On the dynamical properties of an elliptical–oval billiard with static boundary

Some dynamical properties for a classical particle confined inside a closed region with an elliptical–oval-like shape are studied. The dynamics of the model is made by using a two-dimensional nonlinear mapping. The phase space of the system is of mixed kind and we have found the condition that breaks the invariant spanning curves in the phase space. We have discussed also some statistical properties of the phase space and obtained the behaviour of the positive Lyapunov exponent.     

Dynamical and statistical properties of a rotating oval billiard

Some dynamical and statistical properties of a time-dependent rotating oval billiard are studied. We considered cases with (i) positive and (ii) negative curvature for the boundary. For (i) we show the system does not present unlimited energy growth. For case (ii) however the average velocity for an ensemble of noninteracting particles grows as a power law with acceleration exponent well defined. Finally, we show for both cases that after introducing time-dependent perturbation, the mixed structure of the phase space observed for static case is recovered by making a suitable transformation in the angular position of the particle.     

Synchronization in pair-coupled maps with invariant measure

Chaos synchronization, as an important topic, has become an active research subject in non-linear science. By considering a symmetric two-dimensional map that possesses invariant measure in its diagonal and anti-diagonal invariant sub-manifolds, we have been able to introduce the most general pair-coupled map possessing invariant measure at synchronized or anti-synchronized states. Then chaotic synchronization and anti-synchronization are investigated in introduced model. We have calculated Kolmogrov–Sinai entropy and Lyapunov exponent as another tool to study the stability of pair-coupled map at synchronized and anti-synchronization states.     

Dynamics of a two-frequency parametrically driven duffing oscillator

Summary We investigate the transition from two-frequency quasiperiodicity to chaotic behavior in a model for a quasiperiodically driven magnetoelastic ribbon. The model system is a two-frequency parametrically driven Duffing oscillator. As a driving parameter is increased, the route to chaos takes place in four distinct stages. The first stage is a torus-doubling bifurcation. The second stage is a transition from the doubled torus to a strange nonchaotic attractor. The third stage is a transition from the strange nonchaotic attractor to a geometrically similar chaotic attractor. The final stage is a hard transition to a much larger chaotic attractor. This latter transition arises as the result of acrisis, the characterization of which is one of our primary concerns. Numerical evidence is given to indicate that the crisis arises from the collision of the chaotic attractor with the stable manifold of a saddle torus. Intermittent bursting behavior is present after the crisis with the mean time between bursts scaling as a power law in the distance from the critical control parameter; τ ∼ (A-A_c)^-α. The critical exponent is computed numerically, yielding the value α=1.03±0.01. Theoretical justification is given for the computed critical exponent. Finally, a Melnikov analysis is performed, yielding an expression for transverse crossings of the stable and unstable manifolds of the crisis-initiating saddle torus.     

The destruction of tori in volume-preserving maps

Invariant tori are prominent features of symplectic and volume-preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an n-dimensional volume-preserving map, such tori are prevalent when the map is nearly “integrable,” in the sense of having one action and n − 1 angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, ε_c, above which there are no tori. Numerical investigations to find the “last invariant torus” reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at ε_c, and the local phase space near the critical torus contains many high-order resonances.     

Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map P^T is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an invariant circle, on which P^T is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of P^T on this curve.     

Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map P^T is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an invariant circle, on which P^T is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of P^T on this curve. Supported by National Natural Science Foundation of China (10771155, 10571131) and Natural Science Foundation of Jiangsu Province (BK 2006046).     

Moduli of plane curve singularities with a single characteristic exponent

This paper studies the moduli space corresponding to irreducible germs of plane analytic curve with a single characteristic exponent. We stratify the moduli space corresponding to such germs using an analytical invariant introduced by Zariski. Then, we compute the minimum Tjurina number on each stratum as well as the dimension of the strata.

     

A Continuous Archetype of Nonuniform Chaos in Area-Preserving Dynamical Systems

We propose a piecewise linear, area-preserving, continuous map of the two-torus as a prototype of nonlinear two-dimensional mixing transformations that preserve a smooth measure (e.g., the Lebesgue measure). The model lends itself to a closed-form analysis of both statistical and geometric properties. We show that the proposed model shares typical features that characterize chaotic dynamics associated with area-preserving nonlinear maps, namely, strict inequality between the line-stretching exponent and the Lyapunov exponent, the dispersive behavior of stretch-factor statistics, the singular spatial distribution of expanding and contracting fibers, and the sign-alternating property of cocycle dynamics. The closed-form knowledge of statistical and geometric properties (in particular of the invariant contracting and dilating directions) makes the proposed model a useful tool for investigating the relationship between stretching and folding in bounded chaotic systems, with potential applications in the fields of chaotic advection, fast dynamo, and quantum chaos theory.     

