The New Results in the Theory of Periodic Oscillation in Nonlinear Control Systems (original) (raw)
Related papers
Periodic Solutions of Nonlinear Dynamic Systems with Feedback Control
2011
In this paper, sufficient criteria for the existence of multiple positive periodic solutions of a certain nonlinear dynamic system with feedback control are established. This is done by the Avery-Henderson fixed point theorem and the LeggettWilliams fixed point theorem. By using the method of coincidence degree, sufficient conditions are derived ensuring the existence of at least one periodic solution of a more general nonlinear dynamic system with feedback control on time scales.
Selected issues in the theory of nonlinear oscillations
2014
The paper presents results concerning the theory of oscillations in the field of linear extensions of dynamical systems. An overview of the basic results was done, the direction of research was outlined and the results obtained were given in this regard.
On the periodic solutions of the nonlinear oscillators
Journal of Vibroengineering, 2014
In this paper, a new approach is introduced to overcome the difficulty of applying the differential transformation method to the nonlinear oscillators described by x¨t+fx,x˙t,xt=0. The obtained approximate periodic solutions are compared with those in open literatures and the results reveal that the present approach is very effective and convenient for a class of nonlinear oscillators with discontinuities.
Physica D: Nonlinear Phenomena, 2012
We deal with nonlinear T -periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov-Schmidt reduction method applied to the T -Poincaré-Andronov mapping.
Periodic solutions for nonlinear differential systems of equations with a small parameter
Nonlinear Analysis: Theory, Methods & Applications, 2003
We analyse the behaviour of the solution for a small parameter of some nonlinear di erential systems for which the linearized system admits periodic solutions. We obtain the normal variation system, which allows the study of stability of the transformed system, as well as several considerations on the periodicity of the solutions. ?
Periodic Solution for Strongly Nonlinear Oscillators by He’s New Amplitude–Frequency Relationship
International Journal of Applied and Computational Mathematics, 2017
This paper applies He's new amplitude-frequency relationship recently established by Ji-Huan He (Int J Appl Comput Math 3 1557-1560, 2017) to study periodic solutions of strongly nonlinear systems with odd nonlinearities. Some examples are given to illustrate the effectiveness, ease and convenience of the method. In general, the results are valid for small as well as large oscillation amplitude. The method can be easily extended to other nonlinear systems with odd nonlinearities and can therefore be found widely applicable in engineering and other science. The method used in this paper can be applied directly to highly nonlinear problems without any discretization, linearization or additional requirements.
Stability and Existence of Periodic Solutions in Non-linear Differential Equations
Abstract— Our purpose ,in this work, is to obtain periodic solutions of some nonlinear differential equation (NLDE) and to study the stability of these periodic solutions . Then we have studied the existence of limit cycles in NLDE and nonexistence by applying the theorem of (Negative Pointcaré - Bendixson Criterion). So we attempt to solve some nonlinear differential equations by combining catastrophe theory, fundamental theorem of algebra and Krylov-Boogoliubov method And the main result is the following proposition: There is at least one stable periodic solution for the nonlinear differential equation. Keywords—mathematical model, stability of periodic solution, nonlinear differential equations.
Small parameter perturbations of nonlinear periodic systems
In this paper we consider a class of nonlinear periodic differential systems perturbed by two nonlinear periodic terms with multiplicative different powers of a small parameter ε > 0. For such a class of systems we provide conditions that guarantee the existence of periodic solutions of given period T > 0. These conditions are expressed in terms of the behaviour on the boundary of an open bounded set U of R n of the solutions of suitably defined linearized systems. The approach is based on the classical theory of the topological degree for compact vector fields. An application to the existence of periodic solutions to the van der Pol equation is also presented.