Polynomial differential systems : qualitative study and applications

(1)

D

EPARTEMENT DE

M

ATHEMATIQUES

Thèse

Présentée par

GHERMOUL Bilal

Pour l’obtention du diplôme de

Doctorat en Sciences

Filière : MATHEMATIQUES

Spécialité : EDO

Thème

Polynomial differential systems: Qualitative study and applications

Soutenue publiquement le 18/02/2021 Devant le Jury

Président ACHACHE Mohamed Pr. Université Ferhat Abbas Sétif 1

Directeur BENDJEDDOU Ahmed Pr. Université Ferhat Abbas Sétif 1

Examinateurs BERBOUCHA Ahmed Pr. Université Abderrahmane Mira Béjaïa

MERZOUGUI Abdelkrim Pr. Université Mohamed Boudiaf M’sila

Invité BOUREGHDA Abdellatif Dr. Université Ferhat Abbas Sétif 1

Année universitaire 2020/2021

N°……….…………..…….……/ /2021

Université Ferhat Abbas Sétif 1 Faculté des Sciences

ةيبعشلا ةيطارقميدلا ةيرئازجلا ةيروهمجلا

ملعلا ثحبلا و يلاعلا ميلعتلا ةرازو

ي

فيطس ،سابع تاحرف ةعماج

1

ةيلك

مولعلا

(2)

Polynomial Differential Systems: Qualitative Study

And Applications

Bilal Ghermoul

University of Ferhat Abbas Setif 1

(3)

3.2.1 Proof of Theorem 3.1: What is the maximum number of limit cycles for the PWDS formed by a linear and quadratic isochronous centers? 18 3.3 Limit cycles of continuous PWDS formed by two quadratic isochronous centers 21 3.3.1 Proof of Theorem 3.2:What is the maximum number of limit cy-cles for the PWDS formed by the general quadratic isochronous cen-ter ˙x = −y + x2 − y2_{, ˙}_{y = x(1 + 2y), and an arbitrary quadratic} isochronous center? . . . 21

(4)

Contents 5 Appendix 51 5.1 Section6.1 . . . 51 5.2 Section6.2 . . . 51 5.3 Section6.3 . . . 51 5.4 Section6.4 . . . 52 5.5 Section6.5 . . . 52 5.6 Section6.6 . . . 52 5.7 Section6.7 . . . 53 5.8 Section6.8 . . . 53 5.9 Section6.9 . . . 53 5.10 Section6.10 . . . 54 5.11 Section6.11 . . . 55 5.12 Section6.12 . . . 55 5.13 Section6.13 . . . 55 5.14 Section6.14 . . . 55 5.15 Section6.15 . . . 56 5.16 Section6.16 . . . 57 5.17 Section6.17 . . . 58 5.18 Section6.18 . . . 59 5.19 Section6.19 . . . 59 5.20 Section6.20 . . . 59 5.21 Section6.21 . . . 59 5.22 Section6.22 . . . 60 5.23 Section6.23 . . . 60 5.24 Section6.24 . . . 61 5.25 Section6.25 . . . 61 5.26 Section6.26 . . . 61 5.27 Section6.27 . . . 62 5.28 Section6.28 . . . 62 5.29 Section6.29 . . . 63 5.30 Section6.30 . . . 63 5.31 Section6.31 . . . 64 5.32 Section6.32 . . . 64 5.33 Section6.33 . . . 64 5.34 Section6.34 . . . 65 5.35 Section6.35 . . . 66 5.36 Section6.36 . . . 66 5.37 Section6.37 . . . 67 5.38 Section6.38 . . . 67 5.39 Section6.39 . . . 67 5.40 Section6.40 . . . 67 5.41 Section6.41 . . . 68 Bibliography 69

(5)

This work is dedicated to my parents,

my family,

(6)

Acknowledgements

In the name of Allah the Merciful, Praise to Allah, Lord of the Worlds, Praise be to the Lord of all worlds. Prayers and peace be upon our Prophet, Muhammad, his family and all of his companions.

Firstly, I would like to express my sincere gratitude to my supervisor Prof. Ahmed Bendjeddou for the continuous support of my PhD thesis and related research, for his patience and motivation.

Secondly, I would like to thank Prof. Jaume Llibre from Universitat Aut`onoma de Barcelona for his guidance in some problems during preparation of my PhD thesis, without forgetting my colleague Dr. Tayeb Salhi from Mohamed El–Bachir El–Ibrahimi university. Besides all of them, I would like to thank members of my thesis committee: Prof. Mohamed Achacha from Setif1 university, Prof. Ahmed Berboucha from Bejaia university, Prof. Abdelkrim Merzougui from M’sila university, and Dr. Abdellatif Bouraghda from Setif1 university, for their insightful comments, encouragement, and also for questions which gave me various perspectives.

(7)

Introduction

Ordinary differential equations are among the powerful tools used for the understanding and modeling of various scientific and engineering phenomena, and then for the inves-tigation of solutions. Ordinary differential equations are rarely or very difficult, if not impossible to have an exact or an explicit solution, for this reason we have to use other approaches for finding approximate solutions or qualitative properties of solutions. One of the most important of these approaches is the qualitative analysis or the analysis of the phase space which is the set of dynamical variables, which can be applied for studying qualitatively the behavior of solutions for a given nonlinear dynamical system.

In general, a dynamical system geometrically represents a vector field in the phase-space, which defines a phase flow. The theory of dynamical systems goes back to the qualitative study of differential equations by Newton, Euler, Hamilton, Maxwell and others. At the end of the 19th century, Poincar´e [28] initiated another approach to investigate the geo-metric properties of the solutions instead of finding explicit forms for them, Birkhoff coined the term dynamical systems in [9]. Mathematicians have made important contributions to the theory of dynamical systems, the existence of computer algebra systems too, aid for finding solutions to a wide range of dynamical systems and determining their orbits.

In the qualitative theory of planar differential systems a limit cycle which is an isolated periodic solution in the set of all periodic solutions, remained the most sought solutions when modeling physical systems in the plane, the notion of limit cycle appeared in the year 1885 in the work of Poincar´e [29].

Most of the early examples in the theory of limit cycles in planar differential systems were commonly related to practical problems with mechanical and electronic systems, but periodic behavior appears in all branches of sciences. To determine the existence or non– existence of limit cycles is one of the more difficult objects in the qualitative theory of planar differential equations. A large number of references deals with the subject of limit cycles, many of them motivated by the famous Hilbert’s 16th problem, see for details [15, 19, 21]. In particular, our goal is the study of autonomous ordinary polynomial differential systems in two real variables.

(8)

Introduction

Limit cycles of planar polynomial differential systems are not in general algebraic. For instance, the limit cycle for the Van der Pol equation is non-algebraic as shown by Odani [26]. Most limit cycles known in an explicit way are algebraic, see for instance [5, 6, 14]. In 1998, Abdelkadder [1] presented for the first time an example of Li´enard equations with an exact algebraic limit cycle. This example was obtained as a particular case by Bendjeddou and Cheurfa [5] by considering a more general class of planar systems. the first examples of explicit non-algebraic limit cycles were given by Gasull [13], Al-Dossary [2] for n = 5 and by Llibre [7] for n = 3. The first result about coexistence of algebraic and non-algebraic limit cycles goes back to Gin´e and Grau [14] with n = 9.

Since 1930’s the study of the limit cycles also became important in the continuous and discontinuous piecewise differential system(s) (PWDS) separated by a straight line, due to their applications to mechanics, electrical circuits, ... see for instance the books [3, 8, 30] and the references in. Researchers also took an interest in limit cycles as an extension of the second part of the Hilbert’s 16th problem of continuous and discontinuous PWDS.

For more details and as an example showing the importance of searching for limit cycles with a good explanation of its history, Llibre & Zhang in the paper [23], gave a collection of open problems in both smooth and PWDS that are yet to be solved as of today.

In this thesis we seek limit cycles for a particular quintic polynomial differential system. We consider a class of planar quintic differential systems, for which a non-algebraic limit cycle around a non-elementary critical point is given and it is the unique limit cycle and the non-algebraic limit cycle is constructed explicitly by using polar coordinates. This will be given in chapter 2.

The last chapter is devoted to study the continuous planar PWDS separated by the straight line x = 0 formed by a linear isochronous center [22] in x > 0 and an isochronous quadratic center see [24] in x < 0. We prove that these PWDS cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second we study the crossing periodic orbits and the crossing limit cycles of the continuous planar piecewise differential quadratic isochronous centers separated by the straight line x = 0.

(9)

1 Basic concepts and prerequisites

In this chapter we give some prerequisites and preliminaries, the framework for which we restrict our attention is dynamical systems described by two state variables x and y which are defined over a continuous range of time t from past to future, the mathematical model of the behavior for such dynamical systems is described by a set of two first order autonomous nonlinear ordinary differential equations as follows

˙x = P (x, y), ˙

y = Q(x, y), (1.1)

where P and Q are two relatively prime polynomials in x and y with real coefficients of degree N = max{deg P, deg Q} and implicitly depend to time t and the dot “ ˙ ” denotes the derivative with respect to time. In this case system (1.1) is defined on

_R

2 and called polynomial planar differential system which can be always represented by its vector field

F (x, y) = (P (x, y), Q(x, y)). (1.2)

This vector field related to system (1.1) can be represented by the following differential operator

F = P ∂ ∂x + Q

∂

∂y. (1.3)

Even though system (1.1) is first order ordinary differential equations they are not inte-grable except the case when P and Q are polynomial of degree at most 1. Then for these sets of ordinary differential equations (1.1) we cannot in general write an explicitly gen-eral solution for all t. On the other hand we can extract information about system (1.1) without any given closed form of solution. We need then a basic definitions and concepts such as existence and phase portraits and some other key ideas for which we can deal with the subject qualitative behavior of our systems.

