Dissipative homoclinic loops of two‐dimensional maps and strange attractors with one direction of instability

Abstract

We prove that when subjected to periodic forcing of the form certain two-dimensional vector fields with dissipative homoclinic loops generate strange attractors with Sinai-Ruelle-Bowen measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov and applies the recent theory of rank 1 maps developed by Wang and Young. We prove a general theorem and then apply this theorem to an explicit model: a forced Duffing equation of the form © 2011 Wiley Periodicals, Inc.

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Dissipative Homoclinic Loops of Two-Dimensional Maps
and Strange Attractors with One Direction of Instability
QIUDONG WANG
University of Arizona
WILLIAM OTT
Courant Institute
Abstract
We prove that when subjected to periodic forcing of the form
p?;?;!.t/ D ?.?h.x;y/ C sin.!t//;
certain two-dimensional vector fields with dissipative homoclinic loops gener-
ate strange attractors with Sinai-Ruelle-Bowen measures for a set of forcing pa-
rameters .?;?;!/ of positive Lebesgue measure. The proof extends ideas of
Afraimovich and Shilnikov and applies the recent theory of rank 1 maps de-
veloped by Wang and Young. We prove a general theorem and then apply this
theorem to an explicit model: a forced Duffing equation of the form
d2q
dt2C .? ? ?q2/dq
© 2011 Wiley Periodicals, Inc.
dt? q C q3D ?sin.!t/:
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Bibliography
Introduction
Statement of Results
A Model of Afraimovich and Shilnikov
Normal Forms around the Homoclinic Loop
Explicit Computation of M and N
Proof of Theorem 1
Computational Proofs
Application to a Duffing Equation
1
7
11
19
32
40
43
49
56
1 Introduction
This paper is about proving the existence of sustained, observable chaos in ex-
plicit models of dynamical processes. We study the effects of periodic forcing on
Communications on Pure and Applied Mathematics, 0001–0058 (PREPRINT)
© 2011 Wiley Periodicals, Inc.

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2Q. WANG AND W. OTT
certain two-dimensional flows that admit homoclinic orbits. The aim is to formu-
late checkable hypotheses that imply the existence of sustained, observable chaos
for a set of forcing parameters of positive Lebesgue measure. We formulate such
hypotheses for a general class of systems and then apply the general result to an
explicit model: a Duffing equation. By sustained, observable chaos we refer to a
number of precisely defined dynamical, geometric, and statistical properties that
will be made precise in what follows.
1.1 Background: Dynamical Systems
The general theory of hyperbolic dynamics is one of the most important com-
ponents of the modern theory of dynamical systems. Individual orbits are typically
unstable in systems with some degree of hyperbolicity. It is therefore natural to
study such systems from a probabilistic point of view and ask the following ques-
tions:
(Q1) What mechanisms are creating the hyperbolicity?
(Q2) Does the system admit an invariant measure that describes the asymptotic
distribution of a large set (positive Riemannian volume) of orbits? If so,
how many such measures does the system admit?
(Q3) What are the ergodic and statistical properties of the invariant measures
identified in (Q2)? For example, do correlations decay at an exponential
rate?
For a conservative system (a system preserving a measure ? that is equivalent to
Riemannian volume), the Birkhoff pointwise ergodic theorem answers (Q2). If ?
is ergodic, then almost every point with respect to Riemannian volume produces
an orbit that is asymptotically distributed according to ?. The situation is com-
pletely different for dissipative (volume-contracting) systems. For such systems,
question (Q2) is a major challenge. We focus on dissipative systems in this paper.
Let M be a compact Riemannian manifold and let F W M ! M be a C2
embedding. A compact set † satisfying F.†/ D † is called an attractor if there
exists an open set U, called its basin, such that Fn.x/ ! † as n ! 1 for every
x 2 U. The attractor † is said to be
(1) irreducible if it cannot be written as a union of two disjoint attractors;
(2) uniformly hyperbolic if the tangent bundle over † admits a splitting into
two DF-invariant subbundles Esand Eusuch that DFjEsis uniformly
contracting and DFjEuis uniformly expanding. We assume that Euis
nontrivial.
An irreducible, uniformly hyperbolic attractor † supports a unique F-invariant
Borel probability measure ? with the following property: there exists a set S ? U
having full Riemannian volume in U such that for every continuous observable

