Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos

We analytically link three properties of nonlinear dynamical systems, namely sensitivity to initial conditions, entropy production, and escape rate, in z-logistic maps for both positive and zero Lyapunov exponents. We unify these relations at chaos, where the Lyapunov exponent is positive, and at its onset, where it vanishes. Our result unifies, in particular, two already known cases, namely (i) the standard entropy rate in the presence of escape, valid for exponential functionality rates with strong chaos, and (ii) the Pesin-like identity with no escape, valid for the power-law behavior present at points such as the Feigenbaum one.     

Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits

We investigate how the transition to chaos with multiple positive Lyapunov exponents can be characterized by the set of infinite number of unstable periodic orbits embedded in the chaotic invariant set. We argue and provide numerical confirmation that the transition is generally accompanied by a nonhyperbolic behavior: unstable dimension variability. As a consequence, the Lyapunov exponents, except for the largest one, pass through zero continuously.     

The largest Lyapunov exponent of chaotic dynamical system in scale space and its application

The largest Lyapunov exponent is an important invariant of detecting and characterizing chaos produced from a dynamical system. We have found analytically that the largest Lyapunov exponent of the small-scale wavelet transform modulus of a dynamical system is the same as the system's largest Lyapunov exponent, both discrete map and continuous chaotic attractor with one or two positive Lyapunov exponents. This property has been used to estimate the largest Lyapunov exponent of chaotic time series with several kinds of strong additive noise.     

Chaotic phenomena of charged particles in crystal lattices

In this article, we have applied the methods of chaos theory to channeling phenomena of positive charged particles in crystal lattices. In particular, we studied the transition between two ordered types of motion; i.e., motion parallel to a crystal axis (axial channeling) and to a crystal plane (planar channeling), respectively. The transition between these two regimes turns out to occur through an angular range in which the particle motion is highly disordered and the region of phase space spanned by the particle is much larger than the one swept in the two ordered motions. We have evaluated the maximum Lyapunov exponent with the method put forward by Rosenstein et al. [Physica D 65, 117 (1993)] and by Kantz [Phys. Lett. A 185, 77 (1994)]. Moreover, we estimated the correlation dimension by using the Grassberger-Procaccia method. We found that at the transition the system exhibits a very complex behavior showing an exponential divergence of the trajectories corresponding to a positive Lyapunov exponent and a noninteger value of the correlation dimension. These results turn out to be linked to a physical interpretation. The Lyapunov exponents are in agreement with the model by Akhiezer et al. [Phys. Rep. 203, 289 (1991)], based on the equivalence between the ion motion along the crystal plane described as a "string of strings" and the "kicked" rotator. The nonintegral value of the correlation dimension can be explained by the nonconservation of transverse energy at the transition.     

Anti-control of chaos in Bose-Einstein condensates

The spatial structure of a Bose-Einstein Condensate (BEC) loaded into an optical lattice potential is investigated. We suggest a method for generating chaos in BEC by modulating periodic signals to convert the regular states into chaotic states. The maximal Lyapunov exponent is calculated as a function of modulation intensity and modulation frequency respectively, and the chaotic orbits associated with the positive Lyapunov exponents.      

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Lyapunov exponents in unstable systems

We investigate the dynamical behavior of unstable systems in the vicinity of the critical point associated with a liquid-gas phase transition. By considering a mean-field treatment, we first perform a linear analysis and discuss the instability growth times. Then, coming to complete Vlasov simulations, we investigate the role of nonlinear effects and calculate the Lyapunov exponents. As a main result, we find that near the critical point, the Lyapunov exponents exhibit a power-law behavior, with a critical exponent beta=0.5. This suggests that in thermodynamical systems the Lyapunov exponent behaves as an order parameter to signal the transition from the liquid to the gas phase.     

Upper Quantum Lyapunov Exponent and Anosov Relations for Quantum Systems Driven by a Classical Flow

We generalize the definition of quantum Anosov properties and the related Lyapunov exponents to the case of quantum systems driven by a classical flow, i.e. skew-product systems. We show that the skew Anosov properties can be interpreted as regular Anosov properties in an enlarged Hilbert space, in the framework of a generalized Floquet theory. This extension allows us to describe the hyperbolicity properties of almost-periodic quantum parametric oscillators and we show that their upper Lyapunov exponents are positive and equal to the Lyapunov exponent of the corresponding classical parametric oscillators. As second example, we show that the configurational quantum cat system satisfies quantum Anosov properties.     

Chaotic itinerancy generated by coupling of Milnor attractors

We report the existence of chaotic itinerancy in a coupled Milnor attractor system. The attractor ruins consist of tori or local chaos generated from the original Milnor attractors. The chaotic behavior exhibited by a single orbit can be considered a "nonstationary" state, due to the extremely slow convergence of the Lyapunov exponents, but the behavior averaged over randomly chosen initial conditions is consistent with the limit theorem. We present as a possibly new indication of chaotic itinerancy the presence of slow decay of large fluctuations of the largest Lyapunov exponent.     

