# Rank One Chaos

Quidong Wang (2011), Scholarpedia, 6(10):9380. | doi:10.4249/scholarpedia.9380 | revision #91695 [link to/cite this article] |

**The theory of chaotic rank one attractors** originated from the
theory of Benedicks and Carleson on Hénon maps
((Benedicks and Carleson, 1985), (Benedicks and Carleson, 1991)) and the development that followed
((Benedicks and Young, 1993), (Benedicks and Young, 1999)). This theory was introduced by Lai-Sang
Young and myself in a sequence of articles ((Wang and Young, 2001),
(Wang and Young, 2008), (Wang and Young, 2010)). It takes the theory of Hénon-like maps
((Mora and Viana, 1993), (Benedicks and Viana, 2001)) as a particular case. The motivations for this theory are as follows:

(1) There was a geometric structure (namely the
structure of critical regions) that, if added, could make more
transparent the overall purpose of the different components of the
very complicated *tour de force* analysis of Benedicks and
Carleson. It could also induce non-trivial technical
simplifications. Considering the profoundness of the theory
in terms of impact, there was also a need to fill out, with care, the various technical details
skipped in (Benedicks and Carleson, 1991).

(2) It is *highly* desirable to have a theory on
non-uniformly hyperbolic maps that could be applied to the
analysis of specific differential equations. Such a goal could be achieved
in two steps. First, we could introduce a flexible
setting on non-uniformly hyperbolic maps and construct a
comprehensive chaos theory by generalizing previous analysis based
on (Benedicks and Carleson, 1991). Second, since many natural and
engineered systems are described by concrete differential equations
of rank one type, they offer many possibilities
for potential applications for this body of ideas.

We note that before this theory was introduced, the scope of
application of the theory on the Hénon maps had been extended by
Mora and Viana ((Mora and Viana, 1993)) to maps with transversal homoclinic
tangency through the Newhouse theory ((Newhouse, 1974), (Palis and Takens, 1993)). See
also (Diaz *et al.*, 1996).

The rest of this article is divided into two sections. In the first we discuss the content of the theory and in the second we discuss its application to differential equations.

## Contents |

## A dynamics theory on chaotic rank one attractors

### Admissible family of rank one maps

For \(m \geq 2\ ,\) let \({\mathcal A} = S \times D^{(m-1)}\) where \(S\) is the unit circle and \(D^{(m-1)}\) is the unit disk of dimension \(m-1\ .\) We consider a 2-parameter family of maps \(\{ T_{a, b}, \ (a, b) \in (a_0, a_1) \times (0, b_0) \}\) where \(T_{a, b}: {\mathcal A} \to {\mathcal A}\) is a diffeomorphism from \({\mathcal A}\) to its image. Let \((x_1, y_1) = T_{a, b}(x, y)\) be such that \[\tag{1} x_1 = F(x, y, a) + b u(x, y, a, b),\ y_1 = b v(x, y, a, b) \]

where \(x \in S\ ,\) \(y \in D^{(m-1)}\ .\) Assume that \(F(x, y, a), u(x, y, a, b), v(x, y, a, b)\) are \(C^3\) in \((x, y, a, b)\ ,\) and their \(C^3\)-norms are uniformly bounded by a constant \(K_0\) that is independent of \(a\) and \(b\ .\)

We call the 1D map \(f_a: S \to S\) defined by \(f_a(x) = F(x, 0, a)\)
the *1D singular limit* for \(T_{a, b}\ .\) We obtain \(\{f_a \}\)
by letting \(b = 0\) in (1). For small \(b\) we could regard
\(T_{a, b}\) as an m-dimensional map unfolded from the 1D singular
limit \(f_a\ .\) Let \(C(a) = \{ x: f'_a(x) = 0 \}\) be the set of
critical points for \(f_a\ .\) We assume that \(|\partial_y F(x, 0,
a)| \neq 0\) for all \(x \in C(a)\ .\) This is to say that the
unfolding from \(f_a\) to \(T_{a, b}\) is non-degenerate in the
direction of \(y\) on \(C(a)\ .\) We also assume
\[
\frac{\det(DT_{a, b}(z))}{\det(DT_{a, b}(z'))} < K
\]
for all \(z, z' \in {\mathcal A}\) where \(K\) is a constant
independent of \(b\ .\) The last is a regularity condition.

