Chaos Threshold Research Articles

The rate of entropy increase at the edge of chaos

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov–Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann–Gibbs–Shannon entropy is not appropriate. Instead, the non-extensive entropy S q≡ (1− ∑ i=1 W p i q) (q−1) , must be used. The latter contains a parameter q, the entropic index which must be given a special value q ∗≠1 (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q ∗ enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.

Synchronization Among Spin-Wave Amplitudes in Chaotic Nonlinear Ferromagnetic Resonance

A model of chaos in high-power ferromagnetic resonance in coincidence regime, based on three-magnon interactions of the uniform mode with a group of pairs of parametric spin waves, is investigated numerically. The re- sults are interpreted from the point of view of chaotic synchronization theory. If all spin waves are identical, all of them are excited above the first-order Suhl instability threshold and in the chaotic regime their amplitudes show marginal synchronization, i.e. they differ only by a multiplicative factor, con- stant in time. If spin waves have slightly different instability thresholds, only one or few of them are excited. In the latter case, addition of weak thermal noise changes the results qualitatively. For low rf field amplitude, but above the threshold for chaos, still only few spin-wave pairs are excited above the thermal level. For higher rf field amplitude all spin waves in the group are excited, and their amplitudes are not synchronized. These results sug- gest that low correlation dimension of chaotic attractors, observed often in nonlinear ferromagnetic resonance, can be connected with chaotic synchronization among spin-wave amplitudes, in particular just above the threshold for chaos.

Scaling at the chaos threshold for interacting electrons in a quantum Dot

The chaotic mixing by random two-body interactions of many-electron Fock states in a confined geometry is investigated. Two regimes are distinguished in the dependence of the typical number of Fock states that are mixed into an eigenstate on the interaction strength V, the excitation energy varepsilon, and the level spacing Delta. In both regimes the number is large (indicating delocalization in Fock space). However, only the large- V regime is described by the golden rule (indicating chaotic mixing). The crossover region is characterized by a maximum in a scaling function that becomes more pronounced with increasing excitation energy. The scaling parameter that governs the transition is (varepsilonV/Delta(2))ln(Delta/V).

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1 Ȣ q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law 1/(1 − q) = 1/α min − 1/α max, where α min and α max are the extreme values appearing in the multifractal f(α) function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d f equals unity for all z in contrast with q which does depend on z, it becomes clear that d f plays no major role in the sensitivity to the initial conditions.

Discussion

The chaotic dynamics of nonlinear waves in the harmonic-forced fluid-conveying pipe in primary parametrical resonance, is explored analytically and numerically. The multiple scale method is applied to obtain an equivalent nonlinear wave equation from the complicated nonlinear governing equation describing the fluid conveyed in a pipe. With the Melnikov method, the persistence of a heteroclinic structure is shown to be satisfied and its condition is given in functional form. Similarly, for the heteroclinic orbit, using geometric analysis, a condition function of the stable manifold is derived for the orbit to return to the stable manifold from the saddle point. The persistent homoclinic structures and threshold of chaos in the Smale-horseshoe sense are obtained for the fluid-conveying pipe under both conditions, indicating how the external excitation amplitude can change substantially the global dynamics of the fluid conveyed in the pipe. A numerical approach was used to test the prediction from theory. The impact of the external excitation amplitude on the nonlinear wave in the fluid-conveying pipe was also studied from numerical simulations. Both theoretical predications and numerical simulations attest to the complex chaotic motion of fluid-conveying pipes.

Chaos thresholds in finite Fermi systems

The development of Quantum Chaos in finite interacting Fermi systems is considered. At sufficiently high excitation energy the direct two-particle interaction may mix into an eigen-state the exponentially large number of simple Slater-determinant states. Nevertheless, the transition from Poisson to Wigner-Dyson statistics of energy levels is governed by the effective high order interaction between states very distant in the Fock space. The concrete form of the transition depends on the way one chooses to work out the problem of factorial divergency of the number of Feynman diagrams. In the proposed scheme the change of statistics has a form of narrow phase transition and may happen even below the direct interaction threshold.

Controlling Chaos with Parametric Perturbations

We consider the effects of parametric perturbation on the onset of chaos in different dynamical systems. Favoring or suppression of chaos was observed depending on the phase or the frequency of the periodic perturbation. A lowering of the threshold of chaos was observed in an electronic device simulating a Josephson-Junction model and the suppression of chaos was obtained in a bistable mechanical device. We showed that in case of spatial instability in a sample of liquid crystal, the action of the parametric perturbation is to modify the velocity and the onset of the defects. Considering that the emergence of defects precedes the threshold of spatio-temporal chaos, we infer that parametric perturbation can modify the threshold of chaos in this spatial dynamical system.

