CUBO A Mathematical Journal
Vol.15, No¯ 01, (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.

btoni@vsu.edu

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 de-

termine the conditions of existence for several systems including some pseudo-almost

periodic perturbations of the harmonic oscillator and the renowned Liénard 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

Liénard-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 ĺımites, una nueva clase de no-periodicidad

en la interfaz de las teoŕıas de ciclos ĺımites y seudo-casi periodicidad. Determinamos

condiciones de existencia de muchos sistemas, incluyendo algunas perturbaciones seudo-

casi periódicas del oscilador armónico y los sistemas de Liénard. Aplicamos las condi-

ciones para derivar la existencia de ondas solitarias seudo-cuasi periódicas, primero per-

turbando y luego transformando algunas ecuaciones diferenciales parciales hiperbólicas

y parabólicas a ecuaciones del tipo Liénard. También se incluyen preguntas abiertas so-

bre la co-existencia de ciclos ĺımite y estrictamente pseudo-casi periódicos ciclos ĺımite

de partición del espacio de fases, y la existencia de isócrono pseudo-casi ciclos ĺımite .

Keywords and Phrases: Limit cycles. Almost and pseudo-almost periodic orbits. Periodic

waves. Isochronous systems and Isochrons. Liénard systems. Hyperbolic and parabolic equations.

2010 AMS Mathematics Subject Classification: 34C05, 34C07, 34C27, 34K14



132 Bourama Toni CUBO
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1 Introduction

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 Poincaré2 in the famous paper Mémoire sur les courbes

définies par une équation différentielle [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, Poincaré-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 Poincaré has excelled in all fields of knowledge and is often described as a polymath or The Last
Universalist. The famous Poincaré 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 Poincaré 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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Planar Pseudo-almost Limit Cycles and Applications ... 133

frequently than periodic ones. For instance, in the simplest model of harmonic oscillator or math-

ematical 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 pseudo-

almost 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 per-

turbation, 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 pseudo-

almost limit cycle, its properties, several illustrative examples including the so-called linear pseudo-

center, and existence theorems in the case of the well-known Liénard systems. Section 4 shows

the applications of the existence theorems for Liénard 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
∂t

(x,0). (2.3)

The orbit or trajectory of the flow through x ∈ Rn is given by:

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 = t
T0

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 Poincaré oscillator

The linear center or linear isochrone8

ẋ = −y, ẏ = 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)

ṙ = r(1 − r), θ̇ = 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 (ṙ > 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 Poincaré’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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Planar Pseudo-almost Limit Cycles and Applications ... 135

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

ẋ = −y + xp(x,y), ẏ = x + yp(x,y), (2.7)

where

p(x,y) = sin(
1

x2 + y2
)e

− 1
x2+y2 ,

has an infinite number of limit cycles

γn : x
2 + y2 =

1

nπ
, n ∈ N (2.8)

accumulating at the origin.[9]

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
T→∞

1

2T

∫T

−T

f(t,x)dt, (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

t2
)), for |t| ≥ 1, } (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, φ̃(.) := φ(.,x) is
Lebesgue measurable on R with M(|φ̃|) = 0, and M(|φ|) = 0.

For example the function

φ(t) = t| sinπt|t
N

, 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→∞ 12T
∫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
t→∞

m(Cǫ ∩ [−t,t])
2t

= 0, (Cǫ is called an ergodic zero set), (3.5)

(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, (3.6)

(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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Planar Pseudo-almost Limit Cycles and Applications ... 137

such that

lim
T→∞

1

2T

∫T

−T

|φ(t,x)|dt = 0, (3.8)

uniformly in x ∈ Ω.

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

φ1(t) = sint + sin
√
2t +

e−|t|

1 + t2
(3.11)



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has the almost periodic component a(t) = sint+
sin

√
2t, and the ergodic perturbation e(t) =

e
−|t|

1+t2
. We represent it along with its components

in Figure 1.