The rest of our conditions are imposed on the 1D singular limit
\(\{ f_a \}\ .\) We assume that there exists an \(a_* \in (a_0, a_1)\ ,\)
so that \(f_{a_*}\) is a *Misiurewicz map*. Misiurewicz maps are
the simplest 1D maps with non-uniform expansion (Misiurewicz, 1981). For a
Misiurewicz map, all critical orbits are kept a fixed distance
away from the critical set. We also impose a *parameter transversality condition* on \(f_{a}\) at \(a = a_*\) to assure that
changing \(a\) around \(a_*\) changes effectively the dynamics of
\(f_a\ .\) The precise formulation of these two conditions are a
little tedious and they are fully laid out in (Wang and Young, 2008), to which
we would refer the reader who is interested in such detail. What
is important for us is that these are *verifiable* conditions.
For instance, let
\[\tag{2}
f_a(\theta) = \theta + a + L \Phi(\theta)
\]

and assume that \(\Phi(\theta)\) is periodic in \(\theta\) and all critical points of \(\Phi(\theta)\) are non-degenerate. It is proved in (Wang and Young, 2002a) that, for \(L\) sufficiently large, there exists an \(a_*\) so that \(f_{a_*}\) is a Misiurewicz map and the parameter transversality condition holds at \(a = a_*\ .\)

A two parameter family \(T_{a, b}\) that assumes the form of
(1) and satisfies all assumptions listed above is an * admissible family of rank one maps*.

### Existence of chaotic rank one attractors

If the 1D singular limit \(f_a\) were *uniformly expanding*,
then \(T_{a, b}\) would represent an *Axiom A solenoid*
(Smale, 1967). This is to say that \(T_{a, b}\) would stretch \({\mathcal
A}\) in the \(x\)-direction, compress it in the \(y\)-direction. The
image would then be wrapped a few times and put back in \({\mathcal
A}\ .\) In this case, we have a global splitting of stable and
unstable directions hence a *uniformly hyperbolic attractor*.
See Figure 1(a).

For an admissible family of rank one maps, however, the 1D
singular limit is *not* uniformly expanding. See Figure 1(b).
For rank one maps, the direction in which the image is wrapped in
\({\mathcal A}\) *reverses* at the critical values of \(f_a\ ,\)
making it impossible for us to attain a global splitting of stable
and unstable directions.

The dynamics of the maps of Figure 1(b) is very different from
that of the Axiom A solenoid of Figure 1(a) because of the lacking
of a global splitting of stable and unstable directions. For the
maps of Figure 1(b), periodic sinks, representing stable dynamical
behavior, often co-exist with Smale's horseshoe. From a measure
theoretic stand point, periodic sinks are *directly observable* but horseshoes are not. This is to say that a sink
attracts an open, hence a positive measure, set of orbits but the
basin of attraction of a horseshoe is typically a measure zero
set. Is there chaos in a *directly observable* form for the
maps of Figure 1(b)? How about the existence of such maps that
admit a positive Lyapunov exponent on a positive Lebesgue measure
set? Smale's horseshoe is now insufficient to offer affirmative
answers to these questions.

Historically, these questions were asked for the Hénon maps and
were long standing until the ground breaking analysis of Benedicks
and Carleson emerged. Our first major task, in order to acquire a
more flexible theory on maps for future applications, was to
reconstruct the analysis of Benedicks and Carleson for an
admissible family of rank one maps. Therefore the following
statement is not only the first, but also *the main theorem*
for the theory of chaotic rank one attractors.