Universal similarities of transition to chaos of satellite oscillation in elliptic orbit

For the problem on oscillation about mass centre of a satellite moving in the elliptical orbit in the central force field, the scaling functions have been built, which describe the satellite phase trajectory evolution at transition to chaos. Four period-doubling bifurcation sequences are considered, generated both from the stable and unstable steady states. The scaling universal constants for conservative systems are obtained, the multiplicator universal value for the satellite periodical unstable motion around its centroid at an accumulation point on the threshold of chaos is determined as well.

Chaos due to the interaction of high- and low-frequency modes

The results of theoretical and experimental investigations of the interaction of high- and low-frequency oscillators in the presence of a periodic perturbation are discussed. The reasons for threshold of chaos to be low in such a system are clarified in terms of the development of the decay instability and the destruction of quasiperiodic oscillations. Analytical criteria for predicting the onset of chaos are provided.

Metastable chaos in the ammonia ring laser

We report experimental studies of metastable chaos in the far-infrared ammonia ring: laser. When the laser pump power is switched from above chaos threshold to slightly below, chaotic intensity pulsations continue for a varying time afterward before decaying to either periodic or cw emission. The behavior is in good qualitative agreement with that predicted by the Lorenz equations, previously used to describe this laser. The statistical distribution of the duration of the chaotic transient is measured and shown to be in excellent agreement with the Lorenz equations in showing a modified exponential distribution. We also give a brief numerical analysis and graphical visualization of the Lorenz equations in phase space illustrating the boundary between the metastable chaotic and the stable fixed point basins of attraction. This provides an intuitive understanding of the metastable dynamics of the Lorenz equations and the experimental system.

Chaos in rf-driven long Josephson junctions in the presence of an external field.

Chaotic responses induced by an applied rf signal on the long Josephson junction influenced by a constant dc-driven field are investigated by means of a model based on the sine-Gordon equation. We assume that the localized nonlinear excitations present in the system as initial data are the breatherlike mode. Thus, in order to analyze in detail the onset of the complicated irregular behavior (chaos), we employ collective coordinate theory to derive the total Hamiltonian and the dissipative function in terms of the breather width. The appearance of chaos in this system is predicted by the Melnikov theory and a few numerical treatments. A threshold for chaos as a function of the physical parameters of the system is found. Finally, the Poincar\'e sections, which allow us to see clearly the phenomenon in the nondissipative case, is compared to the analytical prediction and good agreement is obtained.

Lorenz like chaos in a Gaussian mode laser with a radially dependent gain

Numerical simulations of the Maxwell-Bloch equations modelling an optically pumped laser with Gaussian shaped radially varying gain profile show regions of instability leading to Lorenz like chaos. A graph showing the relationship between the chaos threshold and the pump power shows a minimum when the diameter of the pump beam is comparable to the diameter of the beam waist of the resonant Gaussian mode of the cavity. For ratios of pump beam diameter to cavity mode diameter different from one, the power necessary to reach chaos threshold becomes very large.

On the threshold of chaos in random boolean cellular automata

AbstractA random boolean cellular automaton is a network of boolean gates where the inputs, the boolean function, and the initial state of each gate are chosen randomly. In this article, each gate has two inputs. Let a (respectively c) be the probability that the gate is assigned a constant function (respectively a noncanalyzing function, i.e., EQUIVALENCE or EXCLUSIVE OR). Previous work has shown that when a>c, with probability asymptotic to 1, the random automaton exhibits very stable behavior: Almost all of the gates stabilize, almost all of them are weak, i.e., they can be perturbed without affecting the state cycle that is entered, and the state cycle is bounded in size. This article gives evidence that the condition a = c is a threshold of chaotic behavior: with probability asymptotic to 1, almost all of the gates are still stable and weak, but the state cycle size is unbounded. In fact, the average state cycle size is superpolynomial in the number of gates.