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HOMOCLINIC LOOPS AND RANK 1 ATTRACTORS3
' W U ! R and for every x 2 S, we have
(1.1)lim
n!1
1
n
n?1
X
iD0
'.Fi.x// D
Z
M
' d?:
This measure is known as the Sinai-Ruelle-Bowen, or SRB, measure of F. We
adopt the point of view that sets of positive Riemannian volume correspond to
observable events. In this sense, the SRB measure on a uniformly hyperbolic at-
tractor is observable because time and space averages of observables coincide for a
set of initial data of full Riemannian volume in the basin. Note that for dissipative
systems, this does not follow from the Birkhoff ergodic theorem because if F is
volume-contracting in the basin, then the SRB measure will be supported on a set
of zero Riemannian volume.
Many modelsof physicaland biologicalprocesses exhibitsome degreeof hyper-
bolicity but are not uniformly hyperbolic in character. The theory of nonuniform
hyperbolicity can potentially be useful for the analysis of such systems. This the-
ory relaxes the assumptions of uniform hyperbolicity by assuming that contraction
and expansion occur only asymptotically in time and only almost everywhere with
respect to an invariant measure. New notions of SRB measure have developed as
the theory of nonuniform hyperbolicity has matured. The following definition is
the state of the art:
DEFINITION 1.1. Let M be a compact Riemannian manifold and let F W M ! M
be a C2embedding. An F-invariant Borel probability measure ? is called an SRB
measure if .F;?/ has a positive Lyapunov exponent ? almost everywhere (a.e.) and
if ? has absolutely continuous conditional measures on unstable manifolds.
These more general SRB measures are important for many reasons, one of
which is that members of a large class of them are observable: if ? is an ergodic
SRB measure with no 0 Lyapunov exponents, then there exists a set S of pos-
itive Riemannian volume such that (1.1) holds for every continuous observable
' W M ! R and for every x 2 S. For dissipative systems, proving the existence of
genuinely nonuniformly hyperbolic dynamics and SRB measures is a major chal-
lenge. SRB measures were first constructed outside of the uniformly hyperbolic
setting by Benedicks and Young relatively recently [6, 7]. Here SRB measures are
constructed for certain Hénon maps. A major advance has been made in the last
decade: the theory of rank 1 maps.
1.2 Theory of Rank 1 Maps
Developed by Wang and Young [48, 52, 53], the theory of rank 1 maps provides
checkable conditions that imply the existence of strange attractors and SRB mea-
sures in parametrized families fFag of dissipative embeddings in dimension N for
any N ? 2. The term “rank 1” refers to the local character of the embeddings:
some instability in one direction and strong contraction in all other directions.
Roughly speaking, the theory asserts that under certain checkable conditions, there

