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CUBO A Mathematical Journal Vol.15, N¯o01, (131–149). March 2013
Planar Pseudo-almost Limit Cycles and Applications to solitary Waves
Bourama Toni Virginia State University, Department of Mathematics & Computer Science, Petersburg VA 23806.
[email protected]
ABSTRACT
We investigate the existence of pseudo-almost limit cycles, a new class of non-periodicity at the interface of the theories of limit cycles and pseudo-almost periodicity. We determine the conditions of existence for several systems including some pseudo-almost periodic perturbations of the harmonic oscillator and the renowned Li´enard systems. We apply to derive the existence of pseudo-almost periodic solitary waves by perturbing first then transforming some hyperbolic and parabolic partial differential equations to Li´enard-type equations. Included also are open questions on the co-existence of limit cycles and strictly pseudo-almost periodic limit cycles partitioning the phase space, and the existence of isochronous pseudo-almost limit cycles.
RESUMEN
Investigamos la existencia de ciclos seudo-casi l´ımites, una nueva clase de no-periodicidad en la interfaz de las teor´ıas de ciclos l´ımites y seudo-casi periodicidad. Determinamos condiciones de existencia de muchos sistemas, incluyendo algunas perturbaciones seudocasi perio´dicas del oscilador armo´nico y los sistemas de Li´enard. Aplicamos las condiciones para derivar la existencia de ondas solitarias seudo-cuasi perio´dicas, primero perturbando y luego transformando algunas ecuaciones diferenciales parciales hiperb´olicas y parab´olicas a ecuaciones del tipo Li´enard. Tambi´en se incluyen preguntas abiertas sobre la co-existencia de ciclos l´ımite y estrictamente pseudo-casi perio´dicos ciclos l´ımite de partici´on del espacio de fases, y la existencia de is´ocrono pseudo-casi ciclos l´ımite .
Keywords and Phrases: Limit cycles. Almost and pseudo-almost periodic orbits. Periodic waves. Isochronous systems and Isochrons. Li´enard systems. Hyperbolic and parabolic equations. 2010 AMS Mathematics Subject Classification: 34C05, 34C07, 34C27, 34K14

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1 Introduction

Bourama Toni

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Limit cycles are used to model the dynamical state of self-sustained oscillations found very often in biology, chemistry, mechanics, electronics, fluid dynamics, etc. See for example [2, 16, 18, 26]. They often arise in many physical systems around a state at which energy generation and dissipation balance. One of the most important limit cycles of our lives is the heartbeat. A spectacular example is the Tacoma Narrows Bridge1. and its 1940 dramatic collapse, where the limit cycle drew its energy from the wind and involved torsional oscillations of the roadbed. In Robotics the stable gait to which the repeated dynamic walking pattern converges is modeled as a stable limit cycle, stability easily lost to even small disturbances, evidence of a narrow basin of attracting of the limit cycle.
Planar limit cycles were defined by Poincar´e2 in the famous paper M´emoire sur les courbes d´efinies par une ´equation diff´erentielle [22], using his so-called Method of sections. However much attention in this century has been drawn to the determination of the number, amplitude and configuration of limit cycles in a general nonlinear system, which is still an unsolved problem. This is part of the so-called Hilbert’s 16th Problem3. A weakened version4 by Arnold called the tangential Hilbert’s problem, concerns the bound on the number of limit cycles which can bifurcate from a first-order perturbation of a Hamiltonian system.[3, 9, 13, 14, 17]
The possibility of a limit cycle on a plane or a two-dimensional manifold is restricted to nonlinear dynamical systems, due to the fact that, for linear systems, kx(t) is also a solution for any constant k if x(t) is a solution. Therefore the phase space will contain an infinite number of closed trajectories encircling the origin, with none of them isolated. Conservative and gradient systems do not have limit cycles, but these systems may exhibit almost or pseudo-almost limit cycles. The most common techniques for predicting the absence or existence of periodicity and limit cycles include the Index Theory, Dulac’s Criterion, Poincar´e-Bendixson Test, Perturbation and Bifurcation theory, Configuration of limit cycles, the Toroidal Principle. These concepts and related examples could be found in [2, 5, 6, 9, 10, 13, 18, 25]. The nonlinear character of isolated periodic oscillations renders their detection and construction challenging. In mechanical terms the appraisal of the regions of the phase plane where energy loss and energy gain occur might reveal a limit cycle.
Let us emphasize that even though in most studies periodicity has been illustrated more frequently, almost and pseudo-almost periodic oscillations or waves actually occur much more
1A wealth of information including historical and anecdotal facts could be found in http://en.wikipedia.org/wiki/Tacoma-Narrows-Bridge(1940)
2Jules Henri Poincar´e has excelled in all fields of knowledge and is often described as a polymath or The Last Universalist. The famous Poincar´e conjecture named after him was finally solved in 2002-2003 by Grigori Perelman who turned down the related prize of $1, 000, 000!
3Determine the maximum number and relative positions of limit cycles in polynomial vector fields of degree n. Stated in 1900, it was only in 1987 that Ecalle and Ilyashenko proved independently the finiteness of that number using the compactification of the phase space to Poincar´e disk
4The number of limit cycles in a small perturbation of a polynomial Hamiltonian system is given by the number of zeroes of Abelian integrals at least far from polycycles.