**Main Theorem** *Let \(\{ T_{a, b}, \ (a, b) \in
(a_1, a_2) \times (0, b_0)\}\) be an admissible family of rank one maps. Then there exists a positive measure set of parameters \(\Delta \subset (a_1, a_2) \times (0, b_0)\ ,\) such that for all \((a, b) \in \Delta\ ,\) \(T_{a, b}\) admits a positive Lyapunov exponent Lebesgue almost everywhere in \({\mathcal A}\ .\)*

The contents of the theory of chaotic rank one attractors are (a) to construct a positive measure set of parameters that fulfills this theorem, and (b) to further understand the dynamics of these chaotic rank one attractors by systematically building a dynamical profile for the maps of the good parameters constructed.

### Inductive construction of good parameters

Let \(\{ T_{a, b} \}\) be an admissible family of rank one maps, and
for \(T = T_{a, b}\ ,\) let
\[
\Omega = \cap_{n=0}^{+\infty} \ T^n({\mathcal A})
\]
be the attractor for \(T\ .\) Our main theorem was proved by
constructing a positive measure set \(\Delta\) of good parameters
through an elaborate inductive process. The good parameter set
\(\Delta\) is so constructed that, for \((a, b) \in \Delta\ ,\) the map
\(T = T_{a, b}\) possesses a well-defined critical set, which we
denote as \({\mathcal C} \subset \Omega\ .\) The critical set
\({\mathcal C}\) is such that (i) every point in \({\mathcal C}\) is a
point of quadratic tangency of well-defined local stable and
unstable manifolds, and (ii) the critical orbits
\(\cup_{n=-\infty}^{\infty} T^n({\mathcal C})\) are *all the tangency* in \(\Omega\) for \(T\ .\)

For the maps of good parameters, the critical set \({\mathcal C}\)
is a well structured Cantor set located around the critical points
of the 1D singular limit. Let \(x_0\) be a critical point of
\(f_{a_*}\ ,\) and let \(Q(x_0) = [x_0 - \delta, x_0+ \delta] \times
D^{(m-1)}\) where \(\delta > 0\) is a fixed small number. A
conceptual way to comprehend the structure of that Cantor set is
to *forever* replace every cylinder by a finite collection of
thinner and shorter cylinders inside, starting with \(Q(x_0)\) for
all \(x_0 \in C(a_*)\ .\) See Figure 2.

The set of good parameters and the critical sets \({\mathcal C}\)
are constructed side by side inductively. Each inductive step
starts with a temporary set of good parameters and a collection of
small cylinders, each of which we denote as *a critical region \(Q^{(k-1)}\)*. We construct inductively new critical regions
\(Q^{(k)}\) inside of \(Q^{(k-1)}\ ,\) deleting parameters that could
potentially ruin the intended quadratic tangency of local stable
and unstable manifolds. We prove that, at each step of the
induction, the measure of the deleted parameters declines
exponentially. At the end we obtain a positive measure set of
parameters with a well-defined critical set in \(Q(x_0)\) for all
\(x_0 \in C(a_*)\ .\)

The induction described above and the proof of the main theorem are complicated and long. See (Wang and Young, 2001), (Wang and Young, 2008) and (Wang and Young, 2010).

### Dynamics of chaotic rank one attractors

Through the inductive process discussed in the previous subsection we have obtained a new class of chaotic attractors. These chaotic attractors are genuinely non-uniformly hyperbolic in the sense that the stable and unstable directions are allowed to mix. Our focus is now to built a dynamical profile so we could claim a comprehensive understanding on the geometric and dynamical structure of these chaotic rank one attractors. In what follows we let \(T = T_{a, b}\) where \((a, b) \in \Delta\) is a good parameter that came out of the long inductive construction. Let \(\Omega = \cap_{n=0}^{+\infty} \ T^n({\mathcal A})\) be the chaotic rank one attractor, and \({\mathcal C}\) be the critical set for \(T\ .\) The theorems listed in this subsection are proved for 2D chaotic rank one attractors in (Wang and Young, 2001), and the corresponding version in higher dimensions is proved in (Wang and Young, 2008) and (Wang and Young, 2010).