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4Q. WANG AND W. OTT
Theory of
1-D maps
?!
Hénon maps
and perturbations
?!
Rank one
attractors
FIGURE 1.1. Progression of ideas leading to the theory of rank 1 maps.
exists a set ? of values of a of positive Lebesgue measure such that for a 2 ?,
Fais a genuinely nonuniformly hyperbolic map that has a strange attractor and
admits an SRB measure. A comprehensive dynamical profile is given for such Fa.
In its strongest form, the profile is as follows.
The map Faadmits a unique SRB measure ?, and ? is mixing. Lebesgue almost
every orbit in the basin of the strange attractor is asymptotically distributed accord-
ing to ? in the sense of (1.1) and has a positive Lyapunov exponent. Thus the chaos
associated with Fais both observable and sustained in time. The system .Fa;?/
satisfies a central limit theorem, and correlations decay at an exponential rate for
Hölder observables. The source of the nonuniform hyperbolicity is identified, and
the geometric structure of the attractor is analyzed in detail.
Figure 1.1 illustrates the progression of ideas that has led to the development of
the theory of rank 1 maps. The theory is ultimately based on theoretical develop-
ments concerning one-dimensional maps with critical points; see, e.g., [4, 9, 17,
24, 42, 43, 44]. We note in particular the parameter exclusion technique of Jakob-
son [17] and the analysis of the quadratic family by Benedicks and Carleson [4].
The analysis of the Hénon family by Benedicks and Carleson [5] provided a break-
through from one-dimensional maps with critical points (the quadratic family) to
two-dimensional diffeomorphisms (small perturbations of the quadratic family).
Mora and Viana [25] generalized the work of Benedicks and Carleson to small per-
turbations of the Hénon family and proved the existence of Hénon-like attractors
in parametrized families of diffeomorphisms that generically unfold a quadratic
homoclinic tangency.
Wang and Young made several advances, two of which we discuss here. First,
the theory of rank 1 maps replaces the formulas of the Hénon diffeomorphisms
with a set of analytic and geometric hypotheses so that the theory can be applied to
concrete systems of differential equations. Second, as discussed earlier, a compre-
hensive dynamical profile is given for the nonuniformly hyperbolic maps. Expo-
nential decay of correlations for these maps is proved by applying the techniques
of [54, 55].
The theory of rank 1 maps has been applied to several concrete models thus
far. Examples include simple mechanical systems [49] and electronic circuits [29,
30, 46, 47]. Guckenheimer, Wechselberger, and Young [14] connect the theory of
rank 1 maps and geometric singular perturbation theory by formulating a general
technique for proving the existence of chaotic attractors for three-dimensional vec-
tor fields with two time scales. Lin [21] demonstrates how the theory of rank 1
maps can be combined with sophisticated computational techniques to analyze the

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HOMOCLINIC LOOPS AND RANK 1 ATTRACTORS5
Wu
Ws
(a) Transverse homoclinic intersections.
Ws
Wu
(b) Invariant manifolds do not intersect.
FIGURE 1.2. Some time-T maps that can occur when a system with a
homoclinic loop is subjected to periodic forcing of period T.
response of concrete nonlinear oscillators of interest in biological applications to
periodic pulsatile drives.
The dynamical scenario studied most extensively thus far is that of weakly
stable structures subjected to periodic pulsatile forcing. Weakly stable equilib-
ria [31], limit cycles [32, 49, 50], and supercritical Hopf bifurcations [50] in finite-
dimensional systems have been treated. Here intrinsic shear in the unforced system
is amplified by the cumulative effects of a pulsatile force followed by a long pe-
riod of relaxation. This amplification produces rank 1 dynamics. Lu, Wang, and
Young [22] use the theory of rank 1 maps and invariant manifold techniques to
prove that certain parabolic partial differential equations undergoing supercritical
Hopf bifurcations admit SRB measures when subjected to periodic pulsatile forc-
ing. In this paper we analyze systems with dissipative homoclinic loops.
1.3 Periodically Forced Homoclinic Solutions
Homoclinic phenomena including homoclinic tangles were first observed by
Poincaré [38, 39, 40] and have been studied extensively. Important systems in
this context include the nonlinear pendulum, the Duffing equation, and the van der
Pol oscillator [3, 11, 13, 19, 20, 45].
Consider a differential equation of the form
(1.2)
dx
dt
D f .x/
on RN. A homoclinic orbit is a solution ˆ of (1.2) that converges to a single
stationary point of saddle type as t ! ˙1. The homoclinic orbit is therefore part
of both the stable and unstable manifolds of the saddle. When a system with such
an orbit is forced periodically, the stable and unstable manifolds that coincide in
the unforced system will typically become distinct.
Figure 1.3 illustrates some of the possibilities. If the stable and unstable mani-
folds intersect transversely as in Figure 1.3(a), homoclinic tangles and horseshoes
are produced (horseshoes are invariant sets on which the dynamics are conjugate
to certain symbolic systems). This scenario has been studied extensively; see