Chaos Solitons Research Articles

Applications of a further extended tanh method

By using a further extended tanh method [Chaos Solitons Fract. 17 (2003) 669] and symbolic computation system Maple, we obtain abundant new explicit exact solutions of the (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional Boussinesq equation. The new solutions including soliton-like solutions, period form solutions and some other solutions.

C-bifurcations and period-adding in one-dimensional piecewise-smooth maps

This paper examines the behaviour of piecewise-smooth, continuous, one-dimensional maps that have been derived in the literature as normal forms for grazing and sliding bifurcations. These maps are linear for negative values of the parameter and non-linear for positive values of the parameter. Both C 1 and C 2 maps of this form are considered. These maps display both period-adding and period-doubling behaviour. For maps with a squared or 3/2 term the stability and existence conditions of fixed points and period-2 orbits in the vicinity of the border-collision are found analytically. These agree with the Feigin classification proposed by di Bernardo et al. [Chaos Solitons and Fractals 10 (1999) 1881]. The period-adding behaviour is examined in these maps, where analytical solutions for the boundaries of periodic solutions are found. Implicit equations for the boundaries of periodic windows for varying power term are also found and plotted. Thus, it is proved that period-adding scenarios are generic in maps of this form.

Bifurcations in a Mathieu equation with cubic nonlinearities: Part II

In a previous paper [Chaos Solitons Fract. 14(2) (2002) 173], the authors investigated the dynamics of the equation: d 2x dt 2 +(δ+ϵ cost)x+ϵ Ax 3+Bx 2 dx dt +Cx dx dt 2+D dx dt 3 =0 We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented.

Distance, correlation and mutual information among portraits of organisms based on complete genomes

Based on the portrait method proposed by Hao et al. (Chaos Solitons Fractals 11 (6) (2000) 825), four parameters (distance, correlation coefficient, entropy and mutual information) are introduced to provide exact measures of the difference between a real genome and the “white noise” genome of it. Here the “white noise” genome of a real genome means a random sequence based on the frequencies of four kinds of nucleotides appearing in the real genome. These parameters can also be used to discuss the classification and evolution among organisms. The evolution trees of some bacteria based on these parameters are given.

Apparently Chaotic Orbits Embedded in Closed Curves

In [S. Bahar, Chaos Solitons Fractals, 5 (1995), pp. 1001--1006; 7 (1996), pp. 41--47], pictures were presented of the forward orbit {zn}n in the plane, where zn satisfies zn+1 = znAn+1 for certain affine transformations An . These orbits (a) appear to lie on closed curves, and (b) assume forms reminiscent of strange attractors. We answer the questions raised in these references and prove that indeed in the aperiodic case the invariant orbits of this "iterated function system" lie on closed curves and that in some sense the maps act "chaotically" on the set of closed curves. A key role is being played by almost periodic functions. Furthermore, we provide an expression for the top Lyapunov exponent, that is, a sum in the periodic case and an integral in the aperiodic case. Our results have important implications in applications of computer visualization and imaging and shed some light on nonlinear dynamical systems.

Spectral correlations in systems undergoing a transition from periodicity to disorder.

We study the spectral statistics for extended yet finite quasi-one-dimensional systems, which undergo a transition from periodicity to disorder. In particular, we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly between the two extreme limits-the approach to Poissonian statistics in the (weakly) disordered case, and the universal expressions derived in T. Dittrich, B. Mehlig, H. Schanz, and U. Smilansky, Chaos Solitons Fractals 8, 1205 (1997); Phys. Rev. E 57, 359 (1998); B. D. Simons and B. L. Altshuler, Phys. Rev. Lett. 70, 4063 (1993); and N. Taniguchi and B. L. Altshuler, ibid. 71, 4031 (1993) for the periodic case. The theoretical results agree very well with the spectral statistics obtained numerically for chains of chaotic billiards and graphs.

On the route to strangeness without chaos in the quasiperiodically forced van der Pol oscillator

The non-chaotic strange attractor of a non-autonomous dissipative dynamical system with a two-frequency-quasiperiodic forcing is studied. Following results from T. Kapitaniak [ Chaos Solitons & Fractals 1, 67 (1991)], numerical observations are refined and the route from torus to strange non-chaotic attractor is described in detail. Possible misinterpretations of experimental observations are discussed.

Fractional Brownian Motions, Fractional Noises and Applications

Fractional Brownian Motions, Fractional Noises and Applications