different discrete models of population dynamics of certain insects have been investigated under various feasible conditions within the framework of nonlinear dynamics. evolutionary phenomena are discussed through bifurcation analysis leading to chaos. some tools of nonlinear dynamics, such as lyapunov characteristic exponents (lce), lyapunov numbers, correlation dimension, etc. are calculated ...

The butterfly-like Lorenz attractor is one of the best known images of chaos. The computations in this paper exploit symbolic dynamics and other basic notions of hyperbolicity theory to take apart the Lorenz attractor using periodic orbits. We compute all 111011 periodic orbits corresponding to symbol sequences of length 20 or less, periodic orbits whose symbol sequences have hundreds of symbol...

In this paper, we consider one-dimensional attractor of a non-autonomous second order strongly damped lattice system with periodic driving force under Neumann boundary condition or periodic boundary condition. We obtain the existence of a global attractor and prove this attractor is homeomorphic to the circle.

In a 1963 paper, Lorenz inferred that the Lorenz attractor must be an infinite complex of surfaces. We investigate this fractal property of the Lorenz attractor in two ways. Firstly, we obtain explicit plots of the fractal structure of the Lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits. The method we derive for multiple precision computation is b...

Riddling bifurcation, i.e., the bifurcation in which one of the unstable periodic orbits embedded in a chaotic attractor becomes unstable transverse to the attractor, leads to the loss of chaos synchronization in coupled identical systems. We discuss here the manifestation of the riddling bifurcation for the case of a small parameter mismatch between coupled systems. We show that for slightly n...

We investigate the sudden destruction of hyperchaotic attractors in symmetrically coupled onedimensional maps. An asynchronous hyperchaotic attractor may appear through a blow-out bifurcation, where the synchronous chaotic attractor on the invariant synchronization line becomes unstable with respect to perturbations transverse to the synchronization line. It is found that such a hyperchaotic at...

We show how to characterize a strange attractor by a set of integers. These are extracted from the chaotic time-series data by first reconstructing the low-period orbits and then determining the template, or knot holder, which supports all periodic orbits embedded in the strange attractor, and the strange attractor itself. The template is identified by a set of integers which therefore characte...

We demonstrate that the pedestal components observed in the power spectra of a directly modulated laser diode, which were interpreted as a sign of instability of the periodic regime, are an indication of the coexistence of a chaotic regime with the periodic one. We present the underlying dynamics behind the rise of these pedestals, showing two different situations in which the pedestals appear....

We study the combined effects of seasonal trends and diseases on the extinction and persistence of discretely reproducing populations. We introduce the epidemic threshold parameter, R0, for predicting disease dynamics in periodic environments. Typically, in periodic environments, R0 > 1 implies disease persistence on a cyclic attractor, while R0 < 1 implies disease extinction. We also explore t...