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frequently than periodic ones. For instance, in the simplest model of harmonic oscillator or mathematical pendulum, as well as for the one-dimensional wave equation, diverse kinds of oscillatory trajectories can be displayed, both periodic and more generally non-periodic.
The theory of almost periodic functions introduced by H. Bohr [6] in the 1920s and extended to pseudo-almost periodicity5 by Zhang [27] in the 1990s is also connected with problems in differential equations, stability theory, dynamical systems, partial differential equations or equations in Banach spaces. There are several results concerning the existence and uniqueness of almost and pseudoalmost periodic solutions for first-order differential equations, e.g., in [7, 11, 12, 15, 20, 21, 23, 24, 27]. But the authors usually derived their results from the existence of bounded solutions.
We extend the theory of limit cycles and pseudo-almost periodicity to that of pseudo-almost limit cycles, isolated pseudo-almost periodic orbits, and we investigate in the current and future work the usual questions of conditions of existence and uniqueness, stability, bifurcation and perturbation, the coexistence of limit cycles and strictly pseudo-almost limit cycles. We also introduce the idea of isochronous pseudo-almost limit cycles and pseudo-almost isochrons6.
Section 2 overviews the theory of limit cycles recalling the definitions and presenting some classic and concrete examples relevant to our study. In section 3, we develop the concept of pseudoalmost limit cycle, its properties, several illustrative examples including the so-called linear pseudocenter, and existence theorems in the case of the well-known Li´enard systems. Section 4 shows the applications of the existence theorems for Li´enard systems to obtain pseudo-almost periodic solitary waves for some hyperbolic and parabolic partial differential equations. Finally in section 5 we discuss some directions for future research, and state some open problems, defining in the process the concept of isochronous pseudo-almost limit cycles and pseudo-almost isochrons.

2 Preliminary Definitions and Examples

Let the multi-dimensional space Rn represents all the possible states of a system modeling nonlinear phenomena. The dynamics of the system are determined by the values in Rn in terms of the time. That is to say we define an evolution map or flow Φ, smooth on the smooth manifold Rn :

Φ : Rn × R −→ Rn,

(2.1)

such that Φ(x, t) = y indicates that the state x ∈ Rn evolved into the state y ∈ Rn after t units of time, together with the usual flow properties

Φ(x, 0) = x, Φ(x, t1 + t2) = Φ(Φ(x, t1), t2).

(2.2)

5Any pseudo-almost periodic function is also a Besicovitch almost periodic function 6The development of the concept of isochrons and the recognition of their significance is due to Winfree (1980)

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The flow Φ then determines a vector field X (conversely as well) such that, for x ∈ M
X (x) := ∂Φ (x, 0). ∂t
The orbit or trajectory of the flow through x ∈ Rn is given by:

(2.3)

O(x) := {Φx(t) := Φ(x, t)|t ∈ R}.