Figure 1: φ1(t) = sint + sin
√
2t + e

−|t|

1+t2

(2) Example 2 We have also the function

φω(t) = I1(t) + I2(t), ω 6= 0, (3.12)

with the almost periodic component

I1(t) =

∫
∞

−∞

h(t − s)(sins + sin
√
2s)ds, h ∈ L1(R) (3.13)

and the ergodic component

I2(t) =

∫
∞

−∞

h(t − s)

s2 + ω2
ds (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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Planar Pseudo-almost Limit Cycles and Applications ... 139

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])

ẋ(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

xα(t) = (K ∗ p)(t) = e−αt
∫t

−∞

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) = sint + 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) = sint + 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

ẍ(t) + x(t) = f(t) (3.18)

where the forcing term is

f(t) = − sin
√
2t +

t2(t2 + 4)

(t2 + 1)3
(3.19)

or equivalently, for ẋ = y

ẋ = y, ẏ = −x + f(t) (3.19a)

Clearly the function explicitly given by

x(t) = sint + sin
√
2t +

1

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 pseudo-

almost 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 ẍ(t) + x(t) = − sin
√
2t +

t
2
(t

2
+4)

(t2+1)3

We further illustrate the theory of pseudo-almost limit cycles with the well-known Liénard

systems.

3.3 Liénard pseudo-almost limit cycles

Liénard 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 Liénard systems

ẍ + f(x)ẋ + g(x) = µh(t), (3.21)



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Planar Pseudo-almost Limit Cycles and Applications ... 141

or equivalently
ẋ = y − F(x),

ẏ = −g(x) + µh(t),
(3.22)

where f, g, are two functions generally nonlinear, continuous and differentiable from R to R, and h

is a time-dependent continuous functions on R, µ ≥ 0 a small real parameter, and F(x) :=
∫x
0
f(s)ds.

For the homogeneous Liénard systems at µ = 0 we recall the following classical result. See

more details in, e.g., [5, 10, 18]

Theorem 3.4. If the homogeneous Liénard systems satisfy the following conditions:

(1) f(x) is continuous, even and f(0) < 0.

(2) g(x) is locally Lipschitz, odd, and such that xg(x) > 0 for x 6= 0.

(3) f(x) has a unique positive zero at x = b, and it increases at ∞ for x > b.

Then there exists a unique stable limit cycle.

Therefore this theorem provides conditions under which there exists, for the unperturbed

Liénard systems, a unique limit cycle, isolated periodic orbit controlling the behavior of neighboring

trajectories. We next show that we could subject some classes of Liénard systems to perturbations

that, in fact, destroy the limit cycles to give birth to strictly pseudo-almost limit cycles under

suitable conditions.

We study system (3.21) or its equivalent form (3.22) under the following additional assump-

tions:

L1 : f(x) > 0, in R, with F(x)sgnx → ∞ as |x| → ∞.

L2 : xg(x) > 0 for x 6= 0, G(x) → ∞ as |x| → ∞, with G(x) :=
∫x
0
g(s)ds.

L3 : |h(t)| ≤ K, and |H(t)| ≤ K, with H(t) =
∫t
0
h(s)ds, t ∈ R, and K a positive constant.

L4 : g
′(x) > 0, and g′′(x) exists and is bounded.

It is known that, under such assumptions, for 0 < µ ≪ 1, there exists in the xy-plane a region
R bounded by a regular simple curve (C1 except possibly at a finite number of points) such that:

(1) For every solution γ(t) = (x(t),y(t)) of system (3.21) there is a value t0 such that γ(t0) ∈ R.

(2) If, for a value t0 of t, we have γ(t0) ∈ R, then we have also γ(t) ∈ R, for t ≥ t0. That is,
solutions entering the set cannot leave it for increasing time.

Moreover the region R depends only on the functions f(x), g(x), h(t), the parameter µ and the

constant K. Equivalently, the region R may be described by the inequalities |x(t)| ≤ x0 |ẋ(t)| ≤ v0,
for a solution x(t) of the equation (3.22), and where x0 and v0 are constants independent of µ. See,



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for example, [7, 15, 21, 23]. In other words, under the above conditions the solutions ultimately

settle in a C1−bounded region R in R2. Actually we obtain

Lemma 3.5. Assume the conditions L1, . . .,L4. Let γ(t) = (x(t),y(t)) be a solution of the system,

and γ̃(t) = (x̃(t), ỹ(t)) either another solution of the system or a solution of an associated system

with a sufficiently small perturbation h̄(t) of the forcing term h(t). Then we have

lim
t→∞

|γ̃(t) − γ(t)| = 0, (3.23)

Moreover there exists a unique solution x(t) for all t ∈ R.

