This paper presents a theory for characterization of the blowout bifurcation by periodic orbits. Blowout bifurcation in chaotic systems occurs when a chaotic attractor, lying in some symmetric invariant subspace, becomes transversely unstable. We present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number ...

In this paper we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincaré maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding 0-input fixed point of the associated...

In this article tapping-mode atomic force microscope dynamics is studied. The existence of a periodic orbit at the forcing frequency is shown under unrestrictive conditions. The dynamics is further analyzed using the impact model for the tip-sample interaction and a spring-mass-damper model of the cantilever. Stability of the periodic orbit is established. Closed-form expressions for various va...

We study the asymptotic behavior of large data solutions to Schrödinger equations iu t + ∆u = F (u) in R d , assuming globally bounded H 1 x (R d) norm (i.e. no blowup in the energy space), in high dimensions d ≥ 5 and with nonlinearity which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that ...

Long periodic orbits of hyperbolic dynamics do not exist as independent individuals but rather come in closely packed bunches. Under weak resolution a bunch looks like a single orbit in configuration space, but close inspection reveals topological orbit-to-orbit differences. The construction principle of bunches involves close self-“encounters” of an orbit wherein two or more stretches stay clo...

One orbit flip and two inclination flips bifurcation is considered with resonant principal eigenvalues. We introduce a local active coordinate system to establish bifurcation equation and obtain the conditions when the original homoclinic orbit is kept or broken. We also prove the existence and the existence regions of double 1-periodic orbit bifurcation. Moreover, the complicated homoclinic-do...

We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil’nikov for ordinary differential equations and of Chow and Deng for...

We consider a deterministic realization of Parrondo games, and use periodic orbit theory to analyze their asymptotic behavior.

We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. Based on the properties of periodic orbit of minimal period, the minimal completely invariant closed set is constructed. Topological characterizations of the renormalizations and α-limit set...

We explore periodic orbits on an equilateral triangular billiard table. We prove that there is exactly one periodic orbit with odd period. For any positive integer n, there exist P d|n μ(d)P ¡ n d ¢ periodic orbits of period 2n, where μ(d) is the Möbius transformation function and P (n) = bn+2 2 c− bn+2 3 c. We count periodic orbits by introducing a new type of integer partition.