**A. Hyperbolic structure and symbolic dynamics** We start with

**Theorem 1**

For any give \(\varepsilon > 0\ ,\) \[ \Lambda_{\varepsilon} = \{ z = (x, y) \in \Omega, \ d(T^n(z), {\mathcal C}) \geq \varepsilon, \ {\rm for \ all} \ n \in {\mathbb Z} \} \] is a uniformly hyperbolic invariant subset in \(\Omega\) for \(T\ .\)

This theorem assures that the critical set \({\mathcal C}\) is in fact the sole source of non-uniform hyperbolicity in \(\Omega\ .\)

We next turn to a special coding of orbits on \(\Omega\ .\) The following notation will be used\[x_1 < x_2 < \cdots < x_q < x_{q+1} = x_1\] are the critical points of \(f_{a_*}\ ,\) and \({\mathcal C}_i\) is the part of \(\mathcal C\) near \((x_i, 0) \in {\mathcal A}\ .\) Let \(\Sigma_q= \Pi_{- \infty}^\infty \{1,2, \cdots, q\}\) be the space of sequences of \(q\)-symbols, and let \(\sigma: \Sigma_q \to \Sigma_q\) be the shift map.

**Theorem 2**

(a) There is a natural partition of \(\Omega \setminus {\mathcal C}\) into disjoint sets \(A_1, A_2, \cdots, A_q\) so that \(z \in A_i\) can be thought of as being "to the right" of \({\mathcal C}_i\) and "to the left" of \({\mathcal C}_{i+1}\ .\)

(b) Under the additional assumption that \(f_{a_*}([x_j, x_{j+1}]) \not \supset S^1\) for any \(j\ ,\) there is a closed subset \(\Sigma \subset \Sigma_q\) with \(\sigma(\Sigma) \subset \Sigma\) and a map \(\pi: \Sigma \to \Omega\) with the property that

- for all \(\mathbf{s}=(s_i) \in \Sigma\ ,\) \(\pi(\mathbf{s})=z\) implies that \(T^iz \in \bar A_{s_i}\) for all \(i\ ;\)

- \(\pi\) is a continuous surjection that is 1-1 except on \(\cup_{i=-\infty}^\infty T^i {\mathcal C}\ ,\) where it is 2-1.

If the assumption in Part (b) does not hold, the statement can be amended by increasing the number of symbols as follows: For \(f_{a_*}\ ,\) use the partition given by \(C(a_*) \cup f_0^{-1}(C(a_*))\) instead of that by \(C(a_*)\ ,\) and do likewise with \(T\ .\) See also (Wang and Young, 2010) for more results that follow immediately from the symbolic coding of Theorem 2.

**B. Statistical properties** Inside a chaotic rank
one attractor, orbits jump around in a seemingly random fashion
and the future of individual orbits appears entirely * unpredictable*. For these chaotic attractors, however, there are,
*laws of statistics* that control the asymptotic distributions
of Lebesgue almost all orbits in \({\mathcal A}\ .\)

The precise meaning of the last statement is as follows. Let us
first partition \({\mathcal A}\) into a collection of disjoint
sub-regions. For a finite orbit of \(T\ ,\) we use this partition to
define a histogram by counting the proportion of points on this
finite orbit in each of the sub-regions. We then take two limits:
first we let the orbit go infinitely long and second we refine the
partition so the maximal size of all sub-regions goes to zero. In
principle, there is no reason *a priori* for the histogram to
converge to anything in any one of the two limit processes. For
chaotic rank one attractors, however, *not only* the
histograms converge, *but also* they converge to only a finite
number of probability distributions for Lebesgue almost all orbits
in \({\mathcal A}\ .\) These limit distributions are the
Sinai-Ruelle-Bowen measures for \(T\ .\)

**Theorem 3**

There are at least one, and at most \(q\ ,\) ergodic SRB measures for \(T\) where \(q\) is the number of critical points of the 1D singular limit.

