Limit Cycles In Dynamical Systems Research Articles

Characterising stochastic fixed points and limit cycles for dynamical systems with additive noise

This study focuses on characterising numerically the attractor volume in the state space of dynamical systems excited by additive white noise. A definition for stochastic attractors is introduced in terms of probability measure and numerical methodologies are presented to characterise them. The study is limited to investigating the effects of additive noise on fixed point and limit cycle attractors of the corresponding noise free system. The effect of noise intensity on the definition and validity of the proposed methodology is also discussed.

Inertial manifolds and limit cycles of dynamical systems in R 2

Inertial manifolds and limit cycles of dynamical systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math>

A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems

AbstractWe propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

Stochastic sensitivity and noise-induced bifurcations of limit cycles

Deformations of limit cycles for dynamical systems forced by random disturbances are studied. A mathematical tool based on stochastic sensitivity analysis is shortly presented. A phenomenon of noise-induced bifurcation of supersensitive limit cycle of randomly forced Brusselator is discussed.

Exact front, soliton and hole solutions for a modified complex Ginzburg–Landau equation from the harmonic balance method

A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg–Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg–Landau equation in terms of Fourier series.

Remarks on Barbashin–Ezeilo Problem on third-order nonlinear differential equations

In this paper, we use the frequency domain criteria to initiate a general result on the Barbashin–Ezeilo Problem on third-order nonlinear differential equation. Earlier ideas of Burkin [I.M. Burkin, Orbital stability of second-kind limiting cycles for dynamical systems with cylindrical phase space, Differential Equations 29 (1993) 1262–1264], Leonov [G.A. Leonov, A frequency criterion for the existence of limit cycles of dynamical systems with cylindrical phase spaces, Differential Equations 23 (1987) 1375–1378] on the problem are being improved upon.

Lyapunov function proof of Poincaré's theorem

One of the most fundamental results in analysing the stability properties of periodic orbits and limit cycles of dynamical systems is Poincaré's theorem. The proof of this result involves system analytic arguments along with the Hartman–Grobman theorem. Using the notions of stability of sets, lower semicontinuous Lyapunov functions are constructed to provide a Lyapunov function proof of Poincaré's theorem.

Some results and open problems concerning symmetries, first integrals and limit cycles of dynamical systems

Some results and open problems concerning symmetries, first integrals and limit cycles of dynamical systems

Some criteria for presence and absence of limit cycles in dynamical systems of the second order

Some criteria for presence and absence of limit cycles in dynamical systems of the second order