Configurations Of Limit Cycles Research Articles

Existence of at most two limit cycles for some non-autonomous differential equations

It is know that the non-autonomous differential equations $ {\rm d}x/{\rm d} t = a(t)+b(t)|x| $, where $ a(t) $ and $ b(t) $ are 1-periodic maps of class $ \mathcal{C}^1 $, have no upper bound for their number of limit cycles (isolated solutions satisfying $ x(0) = x(1) $). We prove that if either $ a(t) $ or $ b(t) $ does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different.

Configuration of Ten Limit Cycles in a Class of Cubic Switching Systems

The bifurcation of limit cycles is an important part in the study of switching systems. The investigation of limit cycles includes the number and configuration, which are related to Hilbert’s 16th problem. Most researchers studied the number of limit cycles, and only few works focused on the configuration of limit cycles. In this paper, we develop a general method to determine the configuration of limit cycles based on the Lyapunov constants. To show our method by an example, we study a class of cubic switching systems, which has three equilibria: (0,0) and (±1,0), and compute the Lyapunov constants based on Poincaré return map, then find at least 10 small-amplitude limit cycles that bifurcate around (1,0) or (−1,0). Using our method, we determine the location distribution of these ten limit cycles.

Realization of Arbitrary Configuration of Limit Cycles of Piecewise Linear Systems

In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of finite number of Jordan curves can be realized by a discontinuous piecewise linear system with two zones separated by a continuous curve.

The smallest upper bound on the number of zeros of Abelian integrals

In this paper, we develop a new method to estimate the smallest upper bound on the number of isolated zeros of Abelian integrals, and give an algebraic criterion for the case of Abelian integrals along energy level ovals of potential systems. As applications of our main result, we first obtain a criterion guaranteeing that two Abelian integrals have the Chebyshev property with accuracy one and give an example of hyperelliptic Abelian integrals. Then we get a criterion that any nontrivial linear combination of three Abelian integrals has at most two isolated zeros. Using this criterion we prove that the smallest upper bound is two for the number of isolated zeros of a non-algebraic Abelian integral along the oval (y2+12)1x2+ln⁡|x|−12=h with 0<h<+∞, which corresponds to the perturbation of a quadratic reversible system with two centers. Moreover, we derive all possible configurations of limit cycles from the Poincaré bifurcation of this perturbation. To our knowledge, these problems can not be solved by other known approaches.

Limit Cycles and Bifurcations in a Class of Discontinuous Piecewise Linear Systems

In this paper, we consider planar piecewise linear differential systems with a line of discontinuity sharing a linear part. We study not only the number of crossing limit cycles, but also the number of sliding ones, and the coexistence of two configurations of limit cycles. In particular, we proved that both numbers of crossing limit cycles and sliding ones are at most [Formula: see text], but the total number of limit cycles is at most [Formula: see text]. Finally, by complete analysis on the number of limit cycles, we show some bifurcations which exist in generic Filippov systems, revealing also two nongeneric bifurcations.

Limit Cycles Bifurcated from SomeZ4-Equivariant Quintic Near-Hamiltonian Systems

We study the number and distribution of limit cycles of some planarZ4-equivariant quintic near-Hamiltonian systems. By the theories of Hopf and heteroclinic bifurcation, it is proved that the perturbed system can have 24 limit cycles with some new distributions. The configurations of limit cycles obtained in this paper are new.

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.

Quadratic Perturbations of a Quadratic Reversible Lotka–Volterra System

We prove that perturbing the two periodic annuli of the quadratic polynomial reversible Lotka–Volterra differential system ẋ = −y + x 2 − y 2, ẏ = x(1 + 2y), inside the class of all quadratic polynomial differential systems we can obtain the following configurations of limit cycles (0, 0), (1, 0), (2, 0), (1, 1) and (1, 2).

Quadratic perturbations of a class of quadratic reversible systems with two centers

Quadratic perturbations of a one-parameter family of quadratic reversible systems with two centers (without other singularities in finite plane) are studied. The exact upper bound of the number of limit cycles, the configurations of limit cycles, and the bifurcation diagrams for different range of the parameter are given.

BIFURCATION OF LIMIT CYCLES IN Z10-EQUIVARIANT VECTOR FIELDS OF DEGREE 9

In this paper, we consider the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. By applying the bifurcation theory of planar dynamic systems and the method of detection functions, with the aid of numerical simulations, we show that a polynomial vector field of degree 9 with Z10 symmetry can have at least 80 limit cycles, i.e. H(9) ≥ 92 - 1.

A Numerical Approach to Bifurcations from Quadratic Centers

A combination of analytical and numerical work is done to analyze bifurcation of limit cycles from non-Hamiltonian codimension-three quadratic centers. The winding curve C3of cyclicity-three separatrix cycles, qualitatively located in earlier work, is determined numerically. Evidence is given that the (2,2), (3,2), and (3,3) configurations of limit cycles do not bifurcate from this class of quadratic centers.

Weak Focus, Limit Cycles, and Bifurcations for Bounded Quadratic Systems

We study the bounded quadratic systems with either two weak foci or a weak focus of order 2. From the first case we obtain (1, 1)-configuration of limit cycles, and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2. Also, we unfold the bounded quadratic system with a center, and the maximum number of limit cycles in this case is two.

Small Amplitude Limit Cycles for Cubic Systems

AbstractIn this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systemsWith a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.