(2.4)

Definition 2.1. The orbit γ = O(x) based at x is called a limit cycle if there is a neighborhood N of γ such that γ is the only periodic orbit contained in N .

The limit cycle7 is stable (unstable) if ω(s) = γ (α(s) = γ) for any s ∈ N , that is, γ is the ω−limit set (α− limit set) of any point in N . In other words, the limit cycle, isolated periodic orbit of some period τ, is stable (resp. unstable) if it has a neighborhood N such that, for some distance function d on Rn, d(Φ(y, t), γ) −→ 0, as t → ∞ (resp. t → −∞), for any y ∈ N .
Note that the phase ϕ = Tt0 of a limit cycle of period T0 refers to the relative position on the orbit, which is measured by the elapsed time (modulo the period) to go from a reference point to the current position on the limit cycle. The most common illustrative examples are from the perturbations of the linear center or linear isochrone.

2.1 Linear center and its perturbations

2.1.1 Poincar´e oscillator

The linear center or linear isochrone8

x˙ = −y, y˙ = x,

(2.5)

where the origin of the plane is surrounded by a continuum of periodic orbits (not isolated) given by x2 + y2 = c > 0, is perturbed into the following system, in polar coordinates (r, θ)

r˙ = r(1 − r), θ˙ = 1

(2.6)

The circle r = 1 is a 2π−periodic orbit and is unique. It is therefore a limit cycle. Moreover r is a monotone function on each orbit (r˙ > 0 inside and < 0 outside) so that all non constant orbits tend towards the limit cycle which is therefore stable.9[2, 18].10

7A limit cycle actually controls the behavior of neighboring orbits, attracting/repelling on both sides, or attracting
on one side and repelling on the other 8The term isochrone refers to the fact that all the periodic orbits in the continuum have the same constant period
normalized to 2π. 9The Poincar´e’s oscillator has been considered a model of biological oscillations, in particular with respect to the
effects of periodic stimulation of cardiac oscillators 10The isochrons here are radial lines from which the trajectories evolve to equal phase

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2.1.2 Limit cycles Annulus

The linear center could also be perturbed into a system to generate several limit cycles as in the following example. The C∞−system

x˙ = −y + xp(x, y), y˙ = x + yp(x, y),

(2.7)

where

1

−1

p(x, y) = sin( x2 + y2 )e x2+y2 ,

has an infinite number of limit cycles

γn : x2 + y2 = 1 , n ∈ N nπ

accumulating at the origin.[9]

(2.8)

2.1.3 Remarks
The linear center is a continuum of periodic orbits encircling a critical point. The perturbation in examples 1 and 2 has in fact destroyed these orbits to give birth to respectively a unique limit cycle in example 1, and an accumulating family of limit cycles in example 2. We will see below that a time-dependent pseudo-almost perturbation could lead to the emergence of the so-called pseudo-almost limit cycles.

3 Pseudo-almost limit cycles

3.1 Introductory Concepts

Let C(R × Ω, Rn), Ω ⊂ Rn open, be the Banach space of bounded continuous functions φ(t, x) endowed with the norm ||φ|| = supt∈R,x∈Ω|φ(t, x)|. The set C(R × Ω, Rn) is a subset of the more general space Lb(R × Ω, Rn) of all Lebesgue measurable and bounded functions.

Definition 3.1. A function f in Lb(R × Ω, Rn) is said to be ergodic if for every compact subset

K ⊂ Ω the mean defined by

M(f) := lim 1 T f(t, x)dt, T →∞ 2T −T

(3.1)

exists uniformly for x ∈ K.

We say that the function has a vanishing mean if M(f) = 0. Let E(R × Ω, Rn) denote the space of all ergodic functions on R × Ω. Note in passing that not all uniformly continuous bounded

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functions on R are ergodic. For instance the function

f(t) = {1 − t2, for |t| < 1, and sin(log( 1 )), for |t| ≥ 1, } t2

(3.2)

is uniformly continuous in R, but not ergodic.