1.1 Existence and uniqueness of solutions

Definition 1. A solution to system (1.1) is a vector-valued function X(t) = (x(t), y(t)) with t in some interval I ⊆

_R

, which satisfies

˙

X(t) = dX(t)

(10)

1 Basic concepts and prerequisites

A solution is called maximal if no such extension exists on an interval J ⊃ I. The interval I is said to be the maximal interval on which a solution to (1.4) exists. If the solutions of differential system have the maximal interval equal to

_R

then it is called complete [4]. Remark 1.1 (Existence and Uniqueness Theorem [27]). All the planar differential systems considering in our work are polynomial, means that we do have always existence and uniqueness of solutions for all initial conditions; i.e.; the initial value problem (1.4) with X(t0) = (x(t0), y(t0)), has a unique solution X(t). Furthermore, F (x, y) is locally

Lipschitz function, then we have always a maximal solution for system (1.1).

By using the existence and uniqueness theorem, a self intersection of a solution for an autonomous systems is impossible except the case when the solution is periodic. We emphasize that this is not the case if we consider a non-autonomous systems.

1.1.1 Different types of solutions

There are three different types of solutions for system (1.1).

1. The first one is a constant solution for any time t, this is called also a critical point which starts at X(t0) and remains unmoving indefinitely during the time t, this is

what gives it also the name an equilibrium point (or fixed point), 2. the second one is a curve in plane which cannot cross itself

3. and the third kind of solutions is the periodic solution which verifies X(t + p) = X(t) where p is the period, this kind of solutions starting at X(t0) and return to the same

state X(t0) in p units of time.

1.2 Flow

A dynamical system can be defined by a function φt :

R

2 →

R

2 that makes X moving (forward) to Xt, which is defined for all t ∈

R

, gives the behaviour of X during the time t

starting at t = 0 (or at any initial condition t = t0) as φ0such that φ0(X) = X. Moving Xt

backward gives naturally φ_−t the inverse of φt. In the same manner, moving Xt again by

another elapsing time s gives φt+s(X) = φt(φs(X)). The formal definition of the function

has these properties is given as follows [16, 31].

Definition 2 (Flow). Let F be the vector field defined in (1.3). The flow is aC1 function

φ : I ×

_R

2 →

_R

2

(t, X) 7→ φ(t, X) = φt(X),

(11)

(a) dφt

dt (X) = F (φt(X)),

(b) φ0(X) = X (Identity function φ0:

R

2 →

R

2), (c) φt◦ φs(X) = φt+s(X), ∀t, s ∈ I (Composition).

1.3 Stability of equilibrium points

The behavior of solutions in the neighbourhood towards or away from the fixed points as t goes to infinity is what stability of equilibrium points means.

Definition 3. For any (x0, y0) ∈ Ω ⊆

R

2 satisfying

P (x0, y0) = Q(x0, y0) = 0,

(x0, y0) is called an equilibrium point, the terms fixed point or critical point are also used.

If there are no equilibrium in the neighbourhood of (x0, y0) other than (x0, y0), then

(x0, y0) is called an isolated critical point.

Remark 1.2. For system (1.1) all equilibrium points are isolated.

Definition 4 (Stable critical point). An isolated critical point (x0, y0) is said to be

stable if for given any neighbourhood V of (x0, y0) every solution passes near (x0, y0)

remains in V as t increasing.

Definition 5 (Asymptotically stable critical point). The critical point (x0, y0) is

said to be asymptotically stable if every solution starts in the neighbourhood of (x0, y0)

approaches (x0, y0) as t goes to infinity.

Remark 1.3. If the critical point (x0, y0) in not stable is said to be unstable.

1.4 Linear and nonlinear systems

Consider a two-dimensional system

˙x = ax + by, ˙

y = cx + dy, (1.5)

with a, b, c and d are constants. In matrix form (1.5) can be written as follows ˙

(12)

here X = (x, y) and A is the matrix given by

A =

"

a b c d

#

.

These linear systems are important, their phase portraits in the plane is completely determined by one of the four types of qualitative behavior: Stable, center, saddle and unstable, depending on the trace and determinant of the matrix A.

Linear systems (1.5) play an important role in studying the fixed points of nonlinear systems.

Definition 6 (Qualitatively equivalent). Two systems of first-order differential equa-tions are said to be qualitatively equivalent if there is a continuous bijection which maps the phase portrait of one onto that of the other in such a way as to preserve the orientation of trajectories.

1.4.1 Linear approximation at equilibrium points

In the neighbourhood of an equilibrium points we can linearize the nonlinear system, i.e.; we approximate our vector field (1.2) using a linear approximation

F (X) ≈ BX + F (0) (1.7)

near the origin. Any other equilibrium point (x0, y0) can be studied in a similar way by

moving it to the origin using a translation of axes. The matrix B into (1.7) is given as follows B =

"

a0 b0 c0 d0

#

, where a0= ∂P ∂x(0, 0), b0 = ∂P ∂y(0, 0), c0 = ∂Q ∂x(0, 0) and d0 = ∂Q ∂y(0, 0),

which are the components of the Jacobian matrix for F . Then the linear approximation for the nonlinear system (1.1) in the neighbourhood of the origin is the following linear system

˙x = a0x + b0y,

˙

y = c0x + d0y.

(1.8)

1.4.2 The linearization Theorem

At the equilibrium point the phase portrait of a nonlinear system can be predicted by its linear system constructed by linearization, this is given in the following theorem [27].

(13)

Theorem 1.4. If we consider the nonlinear system with a simple (simple means that the matrix of linearization system is not singular) equilibrium point at X = (0, 0). Then, in the neighbourhood of the singular point, the phase portraits of nonlinear system and its linearization are qualitatively equivalent provided the linearized system is not a center.

1.5 Limit cycle

Consider system (1.1) and let x = f (t), y = g(t) two periodic solutions as a function of time t, the curve γ = {(x, y) : x = f (t), y = g(t)} is known as a periodic orbit for system (1.1).

Definition 7 (Limit cycle). A limit cycle γ is an isolated periodic orbit, i.e.; for some arbitrarily small neighborhood of the periodic orbit γ there are no other periodic orbits.

As an example, the following quadratic differential system

˙x = x2− 8 x y + y2+ 4 y, ˙

y = x2+ 2 x y − 3 x + y2− 6 y.

(1.9)

Has an non–algebraic limit cycle which is shown in Figure 1.1.

Figure 1.1: Limit cycle formed by system (1.9).

Theorem 1.5 (Equilibrium point criterion). In the interior of any periodic orbit we must have an equilibrium point [25].

One of the most important criterion in dynamical systems that permits us to confirm the absence of limit cycles is the Poincar´e-Bendixon Theorem known as Bendixon’s Criterion [12, 25].

(14)

Theorem 1.6 (Bendixon’s Criterion). Suppose that R is a simply-connected region (there are no holes in it) in the (x, y)−plane such that

∂P ∂x +

∂Q ∂y,

has a constant sign. Then the system (1.1) has no closed orbits and then no limit cycles entirely contained in R.

A generalized form of Theorem 1.6 was introduced by Dulac (see [20]) which can be given as follows

Theorem 1.7 (Dulac’s Criterion). If there exist a function ψ ∈ C1(R) in a simply-connected region R in the (x, y)−plane such that

Z Z

D

∂(ψP ) ∂x + ∂(ψQ) ∂y

dxdy 6= 0,

for every simply-connected subregion D ⊂ R. Then system (1.1) has no closed orbits and then no limit cycles.

The most important thing to deal with is the maximum number of limit cycles for system (1.1) regarding to the following theorem a planar polynomial differential systems can have only a finite number of limit cycles [12, 27].

Theorem 1.8 (Dulac). Any planar polynomial differential system (1.1) has at most a finite number of limit cycles.

1.6 First integral, Invariant curve

For a given systems of the form (1.1), if we know a first integral then the phase portrait of the system is completely determined, as an example Hamiltonian system for which there exists the so-called Hamiltonian such that

˙x = ∂H

∂y , y = −˙ ∂H

∂x,

has always a first integral. Otherwise, for such a general systems (1.1) it is very difficult if it is not impossible to get a first integral for this kind of systems.

Definition 8 (Phase Portrait). The phase portrait or the phase plane is a global picture of a family of solution curves based on the equation

dy dx = ˙ y ˙x = Q(x, y) P (x, y), (1.10)

(15)

Definition 9 (First Integral). Let U be an open subset of

_R

2. A function H : U →

_R

is called a first integral of (1.1) if H is constant on all solution curves of (1.1).

It is clear from Definition 9 that a function H is a first integral of (1.1) if and only if

F (H) = P∂H ∂x + Q

∂H

∂y = 0, (1.11)

where F is the differential operator defined by (1.3). Equation (1.11) means that ∂H

∂xdx + ∂H

∂y dy = dH(x, y) = 0,

and dH is an exact differential form. Rewriting (1.1) as a differential form as follows

ω = Qdx − P dy = 0,

if ω is not an exact differential form, we can always multiplying Qdx − P dy = 0 by what the so called integrating factor R on the open set U ⊂

_R

2 for which the differential form ω becomes an exact one, this is equivalent to

∂(R Q) ∂y = −

∂(R P ) ∂x ,

and by using the differential operator given into (1.3), the last equation is equivalent to the following

F (R) = R div(P, Q). (1.12)

If H is the first integral related to the integrating factor R then system (1.1) is equivalent to

˙x = R P = ∂H

∂y , y = R Q = −˙ ∂H

∂x.

Finally, for any given first integral H for (1.1), integrating factor can be always defined. Integrating factor have the following important property [12].

Proposition 1.9. If system (1.1) has two different integrating factors R1 and R2 on the

open subset U ⊂

_R

2 and have no constant common factors other than 1, then the function R = R1/R2 is also an integration factor on U \ {R2 = 0}.

Proof. Since R1 and R2 are two integrating factors then they must verifying (1.12).

Ap-plying the differential operator F to R1/R2 we get

F

R1 R2

= (F R1) R2− (F R2) R1 R₂2 = 0.