Proof. Let γ(t) = (x(t),y(t)) a solution of the system, and γ̃(t) = (x̃(t), ỹ(t)) either another

solution of the system or a solution of an associated system with a sufficiently small perturbation

h̄(t) of the forcing term h(t).

lim
t→∞

|γ̃(t) − γ(t)| = 0,

is equivalent to

lim
t→∞

|x̃(t) − x(t)| = 0 = lim
t→∞

|ỹ(t) − y(t)|. (3.24).

Upon the change of variables u(t) = x̃(t) − x(t), v(t) = x̃(t) − y(t), we obtain the system

u̇(t) = v(t) − ϕ(t)u(t)

v̇(t) = −ψ(t)u(t) + µ∆h(t), (3.25)

where

ϕ(t) =
F(x2) − F(x1)

x2 − x1
, ψ(t) =

g(x2) − g(x1)

x2 − x1
. (3.26)

Note that the function f, g′ and g′′ are bounded on the compact region R. For sufficiently small

values of the parameter µ ≪ 1, we can construct a Lyapunov-type quadratic form

V(t,u,v) = ψ(t)u2 + v2 − 2cuv, (3.27)

with c > 0 chosen small enough for V(t,u,v) to be positive definite such that

V(t,u,v) ≥ c(u2 + v2), (3.28)

c a positive constant, and such that

V̇(t,u,v) + cV(t,u,v) < 0. (3.29)

Actually we have
dV

dt
(t,u,v) = −2(ϕψ − ψ̇ − 2cψ)u2 − 2cv2 + 2cϕuv, (3.30)

yielding

Ṽ(t,u,v) := V̇(t,u,v) + cV(t,u,v) = −(2ϕψ − ψ̇ − 3cψ)u2 + 2c(ϕ − c)uv − cv2. (3.31)



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Planar Pseudo-almost Limit Cycles and Applications ... 143

The quadratic form Ṽ(t,u,v) can be made negative definite by taking the constant c such that

c <
2ϕψ − ψ̇

3ψ
, c(3ψ + (ϕ − c)2) < 2ϕψ − ψ̇, (3.32)

which entails

V̇(t,u,v) < V(t0)e
−c(t−t0). (3.33)

Therefore V(t) → 0 as t → ∞, implying that u → 0 and v → 0. The constant c is appropriately
chosen so that, when |∆h(t)| = |h̃(t) − h(t)| → 0, we can make V(t) → 0 for t → ∞. That is, the
solutions of the system of the perturbed forcing term ultimately converge to the solutions of the

original system.

Next let γ(t) = (x(t),y(t)) be one of these solutions which settled in R for t ≥ t0. We then
define the sequence of solutions γn(t) = γ(t + n) = (xn(t),yn(t)), t ≥ t0 − n. The sequence
is therefore equicontinuous and uniformly bounded. Consequently we can extract a subsequence

γnk(t) converging uniformly to a solution γ̄(t) = (x̄(t), ȳ(t)) lying completely in R for all t ∈ R.
(limn→∞(t0 + n,∞) = (−∞,∞)). And of course γ̄(t) is unique.

3.3.1 Remarks

Indeed the proof of the theorem actually accomplishes the followings: the solutions of the system

associated to the perturbed forcing term ultimately converge to the solutions of the original system;

moreover, under the assumptions above, only one solution of the system settles in the bounded

region R for all time in R.; as we show below, that single solution will be of the same nature as the

forcing term, when it becomes pseudo-almost periodic.

In a previous work, [24] the case of the pseudo-almost periodic forcing was presented as a

corollary to that of almost periodic forcing; here we present a more elegant and self-contained

proof drawing from the above definitions of pseudo-almost periodicity, definitions not used in the

cited work.

We state and prove

Theorem 3.6. Assume the forcing term h(t) is a pseudo-almost periodic function. Then under

the conditions L1, . . .,L4, the forced Liénard system exhibits a unique asymptotically stable pseudo-

almost limit cycle.

