Limit Cycles from Hopf Bifurcation in Nongeneric Quadratic Reversible Systems with Piecewise Perturbations

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems.

The cyclicity of a class of quadratic reversible centers defining elliptic curves

<p style='text-indent:20px;'>In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [<xref ref-type="bibr" rid="b28">28</xref>] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in <inline-formula><tex-math id="M1">\begin{document}$ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $\end{document}</tex-math></inline-formula>. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to <inline-formula><tex-math id="M2">\begin{document}$ (-2,-\frac{8}{5}) $\end{document}</tex-math></inline-formula>.</p>

The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x ′ = y , y ′ = − x − y z , and z ′ = y 2 − a , where a is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever a is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.

Time-Reversible Chaotic System with Conditional Symmetry

When the polarity reversal induced by offset boosting is considered, a new regime of a time-reversible chaotic system with conditional symmetry is found, and some new time-reversible systems are revealed based on multiple dimensional offset boosting. Numerical analysis shows that the system attractor and repellor have their own dynamics in respective time domains which constitutes the fundamental property in a time-reversible system. More remarkably, when the conditional symmetry is destroyed by a slightly mismatched offset controller, the system undergoes different bifurcations to chaos, and the corresponding coexisting attractors and repellors shape their own phase trajectories.