See (Young, 2002) and the reference therein for more on the theory of SRB measures. In addition to Theorem 3, we have also proved many refined dynamical properties of chaotic rank one attractors, including a central limit theorem, an exponential mixing rate and a principle of large deviations by applying the theories developed previously in (Young, 1998) and (Rey-Bellet and Young, 2008). See (Wang and Young, 2010).

## Application to differential equations

The theory of chaotic rank one attractors has been applied to the
analysis of many naturally and engineered systems described by
concrete differential equations. In this section we present three
applications. The first application is on periodically perturbed
homoclinic solutions (Wang and Ott, 2007). The second and the third
applications are both under the framework of periodically kicked
stable limit cycles (Wang and Young, 2002a). The second is on the dynamics of
periodically kicked equations with Hopf bifurcation (Wang and Young, 2002b) and
the third is an application to PDE arena \cite{LWY}. For more
examples of application see also (Lin and Young, 2010), (Lin, 2006),
(Guckenheimer *et al.*, 2006), (Wang and Oksasoglu, 2008).

### Periodically perturbed homoclinic solutions

Periodically-forced second order equations, including the forced nonlinear pendulum, Duffing equation, and van der Pol oscillator, have been studied extensively in history ((Duffing, 1918), (Cartwright and Littlewood, 1945), (Levinson, 1949) (Guckenheimer and Holmes, 1983), (van der Pol and van der Mark, 1927)). When a homoclinic solution in a given second order equation is periodically perturbed, transversal intersections of stable and unstable manifolds occur within a certain range of the forcing parameter, generating homoclinic tangles and chaotic dynamics ((Poincaré, 1899), (Smale, 1965)). There also exist values of parameter for which the stable and unstable manifolds of the perturbed saddle are pulled apart. For these two cases, Figure 3 schematically illustrates the time-\(T\) maps for the perturbed equations, where \(T\) is the period of the perturbation.

Historically, the homoclinic tangles of Figure 3(a) have been regarded as a major venue for chaos to be manifested in differential equations. Proving chaos through the intersection of the stable and unstable manifolds ((Melnikov, 1963), (Smale, 1965)) has become a standard practice in the study of differential equations. This is to say that we often prove chaos in a given equation via the elimination of the scenario of Figure 3(b).

Afraimovich and Shilnikov, on the other hand, argued for chaos
*in the scenario of Figure 3(b)* in (Afraimovich and Shil'nikov, 1977). They proposed
to study the dynamics of periodically perturbed homoclinic
solutions through the *separatrix map*, a return map around
the unperturbed homoclinic solution in the extended phase space.
Afraimovich and Shilnikov illustrated that, when a homoclinic
solution is periodically perturbed, then horseshoe exists as well
in the scenario of Figure 3(b) provided that the forcing frequency
is sufficiently large.

Following the method proposed in (Afraimovich and Shil'nikov, 1977), Ott and I rigorously
derived the separatrix map for periodically perturbed equation
with a homoclinic saddle in (Wang and Ott, 2007). We verified that, under the
assumption that the saddle fixed point is dissipative and
non-resonant, *the separatrix map is a family of rank one maps* with a 1D singular limit in the form of
\[
f_a (\theta) = \theta + a + \frac{\omega}{\beta} \ln M(\theta)
\]
where \(\omega\) is the forcing frequency; \(\beta\) is the unstable
eigenvalue of the saddle fixed point; \(M(\theta)\) is the classical
Melnikov function, which is *explicitly computable* for a
given equation; and \(a\) is such that
\[
a = \frac{\omega}{\beta} \ln \mu^{-1} + K,
\]
in which \(\mu\) is a parameter representing the magnitude of the
perturbation and \(K\) is a constant from the given equation.
Therefore if the Melnikov function \(M(\theta)\) is such that
\(\min_{\theta} M(\theta) > 0\) (which puts us directly into the
scenario of Figure 3(b)), and all critical points of \(M(\theta)\)
are *non-degenerate*, then the separatrix map is an admissible
family of rank one maps provided that the forcing frequency is
sufficiently large. In this case we obtain chaotic rank one
attractors for a positive measure set of \(\mu\ .\)