Proof. The proof is based on the previous lemma, including the existence of a unique solution

enclosed in R for all time. First assuming the forcing term h(t) is pseudo-almost periodic entails

from the definition above that, for any arbitrary ǫ, there exists δ = δ(ǫ), an ǫ−pseudo-almost

period τ ∈ Dǫ, a relatively dense set in R such that

‖h(t + τ) − h(t)‖ < ǫ, t,t + τ ∈ R − Cǫ (3.34)



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and

|t1 − t2| < δ =⇒ ||h(t1) − h(t2)|| < ǫ, t1,t2 ∈ R − Cǫ, (3.35)

where Cǫ is the ergodic zero set defined above. For such an ǫ− pseudo-almost period consider the

unique solution γ̄(t) given in the previous lemma that settles in R for all time t ∈ (−∞,∞), and
the associated function γ̄(t + τ) = (x̄(t + τ), ȳ(t + τ)). This function is readily a solution of the

following system (Eτ)
ẋ = y − F(x, ẏ = −g(x) + µh(t + τ), (3.36)

Take h(t + τ) as a sufficiently small perturbation of h(t) as above. Therefore, according to the

previous propositions, the solutions γ̄(t) and γ̄(t + τ) converge. Thus we obtain

‖γ̄(t + τ) − γ̄(t)‖ < ǫ, t,t + τ ∈ R − Cǫ. (3.37)

Moreover we also have, for t1,t2 ∈ R − Cǫ,

|γ̄(t2) − γ̄(t1)| ≤ |t2 − t1|supR| ˙̄γ|, (3.38)

which ensures the existence of δ such that

|t1 − t2| < δ =⇒ ||γ̄(t1) − γ̄(t2)|| < ǫ, t1,t2 ∈ R − Cǫ, (3.39)

Therefore we conclude that the unique solution γ̄(t) is pseudo-almost periodic.

Moreover, from the previous lemma, all other solutions of the system that ultimately settle

in R converge to this unique pseudo-almost periodic solution γ̄(t) ∈ R. Therefore the system has a
unique (isolated) almost periodic solution to which any other solution unwinds in the C1−bounded

set R. It is a stable pseudo-almost limit cycle as defined above. Hence the claim.

4 Pseudo-almost periodic Waves

The importance of Liénard systems among nonlinear systems also comes from the fact that several

systems can be transformed into Liénard systems and solved. [1, 19]. We present next some

partial differential equations solvable first by reducing them to some Liénard-type equations, then

by applying the previous theorems.

4.1 Hyperbolic pseudo-almost periodic Wave

Consider systems described by the time-perturbed nonlinear hyperbolic equation

utt = uxx + f0(u)ux + g0(u) + p(t) (H)



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Planar Pseudo-almost Limit Cycles and Applications ... 145

The search of special solutions of the form

u(x,t) = y(x + λt), λ ∈ R (4.1)

defining the wave with speed v = |λ|, yields the Liénard-type equation

(1 − λ2)ÿ + f0(y)ẏ + g0(y) = −p(t) (4.2)

Define f(y) =
f0(y)

1−λ2
, g(y) =

g0(y)

1−λ2)
, and h(t) =

−p(t)

1−λ2
. The functions f0 and g0 are continuously

differentiable chosen together with the speed v = |λ| of the waves u(t,x) such that the function

f, g, and h satisfy the assumptions L1, . . .,L4. Obviously assuming p(t) pseudo-almost periodic

implies h(t) is pseudo-almost periodic. Therefore we conclude under these assumptions

Theorem 4.1. For a pseudo-almost periodic perturbation p(t), the nonlinear hyperbolic equation

(H) has a pseudo-almost periodic solitary wave u(x,t) = y(x + λt), where y(x) is the unique
pseudo-almost limit cycle of the perturbed Liénard-type equation (4.2).

Proof. The proof is immediate and is adapted from theorem (3.6).

We next consider a parabolic partial differential equation describing a reaction-diffusion model.