Properties of cross-well chaos in an impacting system

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing’s equation with negative linear stiffness. Our focus is on the prediction and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under which homoclinic tangles exist, which is a necessary condition for cross-well chaos to occur. We then show how one may manipulate higher harmonics of the excitation in order to affect the range of excitation amplitudes over which fractal basin boundaries between the two potential wells exist. We also experimentally investigate the threshold for cross-well chaos and compare the results with the theoretical results. Second, we consider the rate at which the system crosses from one potential well to the other during a chaotic motion and relate this to the rate of phase space flux in a Poincare map defined in terms of impact parameters. Results from simulations and experiments are compared with a simple theory based on phase space transport ideas, and a predictive scheme for estimating the rate of crossings under different parameter conditions is presented. The main conclusions of the paper are the following: (1) higher harmonics can be used with some effectiveness to extend the region of deterministic basin boundaries (in terms of the amplitude of excitation) but their effect on steady-state chaos is unreliable; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the system as measured by the area of a turnstile lobe in the Poincare map. These results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing’s equation, are influenced by various system and input parameters. The main tools of analysis are a special version of Melnikov’s method, adapted for this piecewise-linear system, and ideas of phase space transport.

Comparison of transient emission of a NH 3 ring laser with the Lorenz-Haken model

Measurements are reported of the transient responses of a 15NH 3 FIR ring laser to sudden changes in pump power above, near and well below the threshold for Lorenz-like chaos. Qualitative agreement with the predictions of the Lorenz-Haken equations is found except for effects which can be attributed to the over-simplification of the Lorenz-Haken model with respect to the real laser system. Quantitatively, the transient laser response permits determination of effective values for the decay constants of the active medium within the framework of the Lorenz-Haken model.

The second-order approximation of threshold value of chaos

For the Melnikov theory of chaos, the second-order approximation is given. Applying the result to the dynamics system with quadratic nonlinear term, it is shown that the threshold of chaos depends on initial condition and can be greater than that of the Melnikov method. The first order variational equations of some nonlinear dynamical systems are all the second-order ordinary differential equations with hyperbolic cosine function, its solution is given.

Modifying the onset of homoclinic chaos: Application to a bistable potential.

We analyze, by means of the Melnikov method, the possibility of modifying the threshold of homoclinic chaos in general one-dimensional problems, by introducing small periodic resonant modulations. We indicate in particular a prescription in order to increase the threshold (i.e., to prevent chaos), and consider then its application to the bistable Duffing-Holmes potential. All results are confirmed both by numerical and by analog simulations, showing that small modulations can in fact sensibly influence the onset of chaos.

Oscillations near the chaos threshold in a system of unidirectionally coupled nonlinear electrical oscillators

Oscillations near the chaos threshold in a system of unidirectionally coupled nonlinear electrical oscillators

Predicting chaos for infinite dimensional dynamical systems: the Kuramoto-Sivashinsky equation, a case study.

The results of extensive computations are presented to accurately characterize transitions to chaos for the Kuramoto-Sivashinsky equation. In particular we follow the oscillatory dynamics in a window that supports a complete sequence of period doubling bifurcations preceding chaos. As many as 13 period doublings are followed and used to compute the Feigenbaum number for the cascade and so enable an accurate numerical evaluation of the theory of universal behavior of nonlinear systems, for an infinite dimensional dynamical system. Furthermore, the dynamics at the threshold of chaos exhibit a self-similar behavior that is demonstrated and used to compute a universal scaling factor, which arises also from the theory of nonlinear maps and can enable continuation of the solution into a chaotic regime. Aperiodic solutions alternate with periodic ones after chaos sets in, and we show the existence of a period six solution separated by chaotic regions.

Subthreshold stochastic diffusion with application to selective acceleration of He-3 in solar flares

view Abstract Citations (22) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Subthreshold Stochastic Diffusion with Application to Selective Acceleration of 3He in Solar Flares Riyopoulos, Spilios Abstract A mechanism of resonant acceleration driven by waves below the threshold for global chaos is discussed. It affects only resonant ions and proceeds through a stochastic network in phase space (Arnold web) characterizing this particular type of wave-particle interaction. A theory for selective acceleration of He-3 and He-3-like ions through resonant subthreshold diffusion is proposed. It relies on the coincidence of two natural resonances, the two-ion (He-4, H) hybrid plasma resonance with the ion cyclotron resonance of He-3. This occurs over a wide range of plasma parameters for relative abundances He-4/H about equal to 0.25. It is argued that existing low-frequency fluctuations or high-frequency noise tends to amplify the selective diffusion of He-3. Publication: The Astrophysical Journal Pub Date: November 1991 DOI: 10.1086/170682 Bibcode: 1991ApJ...381..578R Keywords: Diffusion; Particle Acceleration; Solar Flares; Helium Plasma; Hydrogen Plasma; Plasma Resonance; Stochastic Processes; Wave-Particle Interactions; Solar Physics; DIFFUSION; PARTICLE ACCELERATION; SUN: FLARES full text sources ADS |