Existence and Uniqueness of Limit Cycles in a Class of Planar Differential Systems

Abstract

This paper is an extension to the recent results presented by M. Sabatini about the existence and uniqueness of limit cycles of a certain class of planar differential systems in order to include other new classes. A concrete example exhibiting the applicability of the result is introduced.

1. Introduction

We study planar differential systems defined in the real plane, ̇,̇,𝑥=𝑝𝑥,𝑦𝑦=𝑄𝑥,𝑦 (1.1) where, 𝑃,𝑄∈𝐶¹(𝑅²,𝑅). These differential systems are mathematical models that arise in many fields of application like biology, physics,engineering, and so forth [1]. The study of the dynamics of (1.1) strongly depends on the existence stability properties, number, and location of special solution such as singular points and nonconstant isolated periodic solutions. In particular, if an attracting nonconstant isolated periodic solution exists, then it dominates the dynamics of the system (1.1) in an open connected subset of the plane, its region of attraction, such periodic solution called limit cycles. Studying the number and location of limit cycles is, by no means, a question of Hilbert 16th problem, see [2]. In some cases such a region of attraction can extend to cover the whole plane, with the exception of a singular point. In such a case the limit cycle is unique and dominates the system's dynamics. Uniqueness of limit cycles have been extensively studied in many books and articles, see, for example, [3–7] and references therein. Most of the results obtained for dynamical systems in the plane are concerned with the systems equivalent to the classical Liénard equation 𝑥̈+𝑓(𝑥)𝑥̇+𝑔(𝑥)=0, (1.2) and its generalization such as Lotka-Volterra systems and systems equivalent to Rayleigh equation as special cases, see [1]. This system with some extent of generalization involves Van der Pol system and other systems can be involved as a special case. A recent result on more general class of systems,̇̇(𝑥),𝑥=𝑦,𝑦=−𝑔−𝑦𝜙𝑥,𝑦 (1.3) which is equivalent to the equation𝑥̈+𝑥̇𝜙(𝑥,𝑥̇)+𝑔(𝑥)=0, (1.4) and special cases are concerned and studied, see, for instance, [3, 8].

The main tools used in this paper are the recent results introduced by Sabatini in [4, 5]. We make use of one-to-one transformation in order to transform system (1.1) to system equivalent to (1.3) in the meaning that the phase portrait of the original and the transformed system are equivalent. To that end, we impose some conditions on 𝑃(𝑥,𝑦),𝑄(𝑥,𝑦) in system (1.1) under which it becomes equivalent to system (1.3). Then, consequently, we study the uniqueness and then the existences of the limit cycles which attract every nonconstant solution under two cases, when 𝑔(𝑥) is assumed to be linear and when it is assumed to be nonlinear. We apply the uniqueness results of [5] for limit cycles of (1.3) to find out conditions of uniqueness and then under suitable additional assumptions we obtain conditions of uniqueness and existence of limit cycles that attract every nonconstant solution for systems (1.1).

For the convenience of the reader, we mention the statements of Sabatini's Theorems that we apply in our results and some used definitions.

Let Ω⊆𝑅² be a star-shaped set. We say that a function 𝜙∈𝐶¹(Ω,𝑅) is star shaped if (𝑥,𝑦)⋅∇𝜙=𝑥𝜙_𝑥+𝑦𝜙_𝑦 does not change sign. We say that 𝜙 is strictly star shaped if (𝑥,𝑦)⋅∇𝜙≠0, except at the origin (0,0). The following are the statements of the theorems of M. Sabatini [5] that we apply.

Theorem 1.1 (see, [5]). Let 𝜙∈𝐶¹(𝑅²,𝑅) be a strictly star-shaped function. Then (1.3) has at most one limit cycle.

Let us denote by 𝐷_𝑟 the disk {(𝑥,𝑦)∶𝑥²+𝑦²≤𝑟²} and by 𝜕𝐷_𝑟 its boundary {(𝑥,𝑦)∶𝑥²+𝑦²=𝑟²}.

