Strong isochronism of polynomial reversible dynamical systems on the plane with homogeneous nonlinearities of degree 4

We find the maximum order and initial polar angle of strongly isochronous two-dimensional polynomial reversible systems with homogeneous nonlinearities of the fourth degree.     

Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

We suggest a new approach to studying the isochronism of the system

${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y),$

where p _n and q _n are homogeneous polynomials of degree n. This approach is based on the normal form

${{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y)$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

     

Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities

In this paper we classify the centers, the cyclicity of its Hopf bifurcation and their isochronicity for the polynomial differential systems in R² of arbitrary degree d?3 odd that in complex notation z=x+iy can be written as

     

Limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method

By using the averaging method, we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane. More specifically, we analyze the global shape of limit cycles bifurcating from a Hopf bifurcation and also from periodic orbits with linear center , . The perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms.     

Non-isochronicity of the center at the origin in polynomial Hamiltonian systems with even degree nonlinearities

In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin.     

Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities

In this paper we study the asymptotic behavior of solutions of a first-order stochastic lattice dynamical system with a multiplicative noise.We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attractor.     

Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as where j is either 0 or 1. If j=0 then d?5 is an odd integer and n is an even integer satisfying 2?n?(d+1)/2. If j=1 then d?3 is an integer and n is an integer with converse parity with d and satisfying 0<n?[(d+1)/3] where [⋅] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations.     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations). Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 115–126, January, 1998.     

Asymptotics of solutions of infinite-dimensional homogeneous dynamical systems

In this paper we study the connection between the uniform asymptotic stability and the power-law or exponential asymptotics of the solutions of infinite-dimensional systems (differential equations in Banach spaces, functional differential equations, and completely solvable multidimensional differential equations).     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

Centers for the Kukles homogeneous systems with even degree

For the polynomial differential system $\dot{x}=-y$, $\dot{y}=x +Q_n(x,y)$, where $Q_n(x,y)$ is a homogeneous polynomial of degree $n$ there are the following two conjectures done in 1999. (1) Is it true that the previous system for $n \ge 2$ has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all $n$ even. More precisely, we prove both conjectures in the case $n = 4$ and for $n\ge 6$ even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of $n$ odd was studied in [8].     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities

This paper is devoted to study the planar polynomial system:

$\dot{x}=ax?y+P_n(x,y),\dot{y}=x+ay+Q_n(x,y),$

where $a\in \mathbb{R}$ and $P_n,Q_n$ are homogeneous polynomials of degree $n\ge 2$. Denote $\psi (\theta )=\cos ?(\theta )?Q_n(\cos ?(\theta ),\sin ?(\theta ))?\sin ?(\theta )?P_n(\cos ?(\theta ),\sin ?(\theta ))$. We prove that the system has at most 1 limit cycle surrounding the origin provided $(n?1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.     

Application of chaos degree to some dynamical systems

Chaos degree defined through two complexities in information dynamics is applied to some deterministic dynamical models. It is shown that this degree well describes the chaotic feature of the models.