Planar Differential Systems Research Articles

Limit cycles and chaos in planar hybrid systems

In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.

Global dynamics of a special class of planar sector-wise linear systems

In this article, we study the global dynamics of a special class of planar sector-wise linear differential systems with two subsystems being the same except for their equilibriums. Taking the position of one equilibrium as the bifurcation parameter, we provide a complete analysis about sliding cycle bifurcation and sliding homoclinic bifurcation. Moreover, we obtain the existence of all important separatrix orbits and their dependence on the bifurcation parameter, including sliding heteroclinic orbit, heteroclinic cycle, limit cycle, sliding homoclinic cycle and sliding cycle. For more information see https://ejde.math.txstate.edu/Volumes/2024/57/abstr.html

On the crossing limit cycles created by a discontinuous piecewise differential system formed by three linear Hamiltonian saddles

We study planar discontinuous piecewise differential systems formed by three linear Hamiltonian saddles separated by the non-regular line Σ = { ( x , y ) ∈ R 2 : ( y = 0 ) ∨ ( x = 0 ∧ y ≥ 0 ) } . We prove that when the linear Hamiltonian saddles are homogeneous they have no limit cycles, and when they are non homogeneous they can have at most three limit cycles having exactly one point on each branch of Σ, and at most one limit cycle having four intersection points on Σ. Moreover we show that they can have at most one limit cycle having four intersection points on Σ and three limit cycles having three intersection points on Σ, simultaneously. Thus we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems. Furthermore we show that for the three types of combinations of the limit cycles here studied in two of them the upper bound is sharp by providing examples of these systems with the maximum number of possible limit cycles. For the remaining type of limit cycles the upper bound is four and we have examples with three limit cycles. So at this moment it is an open problem to know if the sharp upper bound is either four or three.

A family of generalized Liénard differential systems with an explicit limit cycle

One of the important problems in the qualitative theory of real planar differential systems is limit cycles i.e., isolated periodic solutions in the set of all periodic this concept was defined at the end of the 19th century by Poincaré, and with the works of van der Pol and Liénard. The significance of these isolated solutions lies in their key role in understanding the dynamics of a certain differential system. Afterward, Hilbert specified a list of 23 problems for the advancement of mathematical science, and from then it started intensive research on these problems throughout the 20th century. Of the 23 problems, only the so-called 16th Hilbert’s problem and the Riemann conjecture remain open until now. The second part of the 16th Hilbert problem has two parts and asks for an upper bound on the number of possible limit cycles and their positions for a planar polynomial differential system of a given degree. The theory of the limit cycle operating as an indispensable mathematical tool has found broad and important applications in modern physics, chemical, biological, ,and other fields. Then the progress of these fields, in turn, promotes limit cycle research. In this paper, we prove the existence and uniqueness of the limit cycle for a family of generalized polynomial Liénard differential systems, the explicit expression of this limit cycle is given. An example illustrating our results is additionally presented.

At Most Five Limit Cycles in a Class of Discontinuous Piecewise Linear Systems with Three Zones

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems.

Non-isochronicity on piecewise Hamiltonian differential systems with homogeneous nonlinearities

In this paper, we prove the non-isochronicity of Σ-centres for a class of planar piecewise smooth differential systems with a straight switching line, whose two sub-systems are Hamiltonian differential systems with a non-degenerated centre and only homogeneous nonlinearities. This result generalizes a classical result of non-isochronicity on the smooth Hamiltonian systems with homogeneous nonlinearities.

Global Phase Portraits of Piecewise Quadratic Differential Systems with a Pseudo-Center

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

Saddle–node canard cycles in slow–fast planar piecewise linear differential systems

By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The obtained results are comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Dumortier and Roussarie in Mem. Am. Math. Soc. 1996, and Krupa and Szmolyan in J. Differ. Equ. 2001. In addition, some novel slow–fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle–node bifurcation of canard limit cycles, one involving headless canard cycles, and the other involving canard cycles with head. This configuration also occurs in smooth systems with N-shaped fast nullcline. However, it has not been previously reported in the Van der Pol system. Our results provide justification for this observation.

The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems

The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles.

Phase Portraits of a Class of Continuous Piecewise Linear Differential Systems

AbstractThe phase portraits of the planar linear differential systems are very well known. This is not the case for the phase portraits of the planar continuous piecewise linear differential systems. In this paper we classify the phase portraits of the class of planar continuous piecewise linear differential systems of the form $$\begin{aligned} {\dot{x}}= a|x|+by+c,\qquad {\dot{y}}= \alpha |x|+\beta y+\gamma , \end{aligned}$$ x ˙ = a | x | + b y + c , y ˙ = α | x | + β y + γ , in the Poincaré disc when $$a\beta -b\alpha \ne 0$$ a β - b α ≠ 0 , and prove the existence and uniqueness of limit cycles. Note that on the straight line $$x=0$$ x = 0 these differential systems are only continuous.