Theorem 1.2 (see, [5]). Let 𝜙∈𝐶¹(𝑅²,𝑅) be a strictly star-shaped function with 𝜙(0,0)<0,𝜙(𝑥,𝑦)≥0 out of some bounded set 𝑈, and 𝜙(𝑥,𝑦) does not vanish identically on any 𝜕𝐷_𝑟 for 𝑟>sup{𝑥²+𝑦²,(𝑥,𝑦)∈𝑈}, then the system (1.3) has exactly one limit cycle, which attracts every nonconstant solution.

Theorem 1.3 (see, [5]). Assume 𝑔∈𝐶¹(𝑅,𝑅), with 𝑔^′(𝑥)>0 and 𝑥𝑔(𝑥)>0 for 𝑥≠0. Let Φ∈𝐶¹(𝑅²,𝑅), with Φ(0,0)<0, Φ(𝑥,𝑦)≥0 out of some bounded set 𝑈 containing (0,0), Φ(𝑥,𝑦) does not vanish identically on any circumference including 𝑈 and satisfy (𝜎(𝑥)2𝐺(𝑥)/𝑔(𝑥))[2𝐺(𝑥)((Φ_𝑥(𝑥,𝑦)𝑔(𝑥)−Φ(𝑥,𝑦)𝑔^′(𝑥))/𝑔²(𝑥))+Φ(𝑥,𝑦)]+𝑦Φ_𝑦(𝑥,𝑦)≠0. Then the system (1.3) has exactly one limit cycle, which attracts every nonconstant solution.

This paper is organized as follows. In Section 2, we give the main results with their proofs. The main tools applied in this article are the theorems presented by Sabatini in [5] which are mentioned above for completeness. In Section 3, we present a concrete example to illustrate the applicability of the results. As far as we know the approach which has been used in the example to study the limit cycles of the systems is new and not presented before.

2. The Main Results

We study system (1.1) under the assumption that the functions 𝑃(𝑥,𝑦) and 𝑄(𝑥,𝑦) satisfy the hypothesesH1:for any given x, z=P(x,y) is 1-1 correspondence between y and z;H2:there exists a function ϕ(x,y) such that for𝑘∈𝑅, 𝑘>0, :∇𝑃(𝑥,𝑦)⋅(𝑃(𝑥,𝑦),𝑄(𝑥,𝑦))+𝑃(𝑥,𝑦)𝜙(𝑥,𝑃(𝑥,𝑦))+𝑘𝑥=0,∀(𝑥,𝑦).

Hypothesis 𝐻1 means that 𝑃(𝑥,𝑦) is invertible in 𝑦 for every 𝑥, that is, there is a function 𝑦=𝑅(𝑥,𝑧) as the inverse of 𝑃(𝑥,𝑦), such that, for every 𝑥, 𝑃(𝑥,𝑅(𝑥,𝑧))=𝑧 and 𝑅(𝑥,𝑃(𝑥,𝑦))=𝑦.

Theorem 2.1. Suppose that P(x,y) in system (1.1) satisfies Hypothesis H1, if there exists a strictly star-shaped function 𝜙∈C¹(R²,R) such that Hypothesis H2 is satisfied, then system (1.1) has at most one limit cycle.