(16)

Definition 10 (Invariant Algebraic Curve). Let f (x, y) = 0 be an algebraic curve, if there exists a polynomial function K in x and y such that

F (f ) = P∂f ∂x + Q

∂f

∂y = Kf. (1.13)

Then, we say that f is an invariant algebraic curve, the polynomial K is called the cofactor of f .

If we move on the points (x, y) of the algebraic curve f (x, y) = 0 we see that

∇f · F = 0,

and since the gradient ∇f is perpendicular to the level curve f = 0, the last equation shows that F is perpendicular to ∇f , means that F is tangent to the level curve f = 0 which explain where the name invariant comes from.

Remark 1.10. Since the system (1.1) has a degree n, it results from (1.13) that any cofactor has at most a degree of n − 1.

Definition 11. Limit cycle of (1.1) is said to be algebraic if it is contained in the zero set of an invariant algebraic curve of this system.

1.7 Continuous PWDS

A continuous piecewise vector field in

_R

2 is a pair of Cr-vector fields X and Y with r ≥ 1, defined on

_R

2 separated by a smooth codimension one manifold Σ. The line of separation Σ is obtained by considering Σ = h−1(0), where h :

_R

2−→

_R

is a differentiable function having 0 as a regular value. Note that Σ is the separating boundary of the regions Σ+ = {(x, y) ∈

_R

2| h(x, y) > 0} and Σ− = {(x, y) ∈

_R

2| h(x, y) < 0}. So a continuous piecewise smooth vector field is provided by

Z(x, y) =







X(x, y), h(x, y) ≥ 0, Y (x, y), h(x, y) ≤ 0,

satisfying that X(x, y) = Y (x, y) if h(x, y) = 0. Of course, a continuous PWDS in

_R

2 is a differential system ( ˙x, ˙y) = Z(x, y), being Z(x, y) a continuous piecewise vector field in

R

2.

1.8 Isochronous center

A center p of a planar differential system is a singular point for which there is a neigh-borhood U such that U \ {p} is filled with periodic orbits. When all the periodic orbits

(17)

surrounding a center have the same period this center is called isochronous. The centers started to be studied by Poincar´e [28] and Dulac [11], but the notion of isochronocity goes back to [18], see also [17].

1.8.1 Linear isochronous centers

It is well known that the linear differential centers are isochronous and that the general expression of such centers is as follows, see for a proof [22].

Lemma 1.11. A linear differential system having a center can be written in the form

˙x = −βx − 4β

2_{+ ω}2

4α y + δ, y = αx + βy + γ,˙ (1.14) with α > 0 and ω > 0.

The linear differential system (1.14) has the first integral

HL(x, y) = 4(αx + βy)2+ 8α(γx − δy) + ω2y2.

1.8.2 Quadratic isochronous centers

We consider the quadratic polynomial differential systems having an isochronous center. This kind of centers were classified by Loud in the paper [24]. Those systems after an affine change of coordinates become one of the following four systems (which are also given in the paper [10] too):

˙x = − y + x2− y2, y =x(1 + 2y),˙ (1.15) ˙x = − y + x2, y =x(1 + y),˙ (1.16) ˙x = − y − 4x 2 3 , y =x˙

1 − 16 3 y

, (1.17) ˙x = − y + 16 3 x 2₋ 4 3y 2_, _{y =x}_˙

_{1 +} 8 3y

. (1.18)

And their first integrals are given

H1.15 = x2+ y2 2 y + 1, H1.16 = x2+ y2 (y + 1)2, H1.17 = (32x2− 24y + 9)2 (3 − 16 y) , H1.18 = −256x2_{+ 128y}2_{+ 96y + 9} (8 y + 3)4 ,

(18)

respectively.

1.9 Crossing periodic orbits and crossing limit cycles

A crossing periodic orbit or a crossing limit cycle is a periodic orbit or a limit cycle which intersects exactly in two points the discontinuity separating boundary h(x, y) = 0 defined in section 1.7.

(19)

2 A non-algebraic limit cycle for a class

of quintic differential systems

In this chapter, we are mainly interested in the study of the existence of a non-algebraic limit cycles for some classes of quintic systems around a non-elementary critical points using the well known Bernoulli ordinary differential equation.

2.1 Limit cycle for a class of quintic differential systems

Our main result is the following theorem.

Theorem 2.1. Consider the following quintic system











˙x = P5(x, y) = bx3+ dmx5− nx4y + cxy2+ dnx3y2

− (2a + n)x2y3+ adxy4− 2ay5, ˙

y = Q5(x, y) = 2mx5+ bx2y + dmx4y + (2m + n)x3y2

+ cy3+ dnx2y3+ nxy4+ ady5.

(2.1)

Then, for bc > 0, ab > 0 and d < 0, system (2.1) has one and only one limit cycle which is a non-algebraic, given in polar coordinates by the formula

r (θ; r∗) =

Z

θ 0 f2(u) g(u) exp

−

Z

u 0 f1(s) g(s)ds

du + r_∗2

!

12 exp 1 2

Z

θ 0 f1(u) g(u) du

!

, (2.2) where

f1(θ) = ad sin4θ + dm cos4θ + dn sin2θ cos2θ

+ (n − 2a) sin3θ cos θ + (2m − n) sin θ cos3θ, f2(θ) = b cos2θ + c sin2θ,

(20)

2 A non-algebraic limit cycle for a class of quintic differential systems and r∗=







Z

2π 0 f2(u) g(u) exp

−

Z

u 0 f1(s) g(s) ds

du exp

−

R

2π 0 f1(u) g(u)du

− 1







1 2 , (2.3) if n2− 4am < 0. (2.4)

Proof of Theorem 2.1. Since all critical points must be included in the level set {y ˙x − x ˙y = 0}, and because

y ˙x − x ˙y = yP5(x, y) − xQ5(x, y)

= 2(x2+ y2)(mx4+ nx2y2+ ay4),

together with condition (2.4), then we have a unique equilibrium point. Furthermore, any limit cycles must be surrounding the origin (0, 0).

In order to search for the limit cycle, we use polar coordinates. System (2.1) becomes







˙r = f2(θ)r3+ f1(θ)r5, ˙ θ = 2g(θ)r4. (2.5)

We can rewrite system (2.5) as the first-order Bernoulli differential equation as follows dr dθ = 1 2r f2(θ) g(θ) + r 2 f1(θ) g(θ) , (2.6)

Using the change of variable ρ = r2, equation (2.6) becomes the following linear first order differential equation

dρ dθ = f1(θ) g(θ)ρ + f2(θ) g(θ) . (2.7)

Now immediately from (2.7) it follows that

r (θ; k) =

Z

θ 0 f2(u) g(u) exp

−

Z

u 0 f1(s) g(s)ds

du + k

!

12 exp 1 2

Z

θ 0 f1(u) g(u) du

!

, (2.8)

solution for (2.6), where k is a constant. The Cartesian coordinate form of (2.8) proves that it is a non-algebraic curve.

(21)

2 A non-algebraic limit cycle for a class of quintic differential systems

It is clear that r (0, k) = r0 > 0, corresponds to k = r₀2, so (2.8) becomes

r (θ; r0) =

Z

θ 0 f2(u) g(u) exp

−

Z

u 0 f1(s) g(s)ds

du + r₀2

!

12 exp 1 2

Z

θ 0 f1(u) g(u) du

!

, (2.9)

Periodic solutions must verify the following condition

r (2π; r0) = r0, (2.10)

solving (2.10) with respect to r0 gives

r2_∗ =

Z

−

Z

u 0 f1(s) g(s)ds

du exp

−

R

2π 0 f1(u) g(u)du

− 1 . (2.11)

In order to show that the right hand side into (2.11) is strictly positive, we can first easily show that

exp

−

Z

2π 0 f1(u) g(u)du

− 1 = e−2πd− 1 > 0,

when d < 0, and because the denominator into (2.11) is positive, the sign of the numerator is the same as the sign of f2(u)/g(u), but we have n2− 4am < 0 this means that am > 0,

which makes f2(u)/g(u) always positive if and only if bc > 0 and ab > 0.

Let’s now consider from (2.9)

˜

g(θ) = r(θ; r∗), (2.12)

r∗ is given by (2.11). From the previous considerations of parameters, we must have ˜g > 0

by construction. Knowing that e −

Z

2π 0 f1(u) g(u)du _{= e}−2πd_,

replacing it into (2.12), with simple calculations, we can easily show that the function ˜g is periodic, i.e.,

˜

g(θ + 2π) = ˜g(θ).

Now we turn to the final step, i.e., the question whether the graph of the function ˜g is indeed a limit cycle. We consider the Poincar´e return map [16, 25, 31], from (2.9), we

(22)

calculate the derivative of r(2π, r0) with respect to r0 at the point r∗, thus

dr dr0 (2π, r0)

r0=r∗ = e πd_r ∗ (G(2π) + r2 ∗) 1 2 , (2.13) where G(2π) =

Z

−

Z

u 0 f1(s) g(s)ds

du. From (2.13) r∗ (G(2π) + r2 ∗) 1 2 < 1, and eπd< 1, therefore dr dr0 (2π, r0)

_r 0=r∗ < 1.

For that reason, limit cycle for ordinary differential equations (2.6) is stable. Finally, system (2.1) has exactly one non-algebraic limit cycle which is the unique limit cycle and there are systems realizing this limit cycle (see section 2.2), which proves that the maximum number of limit cycles is 1.

(23)

2.2 Numerical examples of limit cycles for the class of

systems given in section 2.1

Consider the parameters into system (2.1) to be a = 1, m = 2, n = 1, b = 2, c = 1 and d = −1, then system (2.1) becomes

(

˙x = 2x3− 2x5_{− x}4_{y + xy}2_{− x}3_y2_{− 3x}2_y3_{− xy}4_{− 2y}5_,

˙

y = 4x5+ 2x2y − 2x4y + 5x3y2+ y3− x2_y3_{+ xy}4_{− y}5_. (2.14)

Clearly, conditions of Theorem 2.1 can be easily verified, system (2.14) has one limit cycle as shown in figure 2.1.