In the space L(R × Ω, Rn) of all Lebesgue measurable functions on R × Ω, we consider next
the following subspace L0 of all functions φ : R × Ω → Rn such that ∀x ∈ Ω, φ˜ (.) := φ(., x) is Lebesgue measurable on R with M(|φ˜ |) = 0, and M(|φ|) = 0.

For example the function φ(t) = t| sin πt|tN , N > 6,

(3.3)

is unbounded, Lebesgue measurable with vanishing mean M.

The unbounded and discontinuous function √
φ(t) := { n, n ≤ t ≤ n + 1/n, and 0, otherwise}

(3.4)

is

also

an

element

of

L0.

Indeed

we

have

limT →∞

1 2T

T −T

|φ(t)|dt

=

limn→∞

1 n

n k=1

√1 k

=

0.

Definition 3.2. The orbit O(x0) based at x0 as defined above is called a pseudo-almost limit cycle if it is isolated, and more importantly if the function Φ(.) := Φx0 (.) : R −→ Rn defining the orbit is pseudo-almost periodic in the following sense: ∀ǫ > 0, ∃δ = δ(ǫ) > 0, a relatively dense
subset Dǫ ⊂ R, a subset Cǫ ⊂ R, such that:

(1) For m the Lebesgue measure on R,

lim m(Cǫ ∩ [−t, t]) = 0,

t→∞

2t

(Cǫ is called an ergodic zero set),

(2) Let TτΦ denotes the translate of Φ by τ, that is, (TτΦ(t)) := Φ(t + τ). Then

||(TτΦ)(t) − Φ(t)|| < ǫ, τ ∈ Dǫ, t, t + τ ∈ R − Cǫ,

(3.5) (3.6)

(3) Finally

|t1 − t2| < δ =⇒ ||Φ(t1) − Φ(t2)|| < ǫ, t1, t2 ∈ R − Cǫ.

(3.7)

Denote PA the space of pseudo-almost periodic functions. These functions satisfy the following properties widely available in the relevant literature. [11, 12, 27]

3.1.1 Some properties of pseudo-almost periodicity
We first give an equivalent definition of a pseudo-almost periodic function, in particular in the space C(R × Ω, Rn), with the restriction of L0 to the space E0 containing all functions φ ∈ C(R × Ω, Rn)

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such that uniformly in x ∈ Ω.

lim 1 T |φ(t, x)|dt = 0, T →∞ 2T −T

(3.8)

Definition 3.3. A function f : R × Ω −→ Rn is called pseudo-almost periodic in t uniformly on compact subsets K of Ω if it has a unique decomposition in the form

f(t, x) = a(t, x) + e(t, x),

(3.9)

where the component a is almost periodic, and the component e ∈ E ⊂ L0 is called the ergodic perturbation of f. Recall that a is almost periodic if it satisfies the so-called Bohr’s property. That is: ∀ǫ > 0 ∃l = l(ǫ) such that any interval (t, t + l) ⊂ R contains a number τǫ, the ǫ−almost period or ǫ−translation number, such that:

||f(t + τ, x) − f(t, x)|| < ǫ, t ∈ R, x ∈ Ω.

(3.10)

We have the following properties relevant to our study and details could be found in Zhang [27] and also in [11, 12].

(1) For f ∈ PA, the range f(R, K) := {f(t, x)|t ∈ R, x ∈ K} is bounded for every bounded subset K ⊂ Ω.
(2) The function f(t, .) ∈ PA is uniformly continuous in each bounded subset of Ω uniformly in t.
(3) When the ergodic zero set Cǫ = ∅, the space PA coincides with the space AP of almost periodic functions.
(4) If both functions f and its derivative f′ are pseudo-almost periodic, with f = a + e and f′ = a′ + e′, where a and a′ in PA and e and e′ in L0, then the functions a and e are differentiable with a′ = a and e′ = e.
(5) The space PA is convolution invariant with the space L1(R) of integrable functions on R.

