Isochronous Center Research Articles

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

In this paper, we study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions [Formula: see text] such that the isochronous center lies on the level curve [Formula: see text]. We prove that, in the one-dimensional homology group of the Riemann surface (removing the points at infinity) of level curve [Formula: see text], the vanishing cycle of an isochronous center cannot belong to a subgroup generated by those small loops such that each of them is centered at a removed point at infinity of having one of the two special types described in the paper, where [Formula: see text] is sufficiently close to [Formula: see text]. Besides, we present some topological properties of isochronous centers for a large class of Hamiltonian systems of degree [Formula: see text], whose homogeneous parts of degree [Formula: see text] contain factors with multiplicity of no more than [Formula: see text]. As applications, we study the nonisochronicity for some Hamiltonian systems with quite complicated forms which are usually very hard to handle by the classical tools.

Isochronous cosmological solutions of the Friedmann–Robertson–Walker model

In this paper, the periodic solutions of the equation of Friedmann–Robertson–Walker cosmology with a cosmological constant are investigated. Using variable transformation, the original second-order ordinary differential equation is converted to a planar dynamical system with cosmic time t. Numerical simulations indicate that period function T(h) of this dynamical system is monotonically increasing. However, a new planar dynamical system could be deduced by using conformal time variable [Formula: see text]. We prove that the new planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Explicitly, we find that there exist two families of periodic solutions with equal period for the new planar dynamical system which is derived from the Friedmann–Robertson–Walker model.

Periodic oscillators, isochronous centers and resonance

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators.

On the critical points of the flight return time function of perturbed closed orbits

We deal here with planar analytic systems x˙=X(x,ε) which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by TΣ(q,ε) the time needed by a perturbed orbit that starts from q∈Σ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of x˙=X(x,0) is a continuable critical orbit in a family of the form x˙=X(x,ε) if, for any q∈Γ and any Σ that passes through q, there exists qε∈Σ a critical point of TΣ(⋅,ε) such that qε→q as ε→0. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of x˙=X(x,0) is a continuable critical orbit in any family of the form x˙=X(x,ε). We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x˙=X(x,0).

LIMIT CYCLES FOR TWO CLASSES OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH UNIFORM ISOCHRONOUS CENTERS

In this article, we study the maximum number of limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. Using the first-order averaging method, we analyze how many limit cycles can bifurcate from the period solutions surrounding the centers of the considered systems when they are perturbed inside the class of homogeneous polynomial differential systems of the same degree. We show that the maximum number of limit cycles, <i>m</i> and <i>m</i>+1, that can bifurcate from the period solutions surrounding the centers for the two classes of differential systems of degree 2<i>m</i> and degree 2<i>m</i>+1, respectively. Both of the bounds can be reached for all <i>m</i>.

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

This paper concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quintic homogeneous perturbations, at most 14 limit cycles birfucate from the period annulus of the considered system.

Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture has already been proved for quadratic and quartic systems. Using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields, we are able to describe a very large class of nonisochronous Hamiltonian systems of even arbitrarily large degree.

Isochronicity of a [formula omitted]-equivariant quintic system

The main purpose of this paper is to study the problem of simultaneous existence of two isochronous centers in some families of planar polynomial differential systems with symmetry. More precisely, we present conditions for a family of Z2-equivariant systems of degree 5 to have an isochronous bi-center. We also study their global phase portraits in the Poincaré disk and present considerations about the number of simultaneous centers in Z2-equivariant systems of degree 3 and 5. This study provides examples of quintic systems possessing three and five isochronous centers simultaneously.

Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems

In this paper, we consider the quadratic isochronous centers perturbed inside piecewise polynomial differential systems of arbitrary degree n with the straight line of discontinuity x=0. The main concerns are the number of zeros of the first order Melnikov functions and the estimate of the number of limit cycles bifurcating from the period annuli. For quadratic isochronous centers S1, S2 and S3, we will provide a sharp upper bound for the number of zeros of the first order Melnikov functions. For quadratic isochronous center S4, we give a rough estimate. However, when the problem is reduced to perturbations inside polynomial differential systems, our result for S4 will improve that in Li et al. (2000) [12] significantly. Moreover, we will reveal out some equivalence between the first order Melnikov function method and the first order averaging method for investigating the number of limit cycles of piecewise polynomial differential systems.

Centers and Uniform Isochronous Centers of Planar Polynomial Differential Systems

For planar polynomial vector fields of the form $$\begin{aligned} (-y+X(x,y))\dfrac{\partial }{\partial x}+(x+Y(x,y))\dfrac{\partial }{\partial y}, \end{aligned}$$ where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.

