Coexistence of three limit cycles for a septic polynomial differential systems

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.

One of the main open problems in the qualitative theory of real planar differential systems is the study of limit cycles. In this article, we present an algorithmic approach for detecting how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems via the averaging method. We propose four symbolic algorithms to implement the averaging method. The first algorithm is based on the change of polar coordinates that allows one to transform a considered differential system to the normal form of averaging. The second algorithm is used to derive the solutions of certain differential systems associated to the unperturbed term of the normal of averaging. The third algorithm exploits the partial Bell polynomials and allows one to compute the integral formula of the averaged functions at any order. The last algorithm is based on the aforementioned algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The implementation of our algorithms is discussed and evaluated using several examples. The experimental results have extended the existing relevant results for certain classes of differential systems.

In these last decades, the interest on the discontinuous piecewise differential systems has increased strongly, mainly due to their big number of applications. In their study, the existence or not of limit cycles play a main role. In this paper, we study a class of planar discontinuous piecewise differential systems composed of three equal sectors of angle 2π/3, where each sector is governed by a distinct linear Hamiltonian system. The discontinuity set consists of three rays, which are the boundaries of these three sectors. We prove that such differential systems can exhibit at most three crossing limit cycles. Furthermore, we construct an explicit example illustrating the existence of a discontinuous piecewise differential system that attains this upper bound. So, we have solved the extended 16th Hilbert problem for these classes of discontinuous piecewise differential systems.

In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles play a main role. Up to now, the major part of papers which study the limit cycles of the piecewise linear differential systems consider only two pieces. Here, we consider piecewise linear differential systems with three pieces. In this paper, we study the limit cycles of the discontinuous piecewise linear differential systems in the plane R2 formed by three arbitrary linear centers separated by the set Σ={(x,y)∈R2:y=0{or}x=0{and}y≥0}. We prove that such discontinuous piecewise linear differential systems can have 1, 2, or 3 limit cycles, with 3 the maximum number of limit cycles that such systems can have. Moreover, the limit cycles are nested and must intersect Σ in three or four points. The limit cycles having three intersection points with Σ can reach the maximum number 3. The limit cycles having four intersection points with Σ are at most 1, and if it exists, the systems could simultaneously have 1 or 2 limit cycles intersecting Σ in three points.

In this paper we deal with planar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles α and 2π−α, respectively, for α∈(0,π). We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of α, we prove the existence of systems with four limit cycles up to fifth order and, for α=π/2, we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line.

Quadratic Dynamical Systems and Algebras

Using Einstein’s notation [4]), the polynomial differential systems of finite dimension n and of degree at most k with coefficients in a field k of characteristic zero can be written as dx dt = a + ajα1x α1 + ajα1α2x 1x2 + ajα1···αrx α1 · · · xr (1) where j, α1, αr ∈ {1, . . . , n}, 1 ≤ r ≤ k, and for j = 1, . . . , n and for 2 ≤ r ≤ k, the tensor ajα1···αr (1 time contravariant and r times covariant) is symmetric with respect to the lower subscripts and we consider the action of the general linear group GL(n, k) on the phase space k. In this work, starting from a minimal system of generators of the ideal of algebraic invariants of polynomial differential systems (1) we will develop an algorithmic method to study the local stability of a given cubic differential system with the help of Grobner bases and using the GL(n, k)-invariants of this cubic system [2]. Our algorithmic method is motivated by the important role of the invariant theory in the survey of polynomial differential systems and the numerous works on Groebner basis one of the main practical tools for solving algebraic systems and computing algebraic varieties. Keywords Polynomial differential systems,linear transformations, cubic differential systems,singular points, invariant, covariant, Grobner bases.

Limit cycles in planar continuous piecewise linear systems

First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most one limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with one limit cycles.

In this paper, we deduce the normal forms of polynomial differential systems in ℝ3 having, Fermat cubic surface, Clebsch surface, Goursat surface, Racor surface and 2-dimensional tours as invariant objects. We prove if we have an odd number of polynomial differential equations with the determined inverse Jacobian multipliers then we can make a new polynomial differential system of the first systems so that it’s inverse Jacobian multiplier can be determined by the inverse Jacobian multipliers of the first systems. If new system which is made by them has a limit cycle then we determine it’s position.

We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica.

First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise differential systems we have solved the extension of the 16th Hilbert problem.

A general family of planar piecewise linear ODEs with two zones both having a real focus and separated by a straight line is considered. By analyzing the number of zero points of a new function related to the intersection points of the trajectories of the linear subsystems with the separation line, complete results on the existence and number of limit cycles are obtained. In particular, complete parameter regions for the existence of 1–2 limit cycles are provided with two concrete examples, which will be helpful in studying some kinds of discontinuity-induced bifurcations (i.e., DIBs). Based on the main results, it is obtained that the family of planar piecewise linear ODEs with focus–focus dynamics separated by a straight line can have 3 limit cycles if and only if one subsystem has a real focus and the other one has a virtual focus. Moreover, it is showed that five is the least value of the number of parameters needed in canonical forms of such systems.

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous n -dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold \mathcal{Z}\subset\mathbb{R}^n of periodic solutions satisfying \dim(\mathcal{Z}) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x'=Mx , when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system: x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2 F_2^n(x), in \mathbb{R}^{d+2} , where \varepsilon is a small parameter, M is a (d+2)\times(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d-m non-zero real eigenvalues.

We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.