As a concrete example, we applied the theory of chaotic rank one attractors to the Duffing equation in the form of \[\tag{3} \frac{d^2 q}{dt^2} + (\lambda - \gamma q^2) \frac{d q}{dt} - q + q^3 = \mu \sin \omega t. \]

It is well-known that, for every \(\lambda = \lambda_0 > 0\) sufficiently
small, there exists a \(\gamma = \gamma_0\) such that equation
(3) has a homoclinic solution when \(\mu = 0\ .\) This
homoclinic solution we denote as \(\ell_0\ .\) Our next theorem
asserts the existence of a *directly observable global chaotic attractor* in the neighborhood of \(\ell_0\ .\)

**Theorem 4**

Let \(\lambda_0 > 0\) be sufficiently small and \(\gamma_0\ ,\) \(\ell_0\) be as in the above. Then there exists a positive measure set \(\Delta\) for \((\gamma, \omega, \mu)\) close to \((\gamma_0, \infty, 0)\) such that for \((\gamma, \omega, \mu) \in \Delta\ ,\) the Duffing equation (3) admits a unique chaotic rank one attractor in an open neighborhood of \(\ell_0\ .\)

The results reported in this subsection are due to Ott and myself (Wang and Ott, 2011).

### Periodically kicked super-critical Hopf bifurcations

Our next application is about periodically kicked super-critical
Hopf bifurcations. *Hopf bifurcation* is a dynamical scenario
of historical and practical importance in both theory and
application. External actions in the form of an impulse function
are often modelled by *periodic kicks* in various
applications, including in switch controlled circuits and in
neuroscience and biology.

Let us start with the theory on Hopf bifurcation. For \(x \in {\mathbb R}^m, \ m \geq 2\ ,\) consider the following \(\mu\)-dependent system of differential equations \[\tag{4} \frac{d x}{d t} = A_{\mu} x + f_{\mu}(x) \]

where \(A_{\mu}\) is an \(m \times m\) matrix and \(f_{\mu}(x)\) is a vector valued function of order \(\geq 2\) at \(x = 0\ .\) We assume that, around \(\mu = 0\ ,\) all eigenvalues of \(A_{\mu}\) except a conjugating pair, which we denote as \(\lambda_{1, 2}\ ,\) are with negative real part. We assume that \(\lambda_{1, 2} = a(\mu) \pm \omega(\mu) \sqrt{-1}\) are such that \(a(0) = 0, \ \omega(0) \neq 0\ .\) Corresponding to \(\lambda_{1, 2}\ ,\) equation (4) has a 2-dimensional local center manifold \(W^c\) at \(x = 0\ ,\) and the equation induced on \(W^c\) can be written in a complex variable \(z\) in a normal form as \[\tag{5} \dot z = (a(\mu) + i \omega(\mu)) z + k_1(\mu) z^2 {\bar z} + k_2(\mu) z^3 {\bar z}^2 + \cdots \]

where \(k_1(\mu), k_2(\mu), \cdots\) are complex numbers. We have a
*generic Hopf bifurcation* for equation (4) if
\(Re(k_1(0)) \neq 0\ .\) A Hopf bifurcation is *super-critical* if
\(Re(k_1(0)) < 0\) and it is *sub-critical* if \(Re(k_1(0)) > 0\ .\)
In the rest of this subsection we assume \(Re(k_1(0)) < 0\) so we
are interested in a super-critical Hopf bifurcation, in which a * stable* periodic solution comes out of \(x = 0\ .\)

We now add a time-dependent forcing term to equation (4) to form a new equation \[\tag{6} \frac{dx}{dt} = A_{\mu} x + f_{\mu}(x) + \varepsilon \Phi(x) \sum_{n = -\infty}^{+ \infty} \delta(t - nT) \]

where \(\varepsilon\) is a small parameter representing the magnitude of the forcing, \(\Phi(x)\) is such that \(\Phi(0) = 0\ ,\) and \(\delta(t)\) is the standard \(\delta\)-function.