An inverse approach to the center problem

We consider analytic or polynomial vector fields of the form $${\mathcal {X}}=\left( -y+X\right) \dfrac{\partial }{\partial x}+\left( x+Y\right) \dfrac{\partial }{\partial y},$$ where $$X=X(x,y))$$ and $$Y=Y(x,y))$$ start at least with terms of second order. It is well-known that $${\mathcal {X}}$$ has a center at the origin if and only if $${\mathcal {X}}$$ has a Liapunov–Poincare local analytic first integral of the form $$H=\dfrac{1}{2}(x^2+y^2)+\sum _{j=3}^ {\infty } H_j$$ , where $$H_j=H_j(x,y)$$ is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincare consists in distinguishing when the origin of $${\mathcal {X}}$$ is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which H is a first integral. Moreover, given an analytic function $$V=1+\sum _{j=1}^ {\infty } V_j$$ in a neighborhood of the origin, where $$V_j$$ is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form $$ H=\dfrac{1}{2}(x^2+y^2)\,\left( 1+ \sum _{j=1}^{\infty } \Upsilon _j\right) , $$ in a neighborhood of the origin, where $$\Upsilon _j$$ is a homogenous polynomial of degree j for $$j\ge 1.$$ These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin We extended to analytic or polynomial differential systems the weak conditions of a center given by Alwash and Lloyd for linear centers with homogeneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers.

Centers and isochronous centers of a class of quasi-analytic switching systems

Abstract In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasi-analytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than those of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit conditions are obtained for classifying the centers and isochronous centers of the system.

Asymptotic behaviors of solutions to a reaction–diffusion equation with isochronous nonlinearity

We study the initial boundary value problem for the reaction–diffusion equation with isochronous nonlinearity. We prove that small solutions become spatially homogeneous and is subject to the ODE part asymptotically. We also discuss blow-up of an parabolic system with quadratic nonlinearity having the origin as an uniform isochronous center.

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): x˙=y−12x2+16y2, y˙=−x−16xy, and (r20): x˙=y+4x2, y˙=−x+16xy, and the periodic orbits of the quadratic isochronous centers (S1):x˙=−y+x2−y2, y˙=x+2xy, and (S2):x˙=−y+x2, y˙=x+xy. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system (S1) and (S2) are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line y=0. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively 4n−3(n≥4) and 4n+3(n≥3) counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers (S1) and (S2) are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.

Periodic solutions with equal period for the Friedmann–Robertson–Walker model

In this letter, we study the periodic solutions of the equation of barotropic Friedmann–Robertson–Walker cosmologies. Using variable transformation, the original second order ordinary differential equation is converted to a planar dynamical system. We prove that the planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Consequently, we find that there exist two families of periodic solutions with equal period for the Friedmann–Robertson–Walker model.

Linearizability and critical period bifurcations of a generalized Riccati system

In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.

Inverse Jacobi multiplier as a link between conservative systems and Poisson structures

Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the flow-box theorem we restrict ourselves to neighborhoods of singularities. In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincaré center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka–Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided.

Perturbed rigidly isochronous centers and their critical periods

This paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n. It proves that for n=3,4 and 5, there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively, and in each case this upper bound is sharp. Moreover, for any n>5, there are at most [n−12] critical periods induced by periodic orbits of the unperturbed cubic system. An example is given to show that the upper bound in the case of n=11 can be reached.

Study of the period function of a two-parameter family of centers

In this paper we study the period function of x¨=(1+x)p−(1+x)q, with p,q∈R and p>q. We prove three independent results. The first one establishes some regions in the parameter space where the corresponding center has a monotonous period function. This result extends the previous ones by Miyamoto and Yagasaki for the case q=1. The second one deals with the bifurcation of critical periodic orbits from the center. The third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one-parameter deformation. These three results, together with the ones that we obtained previously on the issue, lead us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family.

Bi-center problem for some classes of [formula omitted]-equivariant systems

We investigate the simultaneous existence of two centers and their isochronicity for two families of planar Z2-equivariant differential systems. First, for a family studied by Liu and Li (2011) we present the necessary and sufficient conditions for the existence of an isochronous bi-center. Next, we give conditions for the existence of a bi-center and study its isochronicity for a planar Z2-equivariant quintic system having two weak foci or centers. We also give two examples of non-Hamiltonian cubic systems with three isochronous centers.