Isochronous Center Research Articles

Hopf Cyclicity and Center Conditions for Some Piecewise Smooth Cubic Differential Systems

In this paper, we consider piecewise smooth differential systems with subsystems having cubic reversible isochronous centers at the origin, where the separation line is a straight line passing through the origin. By computing Lyapunov constants, we obtain Hopf cyclicity at the origin with necessary and sufficient conditions for the origin to be a center in this type of piecewise smooth differential systems.

The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line

The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line

On the number of limit cycles in piecewise planar quadratic differential systems

We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si’s in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.

Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

Global phase portraits of quintic reversible uniform isochronous centers

This paper studies the global phase portraits of uniform isochronous quintic centers at the origin with time reversibility such that their nonlinear part is not homogeneous. By using Poincaré compactification, we obtain all possible phase portraits of this quintic polynomial differential system.

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.

Limit Cycles of Some Families of Discontinuous Piecewise Differential Systems Separated by a Straight Line

In the past years the study of continuous or discontinuous piecewise differential systems has attracted significant interest, due to their wide use to model many natural phenomena. Important questions such as finding an upper bound for the number of limit cycles of such systems and their possible configurations have been considered by many authors. These problems are known as the extension of the second part of the 16th Hilbert’s problem to the piecewise differential systems. In this paper, we solve an extension of the second part of the 16th Hilbert’s problem for two families of discontinuous piecewise differential systems separated by the straight line [Formula: see text]. The first family is formed by a linear center and a cubic Hamiltonian isochronous center, and the second family is formed by cubic Hamiltonian isochronous centers. As a result we prove that the first family can exhibit [Formula: see text] or [Formula: see text] limit cycle, for the second one there are either three limit cycles or no limit cycles. We also show that there are examples of all types of these systems with one or three limit cycles.

On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations

Abstract In this paper, we study the number of limit cycles bifurcated from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations. Using the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, we show that the discontinuous systems have at least 6 more limit cycles than the continuous ones. This study needs some computations that have been verified using Maple.

Piecewise Smooth Perturbations to a Class of Planar Cubic Centers

This paper studies the limit cycle bifurcations of a class of planar cubic isochronous centers. For different values of two key parameters, we give an estimate of the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems under arbitrarily small piecewise smooth polynomial perturbation. The main method and technique are based on the first order averaging theory for discontinuous systems and the Argument Principle in complex analysis.

The isochronous center for Kukles homogeneous systems of degree eight

In this paper, we investigate Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Giné et al. (2015,2017). However, the proofs may have some gaps, hence they are still open. The gaps of Giné et al. (2015,2017) have been shown in Guo et al. (2021). In this paper, we consider isochronous centers in the family of Kukles homogeneous systems symmetric about the y-axis of degree eight. By using Period Abel constants and Gröbner basis of polynomial systems, we obtain that there is only one isochronous center excepting the linear center.

Weak Centers and Local Bifurcation of Critical Periods in a Z2-Equivariant Vector Field of Degree 5

With the help of algebraic manipulator-Mathematica, we identify the order of weak centers at [Formula: see text] and the origin as well as the number of local critical periods in a [Formula: see text]-equivariant vector field of degree 5. We show that [Formula: see text] and the origin can be weak centers of infinite order (i.e. isochronous center) and at most fourth-order weak centers of finite order. Furthermore, we prove that at most four local critical periods bifurcate from the bicenter and the origin, respectively. Our approach is a combination of computational algebraic techniques.

The quantitative analysis of homoclinic orbits from quadratic isochronous systems

In this paper, we investigate a pair of symmetric homoclinic orbits given by a Perturbation-Incremental (PI) method for the quadratic isochronous systems, which can bifurcate from the periodic orbits of the quadratic isochronous centers. The existing researches on the isochronous system are mostly from a qualitative perspective, while we have studied the quadratic isochronous system through the quantitative analysis method. As a quantitative analysis method, the PI method can not only determine the position of the limit cycle and the homoclinic (heteroclinic) orbit, but also provide an analytical expression of the asymptotic solution. At the same time, this method combines the significant characteristics of the perturbation method and the parameter incremental method. By utilizing the perturbation method, we obtain the analytic expressions of the zero-order perturbation solutions of homoclinic orbits when the parameters of systems are small. Then, the approximate analytical expressions of homoclinic orbits can be obtained by using the parameter incremental method when they are iterated from small parameters to large parameters. The numerical results will be compared with those from the Runge–Kutta method. The theoretical results agree very well with numerical simulation.

Bifurcation of Limit Cycles and Isochronous Centers on Center Manifolds for a Class of Cubic Kolmogorov Systems in $${\mathbb {R}}^{3}$$

Bifurcation of Limit Cycles and Isochronous Centers on Center Manifolds for a Class of Cubic Kolmogorov Systems in $${\mathbb {R}}^{3}$$

On the analytic commutator for Λ−Ω differential systems

In this paper, we give the necessary and sufficient conditions for some Ω differential systems to have an analytic commutator, use these properties to judge the origin point of the Ω differential systems to be an isochronous center.

Third-order bifurcation of limit cycles for a perturbed quartic isochronous centre

Third-order bifurcation of limit cycles for a perturbed quartic isochronous centre

Third-Order Bifurcation of Limit Cycles for a Perturbed Quartic Isochronous Center

Third-Order Bifurcation of Limit Cycles for a Perturbed Quartic Isochronous Center

UNIFORM ISOCHRONOUS CENTER OF HIGHER-DEGREE POLYNOMIAL DIFFERENTIAL SYSTEMS

In this paper, we study the uniform isochronous center of a class of more general higher-degree of polynomial differential systems and give the necessary and sufficient conditions for the origin point to be a center. At the same time, we illustrate that under some restrictions, the composition conjecture about these differential systems is valid. As corollaries, the previous results can easily be derived from the current conclusion.

Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centres

These last years an increasing interest appeared in studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. The understanding of the dynamics of these systems has many difficulties. One of them is the study of their limit cycles. In this paper, we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line and formed by combinations of linear centres (consequently isochronous) and cubic isochronous centres with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles.

On Hilbert’s 16th Problem for Some Discontinuous Piecewise Differential Systems

In this paper, we study the number of limit cycles for discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are quadratic isochronous centers in a half-plane and the cubic isochronous centers in the other one. We deal with eight classes of discontinuous piecewise differential systems formed by these types of isochronous centers, and we provide an upper bound for the maximum number of limit cycles for these classes that can exhibit. Also, we show that the corresponding maximum number of limit cycles in four classes is reached.

Non-algebraic crossing limit cycle for discontinuous piecewise differential systems formed by a linear system without equilibrium points and quadratic isochronous centers at the origin

Non-algebraic crossing limit cycle for discontinuous piecewise differential systems formed by a linear system without equilibrium points and quadratic isochronous centers at the origin