4.2 Parabolic pseudo-almost periodic Wave: a reaction-diffusion model

Consider now the time-perturbed parabolic equation describing a reaction-diffusion model

ut = uxx + f0(u)ux + g0(u) + p(t) (RD)

Looking again for special solutions of the form (4.1) leads to the Liénard-type equation

ÿ + [f0(y) − λ]ẏ + g0(y) = 0 (4.3)

As in the previous case we set f(y) = f0(y) − λ, g(y) = g0(y), and h(t) = −p(t). The functions

f0 and g0 are continuously differentiable and determined together with the speed |λ| of the waves

u(t,x) such that the function f, g, and h satisfy the assumptions L1, . . .,L4. Again assuming

p(t) pseudo-almost periodic implies h(t) is also pseudo-almost periodic. We therefore obtain the

equivalent theorems of existence of pseudo-almost solitary waves to the reaction-diffusion equation

as functions of the corresponding Liénard pseudo-almost limit cycles. That is,

Theorem 4.2. For a pseudo-almost periodic perturbation p(t), the nonlinear parabolic equation(RD)
has a pseudo-almost periodic solitary wave u(x,t) = y(x + λt), where y(x) is the unique pseudo-

almost limit cycle of the perturbed Liénard-type equation (4.3).



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5 Outlook and Open Problems

Arnold in [3] states

Une trajectoire fermée nondégénérée ne disparait pas par une petite déformation

du système, mais se déforme légèrement. Donc le système des trajectoires est struc-

turellement stable dans le voisinage de la trajectoire fermée générique

That is, periodic orbits do not just disappear under small perturbation, but they may be

slightly deformed, due to the fact that the system of trajectories is structurally stable in the

neighborhood of a periodic orbit. Many forced systems such as the Liénard ones are actually

small perturbations of systems having periodic orbits (limit cycles) in their unperturbed form, and

many results do imply the disappearance of these orbits upon perturbation. The appearing of

pseudo-almost periodic solutions could result from the deformation/bifurcation of existing orbits.

Therefore one must investigate the relation between the “new” pseudo-almost periodic solutions

appearing upon perturbation and the periodic-type orbits of the unperturbed system, including

the question in the following Open Problem 1.

(1) Open Problem 1: Co-existence of limit cycles and strictly pseudo-almost limit cycles

For parameterized systems, including the above Liénard systems, investigate conditions under

which co-exist limit cycles and strictly almost or pseudo-almost limit cycles partitioning the

phase space.

(2) Open Problem 2: Isochronous pseudo-almost limit cycles

Let γ be a strictly pseudo-almost limit cycle of a flow φ on Rn. A point x1 in R
n has

asymptotic phase with respect to γ if there is a point x0 ∈ γ such that limt−→±∞ |φt(x1) −
φt(x0)| = 0. We say that x1 is in phase with x0.

It is well known that a hyperbolic limit cycle has some neighborhood where every point has

asymptotic phase with respect to the limit cycle, due to the existence of invariant foliation.[8]

Similar question needs to be addressed as well in case of strictly pseudo-almost limit cycles.

Definition 5.1. A strictly pseudo-almost limit cycle is said to be isochronous if there is a

neighborhood of γ in which every point is in phase with a point on γ.

In the case of limit cycles, we have, for instance, the following examples. System

ṙ = −
1

3
(r − 1)4e|r−1|

−3

, θ̇ = 2π (5.1)

has a nonhyperbolic limit cycle at the unit cycle with period 1, attracting for r > 1. The

asymptotic phase of any point (r0,θ0) in its neighborhood is (1,θ0). The limit cycle is

therefore isochronous. For more details see [8].

It would be interesting to:



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Planar Pseudo-almost Limit Cycles and Applications ... 147

(a) Perturb system (5.1), in particular in the angle variable, and study the conditions of

appearance of strictly pseudo-almost limit cycles.

(b) Investigate the conditions of existence of isochronous strictly pseudo-almost limit cycles,

in particular for the forced Liénard systems.

(c) Investigate the bifurcation of pseudo-almost limit cycles from an isochronous period

annulus, as in [25]

(3) Open Problem 3: Pseudo-almost isochrons

As above, we further define:

Definition 5.2. Given x0 ∈ γ where γ is a strictly pseudo-almost limit cycle, a pseudo-
almost isochron I(x0) based at x0 is the set of all point x ∈ Rn in phase with x0.

As in the case of limit cycles we conjecture the existence of pseudo-almost isochrons, and

that they will foliate the neighborhood of pseudo-almost limit cycles. Their determination is

definitely an interesting but difficult question of research. One line of attack might be similar

to Guckenheimer and Winfree investigation of isochrons of limit cycles. [16, 26]

Received: October 2012. Revised: March 2013.

