Related papers: Limit Cycle Analysis of 3-D Nonlinear systems
The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after…
We analyze the critical behavior of isotropic systems with dipole-dipole interaction by renormalization-group methods in fixed space-time dimensions. Working in three-dimensional theory we analytically find three-loop expressions for…
In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic…
We have probed the condition of periodic oscillation in a class of two variable nonlinear dynamical open systems modeled with Lienard-Levinson-Smith(LLS) equation which can be a limit cycle, center or a very slowly decaying center type…
We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of…
In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of…
We study the Poincare-Bendixson theorem for two-dimensional continuous dynamical systems in compact domains from the point of view of computation, seeking algorithms for finding the limit cycle promised by this classical result. We start by…
This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. Let $N=N(p,q,m,n)$ be the maximum number of limit…
In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach…
Limit cycles (attractors for neighbouring periodic orbits in a dissipative dynamical system) have been widely studied but the corresponding generalization for quasi periodic orbits have rarely been discussed. Here we investigate "higher…
We consider three examples of weekly perturbed centers which do not have {\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging…
We show that the number of limit cycles of Lienard vector field L is equal to the index of first order linear PDE defined by L.
The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential…
In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that…
We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the…
Let $x'=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the…
We analyze the complex dynamics dynamics of a family of $\mathbb{Z}_{12}-$equivariant planar systems, by using their reduction to an Abel equation. We derive conditions in the parameter space that allow uniqueness and hyperbolicity of a…
It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional…
We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from…
New criteria are established for upper bounds on the number of limit cycles of periodic Abel differential equations having two periodic invariant curves, one of them bounded. The criteria are applied to obtain upper bounds of either zero or…