Related papers: Limit cycles for two classes of control piecewise …
In this paper, a quadratic system with two parallel straight line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian. Using the field rotation parameters of the constructed…
Lienard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Lienard systems of arbitrary degree on the plane, and develop a new method to obtain a…
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…
Combining local bifurcation analysis with numerical continuation and bifurcation methods we study bifurcations from cylindrical vesicles described by the Helfrich equation with volume and area constraints, with a prescribed periodicity…
In this paper we report some important results that help in analizing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the…
This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation \begin{align*} \frac{dx}{d\theta}=A(\theta)x^p+B(\theta)x^q, \end{align*} where $A$ and $B$ are are piecewise trigonometrical…
This paper deals with a version of the two-timing method which describes various `slow' effects caused by externally imposed `fast' oscillations. Such small oscillations are often called \emph{vibrations} and the research area can be…
We present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular non-invasive control schemes, such as…
In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The `torus' is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same…
In this paper a generalized Rayleigh-Li\'enard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the…
This paper is devoted to the study of the maximum number of limit cycles, $H(m,n)$, of a planar piecewise linear differential system with two zones separated by the curve $y^n-x^m=0$, with $n,m$ being positive integers. More precisely, we…
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…
The quasi-homogeneous (and in general non-homogeneous) polynomial differential systems have been studied from many different points of view. In this paper, Center-focus determination and limit cycles bifurcation for $p:q$ homogeneous weight…
Experiments observing the liquid surface in a vertically oscillating container have indicated that modeling the dynamics of such systems require maps that admit states at infinity. In this paper we investigate the bifurcations in such a…
The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel…
For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here a unique control set (i.e., a maximal set of approximate controllability) with nonvoid…
We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies…
Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near…
Li\'{e}nard equations, $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even continuous function are considered. In the weakly nonlinear regime ($\epsilon\to 0$), the number and an order zero in $\epsilon$ approximation of the amplitude…
In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the…