References

[1] Albarakati, W.A., Lloyd, N.G., J.M. Pearson, Transformation to Liénard form EJDE

2000(76), 1-11 (2000)

[2] Andronov, A.A. et al., Theory of Oscillators Dover, New York (1989)

[3] Arnold, V, Chapites supplémentaires de la Théorie des équations différentielles ordinaires

Editions MIR, Moscou (1978)

[4] Bohr, H.A., Almost Periodic Functions Chelsea, New York (1951)

[5] Brauer, S.G. and Nohel, J.A., The qualitative theory of ordinary differential equations W.A.

Benjamin New York (1968)

[6] Byrnes, C., Topological Methods for Nonlinear Oscillations Notices of the AMS, 57(9), 1080-

1091 (2010)

[7] Cartwright M.L. and Littlewood, J.E., On non-linear differential equations of the second order

II Annals of Maths 48(2), 472-494 (1947)

[8] Chicone, C. and Liu, W., Asymptotic Phase Revisited J.Diff.Equat 204, 227-246 (2004)



148 Bourama Toni CUBO
15, 1 (2013)

[9] Christopher, C. and Li, C., Limit Cycles of Differential Equations Birkhauser Verlag, Basel-

Boston-Berlin (2007)

[10] Coddington, E.A. and Levinson, N, Theory of Ordinary Differential Equations Mc-Graw-Hill,

NY (1953)

[11] Corduneanu, C., Almost Periodic Oscillations and Waves Springer (2009)

[12] Diagana, T., Pseudo Almost Periodic Functions in Banach Spaces Nova Publishers, Inc. New

York (2007)

[13] Dumortier, F., Qualitative Theory of Planar Differential Systems Springer (2006)

[14] Ecalle, J. et al., Non-Accumulation des Cycles Limites I-II C.R.Acad.Sci. Paris. I(304),

375-431 (1987)

[15] Fink, A.M., Convergence and almost periodicity of solutions of forced Liénard equations SIAM

J.Appl.Math. 26(1), 6-34 (1974)

[16] Guckenheimer, J. Isochrons and Phaseless sets J.Math.Biol. 1, 259-273 (1975)

[17] Hilbert, D., Mathematische Probleme The second International Congress of Mathematicians,

Paris, 1900 Nachr.Ges.Wiss. Gottingen Math-Phys Kl 1900 253-297 (1900)

[18] Jordan, D.W. and Smith, P. Nonlinear Ordinary Differential Equations, Fourth Edition,

Oxford University Press (2007)

[19] Lloyd, N.G., Liénard systems with several limit cycles Math.Proc.Camb.Phil.Soc. 102(565)

(1987)

[20] N’Guérékata, G.M., Almost Automorphic Functions and Almost Periodic Functions in Ab-

stract Spaces Kluwer Academic/Plenum Publishers, New York-London-Moscow (2001)

[21] Opial, Z., Sur les solutions périodiques et presque-périodiques de l’équations differentielle

x′′ + kf(x)x′ + g(x) = kp(t). Annales Polonici Mathematici, VII, 309-319 (1960)

[22] Poincaré, H., Mémoire sur les courbes définies par une équation differentielle

J.Maths.Pures.Appli. 7, 375-422 (1881)

[23] Reuters, G.E.H., On certain non-linear differential equaions with almost periodic solutions

Journal London Math.Soc. 26, 215-221 (1951)

[24] Toni, B., Almost and pseudo-almost limit cycles for some forced Liénard systems Nonlinear

Analysis 71, 4718-4724 (2009)

[25] Toni, B., Upper Bounds of Limit Cycles from isochronous period annulus via birational lin-

earization Discrete and Continuous Systems Supp(2005) 846-853 (2005)



CUBO
15, 1 (2013)

Planar Pseudo-almost Limit Cycles and Applications ... 149

[26] Winfree, A.T., Patterns of Phase Compromise in Biological Cycles Journal of Mathematical

Biology, 1, 73-95 (1974)

[27] Zhang, C., Almost Periodic type functions and Ergodicity Science Press (2003)