Figure 2.1: The phase portrait and the vector field for system (2.14), with a convex stable limit cycle included.

If we consider a = 10, m = 2, n = 1, b = 2, c = 10 and d = −10. Then system (2.1) has a concave limit cycle as shown in figure 2.2.

(24)

3 Limit cycles of continuous PWDS

In this chapter we consider continuous PWDS separated by the straight line x = 0 having in x ≤ 0 and in x ≥ 0 linear or quadratic isochronous centers, and we want to study the non–existence, and the existence of crossing periodic orbits and of crossing limit cycles, and in this last case we want also to know the maximum number of crossing limit cycles for these systems.

Here a crossing periodic orbit or a crossing limit cycle is a periodic orbit or a limit cycle which intersects exactly in two points the discontinuity line x = 0.

The continuity of a PWDS separated by the straight line x = 0 formed by two centers means that the vector fields defined by these two centers (linear or quadratic) coincide on the line of discontinuity x = 0. So a continuous PWDS is a continuous differential system in

_R

2 and is an analytic differential system in

_R

2\ {x = 0}.

3.1 Generalized quadratic isochronous centers

We are interested in the general expressions of the quadratic isochronous centers. So we transform their normal forms (1.15), (1.16), (1.17) and (1.18) through the following general affine change of variables

(x, y) → (a1x + b1y + c1, a2x + b2y + c2), (3.1)

with

a1b2− a2b1 6= 0. (3.2)

Generalized isochronous system (1.15). Using the change of variables (3.1) the quadratic system (1.15) becomes

˙x =(−b2c21+ b1c1+ 2b1c2c1+ b2c22+ b2c2 + (2a2b1c1+ 2a1b1c2− 2a1b2c1+ 2a2b2c2+ a1b1+ a2b2)x + (2b2₁c2+ 2b22c2+ b21+ b22)y + (2a2b21+ 2a2b22)xy + (a2₁(−b2) + 2a2a1b1+ a22b2)x2+ (b32+ b21b2)y2)/(a2b1− a1b2), ˙ y =(a2c21− a1c1− 2a1c2c1− a2c22− a2c2+ (−2a21c2− 2a22c2− a21 − a2₂)x + (2a2b1c1− 2a1b1c2− 2a1b2c1− 2a2b2c2− a1b1− a2b2)y

+ (−2a2₁b2− 2a22b2)xy + (−a32− a21a2)x2

+ (a2b21− 2a1b2b1− a2b22)y2)/(a2b1− a1b2).

(3.3)

(25)

3 Limit cycles of continuous PWDS

variables (3.1) we get the following first integral of the generalized isochronous quadratic system (3.3)

H2(x, y) =

(a1x + b1y + c1)2+ (a2x + b2y + c2)2

2 (a2x + b2y + c2) + 1

.

Generalized isochronous system (1.16). System (1.16) is equivalent to the following generalized isochronous system after the linear change of variables (3.1)

˙x = ( b2c21− b1c1− b1c2c1− b2c2 + (−a2b1c1− a1b1c2+ 2a1b2c1− a1b1− a2b2) x + b2₁(−c2) + b2b1c1− b21− b22

y + a1b1b2− a2b21

xy + a2₁b2− a1a2b1

x2

/(a1b2− a2b1), ˙ y = ( a2c21− a1c1− a1c2c1− a2c2+ a21(−c2) + a2a1c1− a21− a22

x + (2a2b1c1− a1b1c2− a1b2c1− a1b1− a2b2) y + a1a2b1− a21b2

xy + a2b21− a1b1b2

y2

/(a1b2− a2b1). (3.4)

The quadratic system (1.16) has the first integral (x2+ y2)/(y + 1)2. Therefore a first integral of system (3.4) is

H3(x, y) =

(a1x + b1y + c1)2+ (a2x + b2y + c2)2

(a2x + b2y + c2+ 1)2

.

Generalized isochronous system (1.17). The quadratic system (1.17) is equivalent to the following generalized quadratic system after the linear change of variables (3.1)

˙x = ( 4b2c21+ 3b1c1− 16b1c2c1+ 3b2c2 + (−16a2b1c1− 16a1b1c2+ 8a1b2c1+ 3a1b1+ 3a2b2) x + −16b2₁c2− 8b2b1c1+ 3b21+ 3b22

y + −16a2b21− 8a1b2b1

xy + 4a2₁b2− 16a1a2b1

x2− 12b21b2y2

/3 (a2b1− a1b2) , ˙ y = ( − 4a2c21− 3a1c1+ 16a1c2c1− 3a2c2 + 16a2₁c2+ 8a2a1c1− 3a21− 3a22

x

+ (−8a2b1c1+ 16a1b1c2+ 16a1b2c1− 3a1b1− 3a2b2) y

+ 16a2₁b2+ 8a2a1b1

xy + 12a2₁a2x2 + 16a1b1b2− 4a2b21

y2

/3 (a2b1− a1b2) . (3.5)

Since (32x2− 24y + 9)2/(3 − 16y) is a first integral of system (3.5), then a first integral of system (3.5) is

H4(x, y) =

(32(a1x + b1y + c1)2− 24(a2x + b2y + c2) + 9)2

3 − 16(a2x + b2y + c2)

(26)

Generalized isochronous system (1.18). Doing the affine change of variables (3.1) the generalized isochronous system for the quadratic system (1.18) is

˙x =(−16b2c21+ 3b1c1+ 8b1c2c1+ 4b2c22+ 3b2c2

+ (8a2b1c1+ 8a1b1c2− 32a1b2c1+ 8a2b2c2+ 3a1b1+ 3a2b2)x

+ (8b2₁c2− 24b2b1c1+ 8b22c2+ 3b21+ 3b22)y

+ (8a2b21− 24a1b2b1+ 8a2b22)xy + (−16a21b2+ 8a2a1b1+ 4a22b2)x2

+ (4b3₂− 8b2₁b2)y2)/3(a2b1− a1b2),

˙

y =(16a2c21− 3a1c1− 8a1c2c1− 4a2c22− 3a2c2

+ (−8a2₁c2+ 24a2a1c1− 8a22c2− 3a21− 3a22)x

+ (32a2b1c1− 8a1b1c2− 8a1b2c1− 8a2b2c2− 3a1b1− 3a2b2)y

+ (−8a2₁b2+ 24a2a1b1− 8a22b2)xy + (8a21a2− 4a32)x2

+ (16a2b21− 8a1b2b1− 4a2b22)y2)/3(a2b1− a1b2).

(3.6)

The quadratic system (1.18) has the first integral (−256x2+ 128y2+ 96y + 9)/(8y + 3)4, which gives the following first integral for system (3.6)

H5(x, y) =

−256(a1x + b1y + c1)2+ 128(a2x + b2y + c2)2+ 96(a2x + b2y + c2) + 9

(8(a2x + b2y + c2) + 3)4

.

3.2 Limit cycles for continuous PWDS formed by linear

and quadratic isochronous centers

It has been proved in [22] that continuous piecewise linear differential systems separated by one straight line formed by two linear centers have no crossing limit cycles. Now we shall see that this result extends to continuous piecewise linear differential systems separated by one straight line formed by one linear center and one quadratic isochronous center have no crossing limit cycles.

Theorem 3.1. The continuous PWDS formed by a linear differential center (which is isochronous) and an isochronous quadratic center separated by the straight line x = 0 have no crossing periodic orbits, and consequently no crossing limit cycles.

3.2.1 Proof of Theorem 3.1: What is the maximum number of limit

cycles for the PWDS formed by a linear and quadratic isochronous

centers?

Proof of Theorem 3.1. The first objective is to study periodic orbits and limit cycles of continuous PWDS formed by a linear center (1.14) and a generalized quadratic system (1.15), (1.16), (1.17) or (1.18). To find the crossing periodic orbits and the crossing limit cycles of such PWDS, we must solve the following algebraic system

(27)

where Hk(x, y) and HL(x, y) are the first integrals of the quadratic isochronous center

and of the linear center respectively, and (0, y1) and (0, y2) with y1 6= y2 are the two

intersection points of the crossing periodic orbits with the straight line x = 0.

In order that we have a continuous PWDS (1.14)-(3.3), we impose that both systems coincide on x = 0, and then both systems must verify the following algebraic system

˙xk− ˙xL|x=0 = 0, y˙k − ˙yL|x=0 = 0, (3.8)

where ˙xL, ˙yL, ˙xk and ˙yk are the derivatives with respect to time t of x and y for linear

system and quadratic system, respectively. Thus we get the following algebraic system −a2b1δ + a1b2δ − b2c21+ b1c1+ 2b1c2c1+ b2c22+ b2c2= 0, 4a2b1β2− 4a1b2β2+ a2b1ω2− a1b2ω2+ 4αb21+ 4αb22+ 8αb21c2+ 8αb22c2 = 0, −b2 b2₁+ b2₂

= 0, −a2b1γ + a1b2γ + a2c2₁− a1c1− 2a1c2c1− a2c2₂− a2c2 = 0, −a2βb1+ a1βb2+ 2a2b1c1− 2a1b1c2− 2a1b2c1− 2a2b2c2− a1b1− a2b2= 0, a2b2₁− 2a1b2b1− a2b2₂= 0.

Since when we evaluate a2b1 − a1b2 in all real solutions (See appendix, section 5.1) for

this algebraic system we obtain zero, we get a contradiction with (3.2). Therefore there are no continuous PWDS (1.14)-(3.3).

In order that the PWDS (1.14)-(3.4) be continuous they must coincide on x = 0, then the algebraic system (3.8) must be satisfied, which gives the following algebraic system

−a2b1δ + a1b2δ − b2c21+ b1c1+ b1c2c1+ b2c2 = 0,

4a2b1β2− 4a1b2β2+ a2b1ω2− a1b2ω2+ 4αb21+ 4αb22− 4αb1b2c1+ 4αb21c2 = 0,

−a2b1γ + a1b2γ + a2c2₁− a1c1− a1c2c1− a2c2, = 0,

−a2βb1+ a1βb2+ 2a2b1c1− a1b1c2− a1b2c1− a1b1− a2b2 = 0,

b1 = 0.