3.1.2 Illustrative Examples

We present some by now classic examples of pseudo-almost periodic functions. See also [12, 27]. We include here their graphics.

(1) Example 1 The function



e−|t|

φ1(t) = sin t + sin 2t + 1 + t2

(3.11)

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has√the almost periodic component a(t) = sin t + sin 2t, and the ergodic perturbation e(t) = 1e+−|tt2| . We represent it along with its components in Figure 1.

(2) Example 2 We have also the function



−|t|

Figure 1: φ1(t) = sin t + sin

2t

+

e 1+t2

φω(t) = I1(t) + I2(t), ω = 0,

(3.12)

with the almost periodic component





I1(t) = h(t − s)(sin s + sin 2s)ds,

−∞

and the ergodic component

∞ h(t − s) I2(t) = −∞ s2 + ω2 ds

h ∈ L1(R)

(3.13) (3.14)

We take h(t) = t2, in L1(R), ω = 1 to illustrate in Figure 2.

Figure 2: φω(t) = I1(t) + I2(t)

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3.2 Existence of Pseudo-almost limit cycles
First note that a periodic or almost periodic function is also pseudo-almost periodic with a zero ergodic perturbation. Consequently a limit cycle is also an almost or a pseudo-almost limit cycle, but not inversely To make the distinction, we will call strictly pseudo-almost limit cycles those pseudo-almost limit cycles that are not limit cycles.
We start with the case of the linear pseudo-almost center.

3.2.1 Linear pseudo-almost center: an example

Let p(t) ∈ PA(R, C) be a complex-valued pseudo-almost periodic function defined on the real numbers, and consider the differential equation (see also [11])

x˙ (t) = −αx(t) + p(t), α > 0.

(3.15)

Define a kernel

K(t) =

0, for t < 0 e−αt, for t ≥ 0

(3.16)

We have K ∈ L1(R, C). Thus the convolution

t
xα(t) = (K ∗ p)(t) = e−αt eαsp(s)ds
−∞

(3.17)

is also in PA(R, C), for every α > 0. Indeed the space PA is convolution invariant with L1. The

equation being linear, it results in the existence of a continuum of parameterized pseudo-almost

periodic solutions which we called linear pseudo-almost center. Therefore these solutions are not

isolated, and are not pseudo-almost limit cycles. A graphical representation for the case K(t) = t2,

√ p(t) = sin t + sin 2t,

α = 1, 2, 3, 4 is

given in Figure 3.

Figure 3: xα(t) = (K ∗ p)(t) = e−αt −t ∞ eαsp(s)ds. K(t) = t2, p(t) = sin t + sin √2t.

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3.2.2 Pseudo-almost periodic perturbations of the harmonic oscillator

Consider the forced oscillations of the harmonic oscillator given by

x¨(t) + x(t) = f(t)

(3.18)

where the forcing term is or equivalently, for x˙ = y

√ t2(t2 + 4) f(t) = − sin 2t + (t2 + 1)3
x˙ = y, y˙ = −x + f(t)

(3.19) (3.19a)

Clearly the function explicitly given by



1

x(t) = sin t + sin 2t + t2 + 1

(3.20)

is the unique solution of the equation and it is one of the classic examples of pseudo-almost periodic function that is not periodic. See also [11]. Therefore we obtain an explicit example of pseudoalmost limit cycle.

Figure 4 gives the phase portrait of (3.19a) and the graph of the pseudo-almost periodic function in (3.20).

Figure

4

x¨(t)

+

x(t)

=



sin

√ 2t

+

t2 (t2 +4)

(t2 +1)3

We further illustrate the theory of pseudo-almost limit cycles with the well-known Li´enard systems.

3.3 Li´enard pseudo-almost limit cycles

Li´enard equation, which also generalizes the famous Van der Pol oscillator, is ubiquitous in the study of nonlinear systems. Consider the one-parameter family of forced Li´enard systems

x¨ + f(x)x˙ + g(x) = µh(t),

(3.21)
Limit CyclesLimit CycleExistenceFunctionsOrbits