Proof. Consider the transformation 𝑧=𝑃(𝑥,𝑦). This transformation is, from Hypothesis 𝐻1, one-to-one correspondence between 𝑦 and𝑧, for any given 𝑥.
System (1.1) will be transformed to the systeṁ̇𝑃̇((.𝑥=𝑧,𝑧=𝑥,𝑅𝑥,𝑧)) (2.1) But 𝑃̇((=𝑃̇⋅𝑃𝜙()𝑥,𝑅𝑥,𝑧))𝑥,𝑦=∇𝑃𝑥,𝑦𝑥,𝑦,𝑄𝑥,𝑦=−𝑃𝑥,𝑦𝑥,𝑃𝑥,𝑦−𝑘𝑥fromHypothesis𝐻2.=−𝑧𝜙𝑥,𝑧−𝑘𝑥. (2.2) Then system (2.1) will be in the form ̇̇()𝑥=𝑧,𝑧=−𝑘𝑥−𝑧𝜙𝑥,𝑧,𝑘∈𝑅,𝑘>0. (2.3) Without loss of generality, possibly performing a time rescaling, we may restrict to the system (2.3), for 𝑘=1, ̇̇().𝑥=𝑧,𝑧=−𝑥−𝑧𝜙𝑥,𝑧 (2.4) Because of the invertibility of the transformation, the phase portrait of system (1.1) is equivalent to the phase portrait of system (2.4) and consequently they have the same number of limit cycles.
Applying Theorem  1 [5], we conclude that system (2.4), and consequently system (1.1), has at most one limit cycle.

Now we focus on the existence of limit cycles for system (1.1). Let 𝑈 be a bounded set, with 𝜎∶=sup{dist((𝑥,𝑦),(0,0)),(𝑥,𝑦)∈𝑈}.

Theorem 2.2. Suppose that P(x,y) in System (1.1) satisfies Hypothesis H1. If there exists 𝜙∈C¹(R²,R) a strictly star-shaped function satisfying Hypothesis H2 with 𝜙(0,0)<0,𝜙(x,y)≥0 out of some bounded set U containing (0,0), and 𝜙(x,y) does not vanish identically on any 𝜕D_r for r>𝜎, then the system (1.1) has exactly one limit cycle, which attracts every nonconstant solution.

Proof. Making use of the same transformation applied in the proof of Theorem 2.1 and applying Theorem  2 of [5], we conclude that the system (2.4) and, consequently, the system (1.1), has exactly one limit cycle, which attracts every nonconstant solution.

On the other hand, there exist classes of differential system models of more general type which are not covered by previous results, that one which is equivalent to (1.4) assuming 𝑥𝑔(𝑥)> 0 for 𝑥≠0. We investigate the uniqueness of limit cycles, and under suitable additional assumptions, its existence which attracts every nonconstant solution. The function 𝑔(𝑥) in (1.4) is considered now nonlinear, more general type, so what is given above is a special case of the following result.

Now, we consider System (1.3) which is equivalent to (1.4). Assume that 𝑥𝑔(𝑥)>0 for 𝑥≠0, 𝑔∈𝐶¹(𝑅,𝑅),𝑔^′(0)≠0.

Let us set 𝐺(𝑥)=^𝑥₀𝑔(𝑠)𝜎(𝑥)=(𝑥)(𝑥)𝑑𝑠,−1𝑥<00𝑥=01𝑥>0𝛼∶𝑅⟶𝑅suchthat𝛼=𝜎(𝑥),𝐸(𝑥)+𝑦2𝐺𝑥,𝑦=𝐺²2. (2.5)  

Theorem 2.3. Suppose that 𝑃(𝑥,𝑦) in System (1.1) satisfies Hypothesis 𝐻1, if there is a function 𝑔∈𝐶¹(𝑅,𝑅) with 𝑔^′(𝑥)>0,𝑥𝑔(𝑥)>0 for 𝑥≠0, and a function Φ∈𝐶¹(𝑅²,𝑅) satisfying the following conditions:
(1)Φ(0,0)<0, (2)Φ(𝑥,𝑦)⩾0 out of some bounded set 𝑈, containing (0,0), and Φ(𝑥,𝑦) does not vanish identically on {(𝑥,𝑦)∶2𝐸(𝑥,𝑦)=𝑟}𝑓𝑜𝑟𝑟>𝜎, where, 𝜎∶=sup{2𝐸(𝑥,𝑦),(𝑥,𝑦)∈𝑈}, (3)(𝜎(𝑥)2𝐺(𝑥)/𝑔(𝑥))[2𝐺(𝑥)((Φ_𝑥(𝑥,𝑦)𝑔(𝑥)−Φ(𝑥,𝑦)𝑔^′(𝑥))/𝑔²(𝑥))+Φ(𝑥,𝑦)]+𝑦Φ_𝑦(𝑥,𝑦)≠0, then the system (1.1) has exactly one limit cycle, which attracts every nonconstant solution.