Existence and visualization of a hyperbolic structure in area-preserving maps

We present a simple method which displays a hyperbolic structure in the phase space of an area preserving map. Using the picture which this method yields one can estimate the entropy of the system and its Lyapunov exponent through the size of “hyperbolic cells” inside the chaotic region.     

Influence of the finite precision on the simulations of discrete dynamical systems

The effect of numerical precision on the mean distance and on the mean coalescence time between trajectories of two random maps was investigated. It was shown that mean coalescence time between trajectories can be used to characterize regions of the phase space of the maps. The mean coalescence time between trajectories scales as a power law as a function of the numerical precision of the calculations in the contracting and transitions regions of the maps. In the contracting regions the exponent of the power law is approximately one for both maps and it is approximately two in the transition regions for both maps. In the chaotic regions, the mean coalescence time between trajectories scales as an exponential law as a function of the numerical precision of the calculations for the maps. For both maps the exponents are of the same order of magnitude in the chaotic regions.     

On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium

In this paper we investigate the similarity solutions of a plane mixed convection boundary layer flow near a semi-vertical plate, with a prescribed power law function of the distance from the leading edge for the temperature, that is embedded in a porous medium. We show the existence and uniqueness of convex and concave solutions for positive values of the power law exponent.     

Control of chaotic instabilities in a spinning spacecraft with dissipation using Lyapunov''s method

Control of chaotic instability in a simplified model of a spinning spacecraft with dissipation is achieved using an algorithm derived using Lyapunov's second method. The control method is implemented on a realistic spacecraft parameter configuration which has been found to exhibit chaotic instability for a range of forcing amplitudes and frequencies when a sinusoidally varying torque is applied to the spacecraft. Such a torque, may arise in practice from an unbalanced rotor or from vibrations in appendages. Numerical simulations are performed and the results are studied by means of time history, phase space, Poincaré map, Lyapunov characteristic exponents and bifurcation diagrams.     

On statistical properties of the lyapunov exponent of the generalized skew tent map

The statistical properties of the Lyapunov exponent of the chaotic generalized skew tent map is studied. Expressions of the mean and the variance of this Lyapunov exponent at each discrete time index are obtained. A sufficient condition for weakly mixing of the chaotic generalized skew tent map is derived, and the asymptotic distribution of its Lyapunov exponent is provided.     

Reverse bifurcation and fractal of the compound logistic map

The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot–Julia set of compound logistic map. We generalize the Welstead and Cromer’s periodic scanning technology and using this technology construct a series of Mandelbrot–Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot–Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.     

Suppressing chaos of a complex Duffing's system using a random phase

The effect of random phase for a complex Duffing's system is investigated. We show as the intensity of random noise properly increases the chaotic dynamical behavior will be suppressed by the criterion of top Lyapunov exponent, which is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. Also Poincaré map analysis, phase plot and the time evolution are carried out to confirm the obtained results of Lyapunov exponent on dynamical behavior including the stability, bifurcation and chaos. Thus excellent agreement between these results is found.     

Diffusion dynamics near critical bifurcations in a nonlinearly damped pendulum system

We numerically study the diffusion dynamics near critical bifurcations such as sudden widening of the size of a chaotic attractor, intermittency and band-merging of a chaotic attractor in a nonlinearly damped and periodically driven pendulum system. The system is found to show only normal diffusion. Near sudden widening and intermittency crisis power-law variation of diffusion constant with the control parameter ω of the external periodic force f sin ωt is found while linear variation of it is observed near band-merging crisis. The value of the exponent in the power-law relation varies with the damping coefficient and the strength of the added Gaussian white noise.     

Visibility graphs of fractional Wu–Baleanu time series

ABSTRACT

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.     

Kirchhoff equations with strong damping

We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the “elastic” operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space, assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space, we prove local existence and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.     

Probability density functions of some skew tent maps

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval F_a,b. We show that F_a,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known (Boyarsky A, Góra P. Laws of chaos: invariant measures and dynamical systems in one dimension. Boston: Birkhauser, 1997), that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.