All solutions (See appendix, section 5.2) of this algebraic system give a2b1− a1b2 = 0,

except the following solution {α = 4a21c21+8a2a1c1+a21ω2+4a22 4a1b2 , β = a1c1+a2 a1 , γ = −a2c21+a1c1+a1c2c1+a2c2 a1b2 , δ = c21−c2 a1 , b1= 0} for which a2b1− a1b2 = −a1b2.

Solving the algebraic system (3.7) for the values of the solution s4 we get y1= y2. So the

(28)

The PWDS (1.14)-(3.5) is continuous if and only if system (3.8) is verified, i.e. −3a2b1δ + 3a1b2δ + 4b2c21+ 3b1c1− 16b1c2c1+ 3b2c2 = 0,

12a2b1β2− 12a1b2β2+ 3a2b1ω2− 3a1b2ω2+ 12αb2₁+ 12αb2₂− 32αb1b2c1

−64αb2

1c22c2= 0,

−3a2βb1+ 3a1βb2− 8a2b1c1+ 16a1b1c2+ 16a1b2c1− 3a1b1− 3a2b2= 0,

−3a2b1γ + 3a1b2γ − 4a2c2₁− 3a1c1+ 16a1c2c1− 3a2c2= 0,

−4b1(a2b1− 4a1b2) = 0,

−4b2

1b2= 0.

But all real solutions (See appendix, section 5.3) of this algebraic system give a2b1−a1b2 =

0, except the solution

{α = 1024a21c21−384a2a1c1+9a21ω2+36a22

36a1b2 , β =

3a2−16a1c1

3a1 , γ =

4a2c21+3a1c1−16a1c2c1+3a2c2

3a1b2 , δ = −4c21−3c2

3a1 , b1 = 0}

which gives a2b1− a1b2= −a1b2.

By solving system (3.7) for the values of this solution we get y1 = y2. Therefore the

continuous PWDS (1.14)-(3.5) cannot have crossing periodic orbits.

To get a continuous PWDS we impose that both systems (1.14)-(3.6) coincide on x = 0 by using (3.8), and we obtain the following algebraic system

−3a2b1δ + 3a1b2δ − 16b2c21+ 3b1c1+ 8b1c2c1+ 4b2c22+ 3b2c2 = 0, 12a2b1β2− 12a1b2β2+ 3a2b1ω2− 3a1b2ω2+ 12αb2₁+ 12αb2₂− 96αb1b2c1+ 32αb2₁c2 +32αb2₂c2= 0, −3a2b1γ + 3a1b2γ + 16a2c2₁− 3a1c1− 8a1c2c1− 4a2c2₂− 3a2c2= 0, −3a2βb1+ 3a1βb2+ 32a2b1c1− 8a1b1c2− 8a1b2c1− 8a2b2c2− 3a1b1− 3a2b2 = 0, 4 4a2b2₁− 2a1b2b1− a2b2₂

= 0, −4b2 b2₂− 2b2₁

= 0.

Since all solutions (See appendix, section 5.4) of this algebraic system give a2b1−a1b2= 0,

then there are no continuous PWDS (1.14)-(3.6). In summary Theorem 3.1 is proved.

(29)

3.3 Limit cycles of continuous PWDS formed by two

quadratic isochronous centers

In what follows we characterize the existence and non-existence of crossing periodic orbits and crossing limit cycles for continuous piecewise linear differential systems separated by one straight line formed by two quadratic isochronous centers.

Theorem 3.2. The following statements hold for the continuous PWDS formed by two generalized isochronous quadratic centers separated by the straight line x = 0.

(a) If the generalized centers are (3.3) and (3.3), then the PWDS can have crossing periodic orbits but they cannot have crossing limit cycles.

(b) If the generalized centers are (3.3) and (3.4), then the PWDS can have crossing periodic orbits but they cannot have crossing limit cycles.

(c) If the generalized centers are (3.3) and (3.5), then the PWDS has at most one crossing limit cycle and there are systems realizing this limit cycle.

(d) If the generalized centers are (3.3) and (3.6), then the PWDS has at most one crossing limit cycle and there are systems realizing this limit cycle.

3.3.1 Proof of Theorem 3.2:What is the maximum number of limit

cycles for the PWDS formed by the general quadratic isochronous

center ˙x = −y + x

2

− y

2

, ˙

y = x(1 + 2y), and an arbitrary quadratic

isochronous center?

In what follows we consider in one side of the straight line x = 0 the generalized isochronous systems of (1.15) and in the other side a generalized isochronous system (1.15), (1.16), (1.17) and (1.18). In the first generalized systems (3.3) we rename the parameters a2, b2

and c2 by α1, β1 and γ1, respectively; and in the second generalized systems (j) for (j=3.3,

3.4, 3.5 or 3.6) we rename the parameters a1, b1, c1, a2, b2 and c2 by a2, b2, c2, α2, β2 and

γ2, respectively. Doing this condition (3.2) becomes

α1b1− a1β1 6= 0 and α2b2− a2β2 6= 0. (3.9)

In order to study the crossing periodic orbits and limit cycles of a PWDS (3.3)-(j) formed by two generalized isochronous systems with j ∈ {3.3, 3.4, 3.5, 3.6}, we must solve the following algebraic system

H2(0, y1) − H2(0, y2) = 0, Hj(0, y1) − Hj(0, y2) = 0, (3.10)

(30)

Proof of statement (a) of Theorem 3.2. In order that the PWDS (3.3)-(3.3) be con-tinuous, they must coincide on x = 0, which means the following algebraic system must be satisfied. −2a2b1β2γ1c1+ 2a1b2β1c2γ2− a2b1β2c1+ a1b2β1c2− a2β1β2γ₁2+ a1β1β2γ₂2 −a2β1β2γ1+ a1β1β2γ2+ a2β1β2c2₁− a1β1β2c2₂+ α1b2β1γ₁2− α1b1β2γ₂2 +α1b2β1γ1− α1b1β2γ2+ α1(−b2) β1c21+ α1b1β2c22+ 2α1b1b2γ1c1 −2α1b1b2c2γ2+ α1b1b2c1− α1b1b2c2 = 0, −2a2β2b21γ1+ 2a1b22β1γ2− a2β2b21+ a1b22β1− 2a2β12β2γ1+ 2a1β1β22γ2 +a1β1β22− a2β12β2− 2α1β22b1γ2+ 2α1b2β12γ1− α1β22b1+ α1b2β12 +2α1b2b21γ1− 2α1b22b1γ2+ α1b2b21− α1b22b1 = 0, −a2b21β2β1+ a1b22β2β1− a2β2β13+ a1β23β1+ α1b2β13+ α1b21b2β1− α1b1β23 −α1b1b22β2 = 0, a2α1β2γ₁2− a1α1β1γ₂2+ a2α1β2γ1− a1α1β1γ2− 2a1α1b2γ1c1+ 2a2α1b1c2γ2 −a1α1b2c1+ a2α1b1c2− a2α1β2c21+ a1α1β1c22+ 2a1a2β2γ1c1− 2a1a2β1c2γ2 +a1a2β2c1− a1a2β1c2− α21b2γ12+ α21b1γ22− α21b2γ1+ α12b1γ2+ α21b2c21 −α2₁b1c2₂= 0, 2a2α1β1β2γ1− 2a1α1β1β2γ2− a1α1β1β2+ a2α1β1β2− 2a1α1b1b2γ1 +2a2α1b1b2γ2− a1α1b1b2+ a2α1b1b2+ 2a1a2b1β2γ1− 2a1a2b2β1γ2 −a1a2b2β1+ a1a2b1β2− 2a1α1b2β1c1+ 2a1α1b2β1c2− 2a2α1b1β2c1 +2a2α1b1β2c2+ 2a1a2β1β2c1− 2a1a2β1β2c2− 2α₁2b2β1γ1+ 2α2₁b1β2γ2 −α2 1b2β1+ α21b1β2+ 2α21b1b2c1− 2α21b1b2c2 = 0, −a1α1β1β₂2+ a2α1β₁2β2− a2α1β2b2₁− 2a1α1b2β1b1+ 2a2α1b2β2b1+ a1α1b2₂β1 +2a1a2β1β2b1− 2a1a2b2β1β2+ α2₁β₂2b1− α2₁b2β₁2+ α2₁b2b2₁− α2₁b2₂b1 = 0, (3.11)

together with conditions (3.9).

From the sixth equation of (3.11) we get

a1 = α1 a2β2 −b21+ 2b2b1+ β12

+ α2b1β22− α2b2 β12+ b1(b2− b1)

β1 2a2(b2− b1) β2+ α2β₂2+ α2(2b1− b2) b2

, (3.12)

if the denominator of a1 is non-zero. If it is zero then β1 = 0 or 2a2(b2− b1) β2+ α2β22+

α2(2b1− b2) b2 = 0. This last equation is equivalent to the following solution

d4 = {b1 =

−2a2β2b2− α2β22+ α2b22

2 (α2b2− a2β2)

}.

by neglecting all solutions (See appendix, section 5.5) which do not satisfying conditions (3.9). So it remains to study this solution.

(31)

a1 = 0, and consequently α1b1− a1β1 = 0, which is a contradiction with first condition in

(3.9).

In summary for solving system (3.11) we consider the following cases.

Case 1: Suppose that b16= (−2a2β2b2− α2β22+ α2b22)/2 (α2b2− a2β2), β1 6= 0 and α16= 0.

Since a1 is defined in (3.12). From the fifth equation of (3.11), we get

c1 = (b1(2γ1+ 1)(α2b2− a2β2) + b2(2α2β1c2− a2(2β1γ2− 4β2γ1+ β1− 2β2)) + β2

+ (α2(−2β1γ22β2γ1− β1+ β2) − 2a2β1c2) + b22(−(2α2γ1+ α2)))/2β1(α2b2− a2β2).