We computed the time-T map *around* the stable periodic
solution coming out of \(x = 0\) for small \(\mu\ .\) It turned out that
the time-T map is a family of rank one maps with a singular 1D
limit, as \(T \to \infty\ ,\) in the form of
\[
f_a(\theta) = \theta + a + L \phi(\theta)
\]
where \(a \approx \omega(0)T \ mod(2 \pi)\ ;\) \(\phi(\theta) =
\phi_0(\theta) + {\mathcal O}(\sqrt{\mu}) + {\mathcal
O}(\varepsilon)\) in which \(\phi_0(\theta)\) is a function of
\(\theta\) determined by \(A_0\) and \(D\Phi(0)\ ;\) and \(L \approx
\varepsilon \cdot \tau\) and
\[
\tau = \left| \frac{Im(k_1(0))}{Re(k_1(0))}\right|.
\]
So to find chaotic rank one attractors for equation
(6),

- (i)] first we compute \(\phi_0(\theta)\ ,\) and verify that all its critical points are non-degenerate;
- (ii)] second we compute the normal form (5) to find \(k_1(0)\ ,\) and we look for equations for which \(\tau\) is large;
- (iii)] let \(\mu\) and \(\varepsilon\) be sufficiently small so that \(\phi(\theta)\) is close to \(\phi_0(\theta)\ ;\) we also let \(L = \tau \cdot \varepsilon\) be sufficiently large;
- (iv)] we then conclude that chaotic rank one attractors exist for a positive measure set of \(T\ .\)

We skip the precise statement and refer the reader to (Wang and Young, 2002b), \cite{LWY} and (Wang and Oksasoglu, 2008) for more details.

To provide a concrete application we applied this theory to a
periodically kicked Chua's circuit. The circuit is shown inFigure 3(a) and the periodic kicks are implemented by switches
being turned on and off periodically. We analyzed the differential
equation for the circuit following the steps (i)-(iv) above, and
found parameters for which the chaotic rank one attractors are
likely to occur. We then physically built the circuit. Figure 3(b)
is a picture of chaotic rank one attractor that came out of lab
simulations. See (Wang and Oksasoglu, 2008) and (Demirkol *et al.*, 2009).

The theory on periodically kicked Hopf bifurcation is due to Young
and myself (Wang and Young, 2002b); the application to Chua's circuit is due to
Oksasoglu and myself (Wang and Oksasoglu, 2008); and the circuit is built by
Demirkol, Ozoguz, and Akgul (Demirkol *et al.*, 2009).

### Application to a parabolic partial differential equation

The *Brusselator*, as described by the equations below, is a
simplified model of an autocatalytic chemical reaction with
diffusion (Lefever and Prigogine, 1968):
\[\tag{7}
u_t = d_1 \Delta u + a - (b+1)u + u^2v,\ v_t = d_2 \Delta v + bu -u^2v.
\]

We consider this model in one physical dimension, i.e. \(u = u(x, t)\) and \(v=v(x, t)\) for \(x \in [0, 1]\ ,\) and let \(\Delta =\partial_{xx}\ .\) Here \(a\) and \(b\) are constants representing concentrations of certain initial substances, \(u\) and \(v\) are variables representing concentrations of two intermediates, and \(d_1, d_2 >0\) are their respective diffusion coefficients. The term \(u^2v\) represents the autocatalytic step in the reaction.