Weak centers and local criticality on planar [formula omitted]-symmetric cubic differential systems

There exist the necessary and sufficient conditions for planar Z2-symmetric cubic differential systems with a bi-center and an isochronous bi-center in the existing literature. Based on these bi-center conditions, we give a complete discussion on the order of weak centers and the local criticality for planar Z2-symmetric cubic systems at the origin and the symmetric equilibria (±1,0) separately. More precisely, we first give the parametric conditions for the origin and the equilibria (±1,0) being weak centers of exact order separately. Then we prove that the local criticality from the weak centers of finite order at the origin is at most 4, and at each of symmetric equilibria (±1,0) is at most 3. Finally, we also obtain that the local criticality from the isochronous center at the origin is at most 3, and at each of equilibria (±1,0) is at least 2. More importantly, these numbers are reachable.

Dynamics in sliding set of planar sector-wise linear systems

For piecewise smooth dynamical systems the existence and properties of some special kinds of sliding points are very important to determine the dynamics (whether local or global), and are also very essential in studying DIBs (i.e. discontinuity induced bifurcations) related to sliding motions. In this paper, we mainly study the inner dynamics of the sliding set of a general planar sector-wise linear differential systems, including the definitions, existence and stability of all special sliding points. Specially, by studying the case when the two zones are separated by straight lines, we obtain explicit dependence on system parameters of the existence, stability and number of all kinds of special sliding points for the planar sector-wise linear systems. Moreover, we provide concrete examples to illustrate our main results and their application in studying DIBs.

Criteria for the nonexistence of periodic orbits in planar differential systems

In this work we summarize some well-known criteria for the nonexistence of periodic orbits in planar differential systems. Additionally we present two new criteria and illustrate with examples these criteria.

A Succinct Characterization of Period Annuli in Planar Piecewise Linear Differential Systems with a Straight Line of Nonsmoothness

We close the problem of the existence of crossing period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness. In fact, a characterization for the existence of such objects is provided by means of a few basic operations on the parameters.

Probability of existence of limit cycles for a family of planar systems

The goal of this work is the study of the probability of occurrence of limit cycles for a family of planar differential systems that are a natural extension of linear ones. To prove our results we first develop several results of non-existence, existence, uniqueness and non-uniqueness of limit cycles for this family. They are obtained by studying some Abelian integrals, via degenerate Andronov-Hopf bifurcations or by using the Bendixson-Dulac criterion. To the best of our knowledge, this is the first time that the probability of existence of limit cycles for a non-trivial family of planar systems is obtained analytically. In particular, we give vector fields for which the probability of having limit cycles is positive, but as small as desired.

Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region

In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincaré half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.

Limit cycles and bifurcations in a class of planar piecewise linear systems with a nonregular separation line

In this paper, we consider a class of planar piecewise linear differential systems with a nonregular separation line, which can be transformed to a normal form with only 5 parameters under some conditions. We study the coexistence of crossing limit cycles and sliding limit cycles by establishing Poincaré maps for two subsystems that have the same Jacobi matrix. Using expressions and properties of Poincaré maps, we can clarify the relationship between trajectories of two subsystems. Our main results reveal that (1) the numbers of crossing limit cycles and sliding limit cycles can be 2; (2) the coexistence number of limit cycles can be 3. Moreover, by a complete discussion of classification and analysis, we show saddle-node bifurcation and critical crossing cycle bifurcation existing in generic Filippov systems. Finally, four numerical examples are given to verify the correctness of the obtained theoretical results.

Periodic solutions to superlinear indefinite planar systems: A topological degree approach

We deal with a planar differential system of the form{u′=h(t,v),v′=−λa(t)g(u), where h is T-periodic in the first variable and strictly increasing in the second variable, λ>0, a is a sign-changing T-periodic weight function and g is superlinear. Based on the coincidence degree theory, in dependence of λ, we prove the existence of T-periodic solutions (u,v) such that u(t)>0 for all t∈R. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or ϕ-Laplacian-type differential operators).

Limit cycles of continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center

<p style='text-indent:20px;'>Due to their applications to many physical phenomena during these last decades the interest for studying the continuous or discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system. Up to now the major part of papers which study the limit cycles of the planar piecewise differential systems have considered systems formed by two pieces separated by one straight line.</p><p style='text-indent:20px;'>Here we consider planar continuous piecewise differential systems separated by a parabola.</p><p style='text-indent:20px;'>We prove that the planar continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center have at most one limit cycle. Moreover there are systems in this class exhibiting one limit cycle. So in particular we have solved the extension of the 16th Hilbert problem to this class of differential systems.</p>

Cubic planar differential systems with non-algebraic limit cycles enclosing a focus

Cubic planar differential systems with non-algebraic limit cycles enclosing a focus