Now we solve simultaneously the second and the third equations of (3.11), and we obtain the only solution

{β1 = α1β2(b 2 2+β22) 2a2(b2−b1)β2+α2β22+α2(2b1−b2)b2, γ2 = 2b2(2γ1+1)(a2β2+α2b1)+β2(β2(2α2γ1−α1+α2)−2a2(2b1γ1+b1))+b 2 2(−(2α2γ1+α1+α2)) 2α1(b22+β22) },

among all real solutions (See appendix, section 5.6) for which b1 6= (−2a2β2b2− α2β₂2+

α2b2₂)/2 (α2b2− a2β2), β1 6= 0, α1 6= 0 and conditions (3.9) are satisfied.

Solving now simultaneously the first and the fourth equations of (3.11). The only real solutions (See appendix, section 5.7) for which b1 6= (−2a2β2b2− α2β22+ α2b22)/2(α2b2−

a2β2), β1 6= 0, α1 6= 0 and conditions (3.9) are satisfied are solutions

s1 = {b1 = b2, α1 = −α2} and s2= {b1 = b2, α1= α2}.

Then we must discuss these two solutions s1 and s2 in the following subcases to detect

whether the continuous PWDS (3.3)-(3.3) have or do not have limit cycles.

Subcase 1.1: Consider s1. Solving system (3.10) we get y1= y2, or the following solution

y1 =

2b1(2γ1c2+ c2) + b2₁(2γ1y2+ y2) + β2 2c2₂− 2γ1(γ1+ 1) + y2(2β2γ1+ β2)

b2₁+ β₂2

(−2γ1+ 2β2y2− 1)

.

Since in order to have a crossing periodic orbit we must have y1 6= y2. Therefore these

continuous PWDS have a continuum of crossing periodic orbits and then no limit cycles. Subcase 1.2: Now we consider s2, so system (3.10) gives y1 = y2, or

y1 = −

2b1(2γ1c2+ c2) + b2₁(2γ1y2+ y2) + β2 −2c2₂+ 2γ1(γ1+ 1) + y2(2β2γ1+ β2)

b2₁+ β₂2

(2γ1+ 2β2y2+ 1)

.

As in the previous subcase no limit cycles.

Case 2: We suppose that b1= (−2a2β2b2−α2β₂2+α2b2₂)/2(α2b2−a2β2) and β1 6= 0. With

this value for b1 the sixth equation of (3.11) gives the only following four real solutions

s1 = {α1= 0}, s2 = {a2 = 0, α2 = 0}, s3 = {b2= 0, β2 = 0} and s4 = {α2= 0, β2= 0}.

The only solution that satisfying conditions (3.9) is s1, which is equivalent to α1 = 0. Now

from the third equation of (3.11), regarding all the real solutions (See appendix, section 5.8), the only solution verifying (3.9) is

(32)

a1 =

4 a2α2b2β23+2 β22(2 a22(b22+β12)−α22b22)−4 a2α2b2β2(b22+2 β12)+α22β24+α22b22(b22+4 β12)

4 β2(b22+β22) (a2β2−α2b2) .

By solving the fifth equation of (3.11) we obtain from all real solutions (See appendix, section 5.9) the following solutions s1, s2 and s8

s1 = {γ2= 2a2b2(2β2γ1−β1+β2)+α2β2(2β2γ1−2β1+β2)+4a2β1β2(c1−c2)−b 2 2(2α2γ1+α2)+4α2β1b2(c2−c1) 4β1(a2b2+α2β2) }, s2 = {a2 = −α2β2/b2, c2= 2b2γ1+b_4β2₁+4β1c1}, and s3 = {b2 = 0, c2 = c1, α2 = 0},

which satisfying (3.9). Then we divide case 2 into the following three subcases.

Subcase 2.1: We consider the solution s1. Solving the second equation of (3.11) we get

the following real solutions u1 = {γ1= −2β1(c1−c_α 2)(α2b2−a2β2)

2(b22+β22)

− 1₂}, u2 = {c2 = c1, α2 = 0} and

u3 = {α2= 0, β1 = 0}.

But u1 and u2 are the only solutions satisfying (3.9) which must be studied separately

into two subcases.

Subcase 2.1.1: Consider u1. Solving simultaneously the first and the fourth equations of

(3.11), we get the only real solution β1 = 0, in contradiction with the hypothesis of Case

2.

Subcase 2.1.2: Consider u2. Then all equations of (3.11) are verified except the first one,

which becomes a2β2 β22− β12

−4b2β1c1(2γ1+ 1) + (2b2γ1+ b2)2+ β12 4c21+ 1

= 0.

Solving this equation we obtain the following set of real solutions v1 = {a2 = 0}, v2 =

{β2 = 0}, v3 = {β2 = −β1} and v4= {β2= β1}.

The solutions for which conditions (3.9) are verified are v3 and v4. Now we must discuss

these two subcases as follows.

Subcase 2.1.2.1: Consider v3. Then we have a continuous PWDS. We prove that there

are a continuum of crossing periodic orbits and no limit cycles, because solving the alge-braic system (3.10) we get y1= y2, or the solution

y1 = −

2b2(2γ1c1+ c1) + b2₂(2γ1y2+ y2) + β1 −2c2₁+ 2γ1(γ1+ 1) + y2(2β1γ1+ β1)

b2₂+ β₁2

(2γ1+ 2β1y2+ 1)

.

(33)

3 Limit cycles of continuous PWDS system (3.10) we obtain y1 = y2, or y1 = − 2b2(2γ1c1+ c1) + b2₂(2γ1y2+ y2) + β1 −2c2₁+ 2γ1(γ1+ 1) + y2(2β1γ1+ β1)

b2₂+ β₁2

(2γ1+ 2β1y2+ 1) .

This gives a continuum of crossing periodic orbits.

Subcase 2.2: Assume that solution s2 holds. Solving the second equation of (3.11) we

get the following sets of solutions {α2 = 0}, {γ2 = (2β2γ1− β1+ β2)/(2β1)} and {β1 =

0, γ1 = −1/2}. The only solution for which conditions (3.9) are satisfied is γ2= (2β2γ1−

β1+ β2)/(2β1). The remaining unsolved equations of (3.11) are the first and the fourth

equations, we solve them simultaneously and we get only one real solution α2 = 0, but this

solution gives a contradiction with (3.9). Then there are no continuous PWDS (3.3)-(3.3) in this case.

Subcase 2.3: Consider the solution s3. Then the remaining unsolved equations of system

(3.11) are the first, the second and the fourth equations. Solving these three equations simultaneously we get one of the following sets of real solutions t1 = {a2= 0}, t2 = {β1 =

0}, t3 = {β2 = −β1, γ2= −γ1− 1} and t4= {β2 = β1, γ2= γ1}.

The sets of solutions t3 and t4are the only for which conditions (3.9) are verified. We now

have a continuous PWDS (3.3)-(3.3). Consider these two cases t3 and t4 separately.

Subcase 2.3.1: Consider solution t3. Then for showing whether the PWDS (3.3)-(3.3)

has a limit cycle or not, we solve the algebraic system (3.10). Here we get y1 = y2, or

y1=

2c2₁− 2γ1(γ1+ 1) − y2(2β1γ1+ β1)

β1(2γ1+ 2β1y2+ 1)

.

We have then a continuum of crossing periodic orbits.

Subcase 2.3.2: Consider the solution t4. I In a similar way solving the algebraic system

(3.10). We get y1= y2, or

y1=

2c2₁− 2γ1(γ1+ 1) − y2(2β1γ1+ β1)

β1(2γ1+ 2β1y2+ 1)

.

Then we get a continuum of crossing periodic orbits.

Case 3: Assume that b16= (−2a2β2b2− α2β22+ α2b22)/2 (α2b2− a2β2), β1= 0. The third

equation of (3.11) becomes −α1b1β2 b22+ β22

= 0, and since β1 = 0 we must have β2= 0,

because b1 = 0 or α1 = 0 gives b1α1− a1β1 = 0. Solving the five remaining equations of

(3.11), the only solutions from all real solutions (See appendix, section 5.10) for which (3.9) and b1 6= (−2a2β2b2− α2β₂2+ α2b₂2)/(2 (α2b2− a2β2)) are verified are s1, s2, s3, s4

and s5 given as follows

(34)

s2 = {a2 = a1, b1 = b2, c1 = c2, α1 = −α2, γ1 = −γ2− 1},

s3 = {b1 = b2, c1= c2, α1 = −α2, γ1 = −1/2, γ2 = −1/2},

s4 = {a2 = a1, b1 = b2, c1 = c2, α1 = α2, γ1 = γ2} and

s5 = {b1 = b2, c1= c2, α1 = α2, γ1 = −1/2, γ2= −1/2}.

For these five cases we have a continuous PWDS (3.3)-(3.3). Now we discuss these five subcases.

Case 3.1: Consider the solution s1. Then the first integrals of systems (3.3)-(3.3) are

H1(x, y) = [8a1b1xy + 8a1c1x + 4a21x2+ 8b1c1y + 4b21y2+ 4c21+ 4α21x2− 4α1x + 1]/[8α1x],

and

respectively. So they are not defined on the y-axis, and consequently we cannot have continuous PWDS.

Case 3.2: Consider the solution s2. Then solving the algebraic system (3.9) we obtain

y1= y2, or y1 = −y2− 2c1/b1. We conclude that there is a continuum of crossing periodic

orbits.

Case 3.3: Consider the solution s3. Then the first integrals of systems (3.3) are

H1(x, y) = −[8a1b2xy + 8a1c2x + 4a21x2+ 8b2c2y + 4b22y2+ 4c22+ 4α22x2+ 4α2x + 1]/[8α2x],

and

H1(x, y) = [8a2b2xy + 8a2c2x + 4a22x2+ 8b2c2y + 4b22y2+ 4c22+ 4α22x2− 4α2x + 1]/[8α2x].

