return map?

The bifurcation diagram of the crossing return map in Figure 5 is incomplete: global bifurcations, yet unknown, occur between the identified scenarios or each time the invariant manifolds of the crossing map, or the images of the boundaries of ES, become tangent to the folds.