Proof. The proof completes by applying Theorem  3 [5].

Hypothesis 𝐻1 implies that the function 𝑃(𝑥,𝑦) may be in the form𝑃𝑥,𝑦=𝑎₀(𝑥)+𝑦+^𝑁_𝑖=1𝑎_2𝑖+1(𝑥)𝑦^2𝑖+1, (2.6) where, 𝑎₀(0)=0, and all 𝑎_𝑘(𝑥) must be positive for all 𝑥.

3. Examples

The following is a concrete example illustrating the applicability of Theorem 2.2.

3.1. Example

Consider the system𝑥̇=𝑥²+𝑦+𝑥²𝑦³𝑦̇=−𝑥+𝑦+𝑥²−2𝑥𝑦−2𝑥³−𝑥²𝑦−𝑦³−𝑥⁴−𝑥³𝑦−4𝑥²𝑦²−𝑥⁵−4𝑥⁴𝑦+𝑥²𝑦³−2𝑥𝑦⁴−𝑥⁶−4𝑥³𝑦³−𝑥³𝑦⁴−3𝑥²𝑦⁵−𝑥⁴𝑦³−2𝑥⁵𝑦³−7𝑥⁴𝑦⁴−3𝑥⁶𝑦³−2𝑥³𝑦⁶−𝑥⁵𝑦⁶−3𝑥⁴𝑦⁷−3𝑥⁶𝑦⁶−𝑥⁶𝑦⁹/1+3𝑥²𝑦². (3.1) The function 𝑃(𝑥,𝑦) satisfies Hypothesis𝐻1. The second equation of the system can be rewritten in the form𝑃_𝑥𝑥,𝑦𝑥̇+𝑃_𝑦𝑥,𝑦𝑦̇=−𝑥−𝑃𝑥,𝑦−1+3𝑥²+2𝑥²𝑦+𝑦²+𝑥⁴+2𝑥²𝑦³+2𝑥²𝑦⁴+2𝑥⁴𝑦³+𝑥⁴𝑦⁶. (3.2) Choose 𝜙(𝑥,𝑦) such that 𝜙(𝑥,𝑃(𝑥,𝑦)) equals to the multiplier of 𝑃(𝑥,𝑦) in the right-hand side, that is,𝜙𝑥,𝑃𝑥,𝑦=−1+3𝑥²+2𝑥²𝑦+𝑦²+𝑥⁴+2𝑥²𝑦³+2𝑥²𝑦⁴+2𝑥⁴𝑦³+𝑥⁴𝑦⁶. (3.3)

In order to find an expression of the desired function 𝜙(𝑥,𝑦), we may assume a form of quadratic polynomial in 𝑥,𝑦 with undefined coefficients then impose it in the above equation, from which we identify the coefficients. Consequently, an expression of the function 𝜙 may be in the following form:𝜙𝑥,𝑦=−1+𝑥²+𝑥𝑦+𝑦². (3.4) Clearly, Hypothesis 𝐻2 is satisfied (with 𝑘=1). Moreover,𝑥,𝑦.∇𝜙𝑥,𝑦=2𝑥²+𝑥𝑦+𝑦²>0,∀𝑥,𝑦≠(0,0). (3.5) This means that 𝜙 is, strictly star-shaped function with 𝜙(0,0)  <  0,𝜙(𝑥,𝑦)≥0 out of some bounded set𝑈 containing (0,0), and 𝜙(𝑥,𝑦) does not vanish identically on any circumferences, hence the system has a unique limit cycle.

Acknowledgment

The author would like to extend his gratitude to the University of Sharjah for its support.