The same conclusion than in case 3.1.

Case 3.4: Consider the solution s4. Now solving the algebraic system (3.9) we obtain

y1= y2, or y1 = −y2− 2c2/b2. There are a continuum of crossing periodic orbits.

Case 3.5: Consider the solution s5. Then the first integrals of systems (3.3) are

and

H1(x, y) = [8a2b2xy + 8a2c2x + 4a22x2+ 8b2c2y + 4b22y2+ 4c22+ 4α22x2− 4α2x + 1]/[8α2x].

(35)

Case 4: Assume that b1= (−2a2β2b2− α2β₂2+ α2b2₂)/(2 (α2b2− a2β2)) and β1 = 0. Then

the sixth equation of (3.11) becomes

α1 α2β22+ 2a2β2b2− α2b22

2

4(a2β2− α2b2)

= 0.

Since we have β1= 0 we cannot take α1= 0 otherwise we get a contradiction with the first

condition in (3.9). So we must take α2β22+ 2a2β2b2− α2b22= 0. By solving this equation,

we obtain a2 = (b22α2− α2β22)/2b2β2, and by replacing a2 in b1 = (−2a2β2b2− α2β₂2+

α2b2₂)/(2 (α2b2− a2β2)) we get b1 = 0, which is a contradiction with the first condition in

(36)

Proof of statement (b) of Theorem 3.2. Before starting, we give the following flow chart for our proof.

β1= 0 b2= b1 b1c2+ β26= 0 b1c2+ β2= 0 β26= 0 β2= 0 γ1(γ1+ 1) 6= 0 γ1(γ1+ 1) = 0

no crossing periodic orbits

no crossing periodic orbits 1 + 2γ16= 0 and c2= c1

1 + 2γ1 = 0 and c2 6= c1

1 + 2γ1= c2− c1= 0

continuum of periodic orbits non-continuous piecewise non continuous piecewise

c26= 0

c2= 0 c1= 1 + 2γ1 = 0 non-continuous piecewise

c1= 0

c16= 0

non-continuous piecewise

no crossing periodic orbits c16= 0 and 1 + 2γ1= 0

c1= 0 and 1 + 2γ16= 0

non-continuous piecewise

continuum of periodic orbits

In order that the PWDS (3.3)-(3.4) be continuous they must coincide on x = 0, so the following algebraic system must be satisfied

−2a2β2b1c1γ1+ a1β1b1c2γ2+ a1β1b1c2− a2β2b1c1− a2β1β2γ12− a2β1β2γ1 +a1β1β2γ2+ a2β1β2c21− a1β1β2c22+ α2β1b1γ12+ α2β1b1γ1− α1β2b1γ2 −α2β1b1c2₁+ α1β2b1c2₂+ 2α2b2₁c1γ1− α1b2₁c2γ2+ α2b2₁c1− α1b2₁c2 = 0, −2a2β2b21γ1+ a1β1b21γ2+ a1β1b21− a2β2b21− a1β1β2b1c2− 2a2β12β2γ1+ a1β1β22 −a2β₁2β2+ 2α2β₁2b1γ1+ α2β₁2b1− α1β₂2b1+ 2α2b3₁γ1− α1b3₁γ2+ α2b3₁ −α1b3₁+ α1β2b2₁c2 = 0, −β1 b21+ β12

= 0, 2a2α1β1β2γ1+ a2α1β1β2− a1α2β1β2+ a2α1b1b2γ2− 2a1α2b1b2γ1+ a2α1b1b2 −a1α2b1b2+ 2a1a2b1β2γ1− a1a2b2β1γ2− a1a2b2β1+ a1a2b1β2− 2a2α1b1β2c1 +a2α1b1β2c2− 2a1α2b2β1c1+ 2a1α2b2β1c2+ 2a1a2β1β2c1− a1a2β1β2c2 −2α2α1b2β1γ1− α2α1b2β1+ α2α1b1β2+ 2α2α1b1b2c1− 2α2α1b1b2c2 = 0, a2α1β2γ12+ a2α1β2γ1− a1α2β1γ2− 2a1α2b2γ1c1+ a2α1b1c2γ2− a1α2b2c1 +a2α1b1c2− a2α1β2c2₁+ a1α2β1c2₂+ 2a1a2β2γ1c1− a1a2β1c2γ2+ a1a2β2c1 −a1a2β1c2− α1α2b2γ₁2− α1α2b2γ1+ α1α2b1γ2+ α1α2b2c2₁− α1α2b1c2₂ = 0, −2a1β1b1+ a1b2β1− α1β12+ α1b21− α1b2b1 = 0, (3.13) with α1b1− a1β1 6= 0 and α2b2− a2β2 6= 0.

(37)

We see easily that a necessary condition for a continuous piecewise system is β1 = 0

and this implies that α1 6= 0 and b1 6= 0, otherwise we have a contradiction with the

condition α1b1− a1β1 6= 0. Substituting β1 = 0 into the last equation of (3.13) we obtain

b1(b1− b2)α1 = 0, and consequently b2 = b1, because b1 and α1 cannot be zero. Now in

all cases of this proof we take β1= 0, b2= b1, α1 6= 0 and b16= 0.

Case 1: b1c2+ β2 6= 0, β2 6= 0 and γ1(γ1+ 1) 6= 0. Then from the first equation of (3.13)

we get

γ2 =

−2a2β2c1γ1− a2β2c1+ 2α2b1c1γ1− α1b1c2+ α2b1c1+ α1β2c22

α1(b1c2+ β2)

.

From the second equation of (3.13) we obtain

α1=

b1(2γ1+ 1) (b1(c2− c1) + β2) (α2b1− a2β2)

β2 b2₁+ β₂2

.

Using the fifth equation of (3.13) we have

a1 = −b1(−a2b1β2− 2a2b1β2c2₂+ 4a2b1β2c1c2− 2a2b1β2c12− α2β₂2+ a2β₂2c1− a2β₂2c2

−α2b1β2c1+ α2b1β2c2+ 2α2b2₁c2₁+ 2α2b2₁c2₂− 4α2b2₁c1c2)/(β2(b2₁+ β₂2)).

Finally from the fourth equation of (3.13) we get

β2 =

b1(c1− c2)(c21− 2c2c1+ c22+ γ12+ γ1+ 1)

γ1(γ1+ 1)

.

With these values of the parameters the PWDS (3.3)-(3.4) are continuous. Now we must solve the system

H2(0, y1) − H2(0, y2) = 0, H3(0, y1) − H3(0, y2) = 0, (3.14)

for studying the crossing periodic orbits and the crossing limit cycles of these continuous PWDS.

The first equation of (3.14) in this case is

b1(y1− y2)(2c1+ b1y1+ b1y2)

1 + 2γ1

= 0. (3.15)

Since in order to have a crossing periodic orbit we must have y1 6= y2, we obtain that

y2 = −(2c1 + b1y1)/b1. Substituting y2 into the second equation of (3.14) we get that

y1= y2= −c1/b1. Therefore this PWDS has no crossing periodic orbits.

Case 2: Assume that b1c2+ β2 6= 0, β2 6= 0 and γ1(1 + γ1) = 0. Then from the case 1 we

(38)

the fourth equation of (3.13) we should have c2 = c1 otherwise we have non–continuous

PWDS. With these values the PWDS (3.3)-(3.4) are continuous. Again from (3.15) we obtain y2 = −(2c1+ b1y1)/b1. Substituting y2 into the second equation of (3.14) we get

that y1= y2= −c1/b1. So no crossing periodic orbits.

Case 3: b1c2+ β2 6= 0 and β2 = 0. Then c26= 0. Now as in the case 1, from first equation

of system (3.13) we obtain γ2= (c1α2(1 + 2γ1) − c2α1)/(c2α1). Then the second equation

of (3.13) becomes − b 3 1(c2− c1)α2(1 + 2γ1) c2 = 0. (3.16)

Which α2 and b1 cannot be zero from the condition of (3.9). So in this case we have three

subcases satisfy (c2− c1)(1 + 2γ1) = 0.

Subcase 3.1: 1 + 2γ1 6= 0. Then c2 = c1 to verify equation (3.16). The fifth equation

of (3.13) becomes −(a1 − a2)b2₁α2(1 + 2γ1) = 0, so we must take a2 = a1, otherwise

the PWDS cannot be continuous. Finally from the fourth equation of (3.13) we get α2 = (α1(1 + γ1+ γ12))/(1 + 2γ1). With these values of the parameters the PWDS

(3.3)-(3.4) are continuous. Now the first and second equation of (3.14) become b1(y1− y2)(2c1+ b1y1+ b1y2)

1 + 2γ1

= 0, b1(y1− y2)(2c1+ b1y1+ b1y2) (1 + γ1+ γ12)2

= 0,

respectively. Since the numerator of both equations coincide we have a continuum of crossing periodic orbits in this subcase.

Subcase 3.2: c2 6= c1. Then we must take 1 + 2γ1 = 0 to satisfy the equation (3.16).

Using the fifth equation of (3.13) we obtain 2b2₁(c1 − c2)α1α2 = 0, we see that cannot

verify this equation under the conditions of this subcase. Then we cannot have continuous PWDS in subcase.

Subcase 3.3: 1 + 2γ1 = c2− c1 = 0. Then the fifth equation of (3.13) is satisfied, and the

fourth equation of (3.13) becomes −3b1α1α2/4 = 0, which cannot be satisfied otherwise

we have a contradiction with (3.9). So in this subcase the PWDS cannot be continuous. Case 4: b1c2 + β2 = 0 and c2 6= 0. The first equation of (3.13) gives α1 = (a2c1c2 +

c1α2)(1 + 2γ1)/(c2(1 + c2₂)). The second equation of (3.13) becomes

b3₁(a2c2+ α2)(1 + 2γ1)(c2+ c3₂− c1(1 + 2c2₂+ γ2))

c2(1 + c2₂)

= 0.

