Related papers: The period function and the harmonic balance metho…
The purpose of this paper is to study various monotonicity conditions of the period function $T(c)$ (energy-dependent) for potential systems $\ddot x + g(x)=0$ with a center at the origin 0. We had before identified a family of new criteria…
In this paper, we study the Hamiltonian differential systems with homogeneous nonlinearity parts on $\mathbb{C}^2$. Firstly, we present a series of topological properties of polynomial Hamiltonian functions, with a particular focus on the…
In this paper we discuss the application of the harmonic balance method for the global analysis of the classical phase-locked loop (PLL) circuit. The harmonic balance is non rigorous method, which is widely used %,often without rigorous…
This paper investigates the monotonicity of the period function associated with planar Hamiltonian systems of the form $H(x,y) = F(x) + G(y)$. We establish sufficient conditions ensuring the monotonicity of the period function corresponding…
We study the existence of centers of planar autonomous system of the form $$(S) \quad \dot x=y,\qquad \dot y = -h(x) - g(x)y - f(x)y^2.$$ We are interested in the period function $T$ around a center 0. A sufficient condition for the…
This paper is concerned with the analytic behaviors (monotonicity, isochronicity and the number of critical points) of period function for potential system $\ddot{x}+g(x)=0$.We give some sufficient criteria to determine the monotonicity and…
The examples of rhythmical signals with variable period are considered. The definition of periodic function with the variable period is given as a model of such signals. The examples of such functions are given and their variable periods…
We give a sufficient condition for systems with symmetries to have periodic solutions with equal periods. We show that the main result can be applied both to Hamiltonian and to non-Hamiltonian systems. We apply the main results to produce…
We are interested at first in the study of the monotonicity for the period function of the conservative equation \ $(1)\quad \ddot x + g(x) = 0.$\quad Some refinements of known criteria are brought. Moreover, we give necessary and…
Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the…
The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure…
It is proved that harmonic functions are characterized by harmonicity of their spherical means, for which purpose the iterated spherical means are used. The similar characterization of solutions to the modified Helmholtz equation…
Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions…
A function on a (generally infinite) graph $\G$ with values in a field $K$ of characteristic 2 will be called {\it harmonic} if its value at every vertex of $\G$ is the sum of its values over all adjacent vertices. We consider binary…
The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection…
This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form…
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,…
This paper is concerned with the monotonicity of the period function for closed orbits of systems of the Li\'enard II type equation given by $\ddot{x} + f(x)\dot{x}^{2} + g(x) = 0$. We generalize Chicone's result regarding the monotonicity…
In this paper, the existence of subharmonic solutions for a class of non-autonomous first-order Hamiltonian systems is investigated. We also study the minimality of periods for such solutions. Our results which extend and improve many…
We study harmonic functions associated to systems of stochastic differential equations of the form $dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots+A_{id}(X_{t-})dZ_t^d$, $i\in\{1,\dots,d\}$, where $Z_t^j$ are independent one-dimensional symmetric…