Lyapunov exponent diagrams of a 4-dimensional Chua system

We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.     

Chaoticity analysis of the current through pure, hydrogenated and hydrophobically modified PEG-Si thin films under varying relative humidity

Polyethylene Glycol has an irregular current characteristic under constant voltage and slowly varying relative humidity. The current through a thin film of Gamma-isocyanatopropyltriethoxysilane added Polyethylene glycol (PEG-Si), its hydrogenated and hydrophobically modified forms, as a function of increasing relative humidity at equal time steps is analyzed for chaoticity. We suggest that the irregular behavior of current through PEG-Si thin films as a function of increasing relative humidity could best be analyzed for chaoticity using both time series analysis and detrended uctuation analysis; the relative humidity is kept as a slowly varying parameter. The presence of more then one regime is suggested by the calculation of the maximal Lyapunov exponents. Furthermore, the maximal Lyapunov exponent in each of the regimes was positive, thus confirming the presence of low dimensional chaos. DFA also confirms the presence of at least two different regimes, in agreement with the behavior of the maximal Lyapunov exponent in the time series analysis. We also suggest that the irregular behavior of the current through PEG-Si can be reduced by hydrogenating and hydrophobically modifying PEG-Si and the improvement in stability can be confirmed by our study.      

Crisis and unstable dimension variability in the bailout embedding map

The dynamics of inertial particles in 2-d incompressible flows can be modeled by 4-d bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent β = −1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.      

Analysis of Chaotic Dynamics by the Extended Entropic Chaos Degree

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics.     

Convective lyapunov exponents and propagation of correlations

We conjecture that in one-dimensional spatially extended systems the propagation velocity of correlations coincides with a zero of the convective Lyapunov spectrum. This conjecture is successfully tested in three different contexts: (i) a Hamiltonian system (a Fermi-Pasta-Ulam chain of oscillators); (ii) a general model for spatiotemporal chaos (the complex Ginzburg-Landau equation); (iii) experimental data taken from a CO2 laser with delayed feedback. In the last case, the convective Lyapunov exponent is determined directly from the experimental data.     

Temporal chaos versus spatial mixing in reaction-advection-diffusion systems

We develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction. The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time-independent flows and equal Pe clet numbers of different components, is demonstrated to work accurately for time-dependent flows and different Pe clet numbers.     

Quantifying chaos: A tale of two maps

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This Letter presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.     

Generation of higher degree chaos by controlling harmonics of the modulating signal in EDFRL

Erbium doped fiber ring lasers (EDFRL) are being used to generate optical chaos for secure communication by modulating the cavity loss/pump power or exploiting nonlinearities. The security level in chaotic communication depends on degree of chaos quantified by the Lyapunov exponent and its variability which is determined by the number of tuneable system parameters which were limited to five main parameters, i.e. modulation index, modulation frequency, pump power, cavity gain and loss. In this study we have increased the number of tuneable parameters using square, triangular and sum of harmonics waveforms. We have analysed the effect on degree of chaos of phase and duty cycle of square modulating signal with gradual addition of harmonics. For the given cavity parameters, the Lyapunov exponents can be increased by more than fifteen times using square wave modulating signal and a duty cycle of 60%. The electrical parameters identified make generation of new chaotic sequences more flexible in a field deployed EDFRL chaotic system.     

Synchronizing chaotic systems with positive conditional Lyapunov exponents by using convex combinations of the drive and response systems

We introduce a new method for synchronizing chaotic systems with positive conditional Lyapunov exponents, i.e., systems that do not synchronize in the Pecora-Carroll sense. This method works by considering a convex combination of the drive and response systems as a new driving signal. In this combination, the compoent associated with the response system acts as a chaos suppression method stabilizing the dynamics of the response system. This allows the chaotic component from the drive signal to synchronize both systems. The method is applied to synchronize some connections of the Rössler, Lorenz and van der Pol-Duffing systems that do not synchronize using the Pecora-Carroll scheme.     

Spatiotemporal chaos in an electric current driven ionic reaction-diffusion system

Two types of transitions from the time-periodic spatiotemporal patterns to chaotic ones in the spatially one-dimensional ionic reaction-diffusion system forced either with direct or alternating electric field are described and analyzed by numerical techniques. An ionic version of the Brusselator kinetic scheme is considered. The Karhunen-Loeve decomposition technique is shown to be a possible tool for the global representation of dynamic behavior, but fails as a tool in the identification of the route of transition to chaos in the case of direct current forcing. Higher dimensional chaos with two positive Lyapunov exponents has been identified for the case of alternating current forcing. Results of the Karhunen-Loeve analysis are compared to results of classical analysis of local time series (attractor dimensions, Lyapunov exponents).