From the condition b1c2+ β2 = 0 of this case we get c2 = −β2/b1 and from the starting

of the proof of this statement we have b1 = b2, by substituting in a2c2+ α2 = 0 gives

a2β2− b2α2 = 0; which is a contradiction with second condition in (3.9). If 1 + 2γ1 = 0,

(39)

which is a contradiction with first condition in (3.9). So a2c2+ α2 and 1 + 2γ1 cannot be

zero, we must take

c2+ c32− c1(1 + 2c22+ γ2) = 0. (3.17)

Subcase 4.1: c1= 0. Then from the second equation of (3.13) we have

b3₁(a2c2+ α2)(1 + 2γ1) = 0,

Since a2c2+ α2 and 1 + 2γ1cannot be zero as we have seen above this last equation cannot

be satisfy, and consequently the PWDS cannot be continuous.

Subcase 4.2: c16= 0. Then from (3.17) we get γ2 = (−c1+ c2− 2c1c2₂+ c3₂)/c1. The fifth

equation of (3.13) is

b2₁(a2c2+ α2)(a2c2(1 + 2c2₁− 3c1c2+ c2₂) − a1(c2+ c3₂) + c1(2c1− 3c2)α2)(1 + 2γ1)

c2(1 + c2₂)

= 0.

As we have seen in the previous case a2c2+ α2 and 1 + 2γ1 cannot be zero. So we should

have

a2c2(1 + 2c21− 3c1c2+ c22) − a1(c2+ c32) + c1(2c1− 3c2)α2 = 0.

Solving this equation we get

a1 =

a2c2+ 2a2c2₁c2− 3a2c1c₂2+ a2c3₂+ 2c₁2α2− 3c1c2α2

c2(1 + c2₂)

.

And solving the fourth equation of (3.13) we obtain

γ1 = −c1±

p

−c1(3c1+ 4c3₁− 4c2− 12c₁2c2+ 12c1c2₂− 4c3₂) 2c1 .

We assume that the squareroot which appears in the expression of γ1 is real, otherwise

the PWDS is not continuous.

With these values of the parameters we have satisfied all equations of system (3.13). Then the PWDS (3.3)-(3.4) are continuous. Now the first equation of (3.14) becomes

−

_p

b1c1(y1− y2)(2c1+ b1y1+ b1y2)

−c1(3c1+ 4c3₁− 4c2− 12c₁2c2+ 12c1c2₂− 4c3₂)

= 0.

In order to get crossing periodic orbits we need that 2c1+ b1y1+ b1y2= 0, or equivalently

y2= (−2c1− b1y1)/b1. Substituting y2 into the second equation of (3.14) we get

4c3₁(1 + c2₂)(c1+ b1y1)3

c2₂(−1 + 2c1c2− c₂2+ b1c1y1)2(1 + 2c2₁− 2c1c2+ c₂2+ b1c1y1)2

(40)

By solving this equation we get y1 = −c1/b1 and yield to y2 = −c1/b1. Then in this case

the PWDS cannot have crossing periodic orbits.

Case 5: b1c2+ β2 = 0 and c2 = 0. Then β2 = 0, and the first equation of system (3.13)

becomes b2₁c1α2(1 + 2γ1) = 0. Since b1 and α2 cannot be zero, so we have the cases verify

c1(1 + 2γ1) = 0, otherwise we have non-continuous PWDS piecewise. Then we have the

two following subcases.

Subcase 5.1: c1= 0 and 1 + 2γ16= 0. Then the second equation of system (3.13) becomes

b3₁(α2+ 2α2γ1− α1(1 + γ2)) = 0. Since b1 cannot be zero we have that α2+ 2α2γ1− α1(1 +

γ2) = 0. By solving this equation we get γ2 = (−α1+ α2+ 2α2γ1)/α1. Thus system (3.13)

is satisfied in this subcase and the PWDS is continuous. Now the first and second equation of (3.14) become

b2₁(y1− y2)(y1+ y2) 1 + 2γ1 = 0 and b 2 1(y1− y2)(y1+ y2) 1 + γ1+ γ₁2 2 = 0

respectively. This system has the two solutions y1 = y2 and y1 = −y2. The last solution

means that the PWDS has a continuum of periodic orbits. Then no crossing limit cycles. Subcase 5.2: c1 6= 0 and 1 + 2γ1 = 0. The second equation of system (3.13) becomes

−b3

1α1(1 + γ2) = 0. Since b1 and α1 cannot be zero, we have that γ2 = −1. The fifth

equation becomes √3b2₁α1α2 = 0, but this is a contradiction because all the parameters

which appear in it are nonzero. Therefore the PWDS cannot be continuous. Subcase 5.3: c1= 1 + 2γ1 = 0. The same proof and result as in subcase 5.2.

(41)

Proof of statement (c) of Theorem 3.2. For studying the maximum number of limit cycles of the PWDS (3.3)-(3.5), we must solve system (3.10). Solving the first equation of (3.10) with respect to y1 we obtain y1= y2 and

y1 = −

2b1(2γ1c1+ c1) + b2₁(2γ1y2+ y2) + β1 −2c2₁+ 2γ1(γ1+ 1) + y2(2β1γ1+ β1)

b2₁+ β₁2

(2γ1+ 2β1y2+ 1)

. (3.18)

In order that the PWDS (3.3)-(3.5) be continuous they must coincide on x = 0, which means the following algebraic system must be satisfied.

−6a2b1β2γ1c1− 16a1b2β1c2γ2− 3a2b1β2c1+ 3a1b2β1c2− 3a2β1β2γ21 −3a2β1β2γ1+ 3a1β1β2γ2+ 3a2β1β2c21+ 4a1β1β2c22+ 3α2b2β1γ12 +3α2b2β1γ1− 3α1b1β2γ2− 3α2b2β1c21− 4α1b1β2c22+ 6α2b1b2γ1c1 +16α1b1b2c2γ2+ 3α2b1b2c1− 3α1b1b2c2 = 0, −6a2β2b2₁γ1− 16a1b2₂β1γ2− 3a2β2b2₁+ 3a1b2₂β1− 8a1b2β1β2c2− 6a2β₁2β2γ1 +3a1β1β₂2− 3a2β₁2β2+ 6α2b2β₁2γ1− 3α1β₂2b1+ 3α2b2β₁2+ 6α2b2b2₁γ1 +16α1b2₂b1γ2+ 3α2b2b2₁− 3α1b2₂b1+ 8α1b2β2b1c2= 0, a2b2₁β2β1+ 4a1b2₂β2β1+ a2β2β₁3− α2b2β₁3− α2b2₁b2β1− 4α1b1b2₂β2= 0, 3a2α1β2γ12+ 3a2α1β2γ1− 3a1α2β1γ2− 6a1α2b2γ1c1− 16a2α1b1c2γ2 −3a1α2b2c1+ 3a2α1b1c2− 3a2α1β2c21− 4a1α2β1c22+ 6a1a2β2γ1c1 +16a1a2β1c2γ2+ 3a1a2β2c1− 3a1a2β1c2− 3α1α2b2γ12− 3α1α2b2γ1 +3α1α2b1γ2+ 3α1α2b2c21+ 4α1α2b1c22= 0, 6a2α1β1β2γ1+ 3a2α1β1β2− 3a1α2β1β2− 16a2α1b1b2γ2− 6a1α2b1b2γ1 +3a2α1b1b2− 3a1α2b1b2+ 6a1a2b1β2γ1+ 16a1a2b2β1γ2− 3a1a2b2β1 +3a1a2b1β2− 6a2α1b1β2c1− 16a2α1b1β2c2− 6a1α2b2β1c1− 8a1α2b2β1c2 +6a1a2β1β2c1+ 16a1a2β1β2c2− 6α2α1b2β1γ1− 3α2α1b2β1+ 3α2α1b1β2 +6α2α1b1b2c1+ 8α2α1b1b2c2 = 0, 3a2α1β₁2β2− 3a2α1β2b2₁− 6a1α2b2β1b1− 16a2α1b2β2b1− 4a1α2b2₂β1 +6a1a2β1β2b1+ 16a1a2b2β1β2− 3α1α2b2β₁2+ 3α1α2b2b2₁+ 4α1α2b2₂b1 = 0, (3.19)

together with conditions (3.9).

Solving the sixth equation of (3.19) we get

a1 = a2α1β2 −3b21− 16b2b1+ 3β12

+ α1α2b2 3b21+ 4b2b1− 3β12

2α2b2(3b1+ 2b2) β1− 2a2(3b1+ 8b2) β1β2 , (3.20)

Polynomial differential systems : qualitative study and applications

D

M

Thèse

GHERMOUL Bilal

Doctorat en Sciences

Filière : MATHEMATIQUES

Spécialité : EDO

Thème

Polynomial differential systems: Qualitative study and applications

ةيبعشلا ةيطارقميدلا ةيرئازجلا ةيروهمجلا

ملعلا ثحبلا و يلاعلا ميلعتلا ةرازو

ي

فيطس ،سابع تاحرف ةعماج

ةيلك

مولعلا

Polynomial Differential Systems: Qualitative Study

And Applications

Bilal Ghermoul

Contents

This work is dedicated to my parents,

my family,

Acknowledgements

Introduction

1 Basic concepts and prerequisites

R

1.1 Existence and uniqueness of solutions

R

R

1.1.1 Different types of solutions

1.2 Flow

R

R

R

R

R

R

R

1.3 Stability of equilibrium points

R

1.4 Linear and nonlinear systems

"

#

1.4.1 Linear approximation at equilibrium points

"

#

1.4.2 The linearization Theorem

1.5 Limit cycle

Z Z





1.6 First integral, Invariant curve

R

R

R

R





1.7 Continuous PWDS

R

R

R

R

R

R







R

R

1.8 Isochronous center

1.8.1 Linear isochronous centers

1.8.2 Quadratic isochronous centers









1.9 Crossing periodic orbits and crossing limit cycles

2 A non-algebraic limit cycle for a class

of quintic differential systems

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R

_R