One sees immediately that \((u(t),v(t)) \equiv (a, ba^{-1})\) is a stationary solution. Letting \(U=u-a\ ,\) \(V=v-ba^{-1}\ ,\) and \(\mathbf{u}=(U,V)\ ,\) we write (7) as an evolutionary equation \[\tag{8} \dot \mathbf{u}=\mathcal{A}_{a,b} \mathbf{u} + f_{a,b}(\mathbf{u}), \]

where
\[
\mathcal{A}_{a,b}=\begin{pmatrix}
d_1 \frac{\partial^2}{\partial x^2} + (b-1)& a^2\\
-b & \theta d_1\frac{\partial ^2}{\partial x^2} - a^2
\end{pmatrix}, \qquad \theta = \frac{d_2}{d_1}, \ \ d_1 \neq 0;
\]
and
\[
f_{a, b}(\mathbf{u}) = \begin{pmatrix}
UV^2 + ba^{-1} U^2 + 2aUV \\
-UV^2 - ba^{-1} U^2 - 2a UV
\end{pmatrix}.
\]
It is well known that Hopf bifurcation occurs as the parameters
\(a\) and \(b\) are varied. See (Hassard *et al.*, 1981).

We study periodically kicked equations satisfying the Neumann boundary condition and the Dirichlet boundary condition. The forcing functions are chosen to respect the corresponding boundary condition.

**Neumann boundary condition.**
\[
u_t = d_1 \Delta u + a - (b+1)u + u^2v +
\rho (1+\cos \pi x) \ p_{T,\iota}(t),
\]
\[
v_t = d_2 \Delta v + bu -u^2v;
\]
\[\tag{9}
\partial_x u(0,t) = \partial_x u(1,t) = 0, \qquad \partial_x v(0,t) = \partial_x v(1,t) = 0.
\]

**Dirichlet boundary condition.**
\[
u_t = d_1 \Delta u + a - (b+1)u + u^2v + \rho \sin \pi x \ p_{T,\iota}(t),
\]
\[
v_t = d_2 \Delta v + bu -u^2v;
\]
\[\tag{10}
u(0,t) = u(1,t) = a, \qquad v(0,t) = v(1,t) = ba^{-1}.
\]

In both cases, \(\rho \geq 0\) and \(0<\iota<<1<<T\) are constants and
\[
p_{T,\iota}(t)= \sum_{n=-\infty}^{\infty} p_\iota (t - nT)
\quad {\rm with} \quad p_{\iota}(t) = \left\{ \begin{array}{lll}
\iota^{-1} & & 0 \leq t < \iota,
\\ 0 & & \rm{elsewhere}.
\end{array} \right.
\]
We use \(p_{T,\iota}(t)\) in the place of a periodic delta function
to avoid certain regularity problems. Our plan in \cite{LWY} is
to first develop a theory on chaotic rank one attractors for
periodically kicked super-critical Hopf bifurcation in the context
of infinite dimensional systems. We then apply this theory to
equations (9) and (10).

**Theorem 5 (Neumann Boundary condition)**

Let \(d_1>0\ ,\) \(\theta<1\) be fixed. Then for \(a\) sufficiently large, there is an open set of \(\rho, \iota\) and \(b\) (depending on \(d_1\ ,\) \(\theta\) and \(a\)), \(b \approx a^2+1\ ,\) for which the time-T map of equation (9) has a chaotic rank one attractor for a positive measure set of large \(T\ .\)

The computation needed for the Dirichlet boundary condition is considerably more involved than those for Neumann boundary condition. We limited ourselves in \cite{LWY} to the special case \(d_1=\pi^{-2}\ .\)

**Theorem 6 (Dirichlet boundary condition)**

Let \(d_1=\pi^{-2}\ ,\) \(0< \theta <<1\ .\) Then there are open sets of \(a, b, \rho\) and \(\iota\ ,\) for which the time-T map of equation (10) has a chaotic rank one attractor for a positive measure set of large \(T\ .\)

We refer the interested reader to \cite{LWY} for details on the infinite dimensional theory and the application. The results reported in this subsection are due to Lu, Young and myself \cite{LWY}.

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