International Journal of Bifurcation and Chaos

We examine and assess deterministic chaos as an observable. First, we present the development of model ecological systems. We illustrate how to apply the Kolmogorov theorem to obtain limits on the parameters in the system, which assure the existence of either stable equilibrium point or stable limit cycle behavior in the phase space of two-dimensional (2D) dynamical systems. We also illustrate the method of deriving conditions using the linear stability analysis. We apply these procedures on some basic existing model ecological systems. Then, we propose four model ecological systems to study the dynamical chaos (chaos and intermittent chaos) and cycles. Dynamics of two predation and two competition models have been explored. The predation models have been designed by linking two predator–prey communities, which differ from one another in one essential way: the predator in the first is specialist and that in the second is generalist. The two competition models pertain to two distinct competition processes: interference and exploitative competition. The first competition model was designed by linking two predator–prey communities through inter-specific competition. The other competition model assumes that a cycling predator–prey community is successfully invaded by a predator with linear functional response and coexists with the community as a result of differences in the functional responses of the two predators. The main criterion behind the selection of these two model systems for the present study was that they represent diversity of ecological interactions in the real world in a manner which preserves mathematical tractability. For investigating the dynamic behavior of the model systems, the following tools are used: (i) calculation of the basin boundary structures, (ii) performing two-dimensional parameter scans using two of the parameters in the system as base variables, (iii) drawing the bifurcation diagrams, and (iv) performing time series analysis and drawing the phase space diagrams. The results of numerical simulation are used to distinguish between chaotic and cyclic behaviors of the systems.

The conclusion that we obtain from the first two model systems (predation models) is that it would be difficult to capture chaos in the wild because ecological systems appear to change their attractors in response to changes in the system parameters quite frequently. The detection of chaos in the real data does not seem to be a possibility as what is present in ecological systems is not robust chaos but short-term recurrent chaos. The first competition model (interference competition) shares this conclusion with those of predation ones. The model with exploitative competition suggests that deterministic chaos may be robust in certain systems, but it would not be observed as the constituent populations frequently execute excursions to extinction-sized densities. Thus, no matter how good the data characteristics and analysis techniques are, dynamical chaos may continue to elude ecologists. On the other hand, the models suggest that the observation of cyclical dynamics in nature is the most likely outcome.

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map T_Z: z′ = z² - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x² - c with x a real variable. Here, the second part deals with an embedding of T_Z into the two-dimensional noninvertible map ; y′ = γ y + 4x²y, γ ≥ 0. For is semiconjugate to T_Z in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0.

This article describes a method — called here "the method of Stretching Along the Paths" (SAP) — to prove the existence of chaotic sets in discrete-time dynamical systems. The method of SAP, although mathematically rigorous, is based on some elementary geometrical considerations and is relatively easy to apply to models arising in applications. The paper provides a description of the basic mathematical ideas behind the method, as well as three applications to economic models. Incidentally, the paper also discusses some questions concerning the definition of chaos and some problems arising from economic models in which the dynamics are defined only implicitly.

Ordinal patterns bring in the dynamic novel aspects, difficult questions, and useful tools for entropy estimation, time series analysis and other applications. In [Amigó et al., 2008a], the structure of the ordinal patterns of the one- and two-sided shifts on N symbols was elucidated with great detail. The results are applied via order isomorphy to the ordinal structure of the sawtooth maps x ↦ Nx mod 1 and the baker map. In this paper, we generalize the technique employed there, to cope with the so-called signed shifts on N symbols and signed sawtooth maps.

By using a time-domain approach, this paper analyzes the chaotic dynamics of the fractional-order system obtained by coupling two fractional Lorenz systems. The work exploits the Adomian decomposition method (ADM), shows that the proposed fractional system with order as low as 2.88 exhibits a novel four-wing chaotic attractor. The existence of chaos is also confirmed by the application of the "0-1 test" for chaos. Finally, an analysis of the system equilibria is conducted, which shows that the fractional-order four-wing attractor is truly the counterpart of the recently introduced integer-order four-wing attractor.

We discuss a system with a discontinuity moments generator to obtain the solutions. It is proven that the system is chaotic and has shadowing property [Anosov, 1967; Bowen, 1975; Hammel et al., 1987; Robinson, 1995] if the generator possesses these characteristics. Illustrative examples are provided.

Both classical Fourier analysis and continuous wavelets transformation are applied to study non-linear vibrations of infinitely long flexible panels subject to longitudinal sign-changeable external load actions. First the governing PDEs are derived and then the Bubnov–Galerkin method is applied to yield 2N first order ODEs. The further used Lyapunov exponent computation is described. Transition scenarios from regular to chaotic dynamics of the being investigated plate strip are analyzed using different wavelets, and their suitability and advantages/disadvantages to nonlinear dynamics monitoring and quantifying are illustrated and discussed. A few novel results devoted to the beam nonlinear dynamics behavior are reported. In addition, links between the largest Lyapunov exponent computation and the wavelet spectrum numerical estimation are also illustrated and discussed.

In the current paper, a class of stochastic cellular neural networks with reaction–diffusion effects, both discrete and distributed time delays, is studied. Several sufficient conditions guaranteeing the almost sure and pth moment exponential stability of its equilibrium solution are respectively obtained through analytic methods such as employing Lyapunov functional, applying Itô's formula, inequality techniques, embedding in Banach space, Matrix analysis and semimartingale convergence theorem. The yielded conclusions, which are independent of diffusion terms and delays, assume much less restrictions on activation functions and interconnection weights, and can be applied within a broader range of neural networks. Moreover, through the obtained results, it could be noted that noise will affect the exponential stability of the system. For illustration, two examples are given to show the feasibility of results.

The exponential stability of a class of stochastic fuzzy cellular neural networks with distributed delays is investigated in this paper. By using analytic methods such as Lyapunov functional, Itô's formula, inequality techniques and non-negative semimartingale convergence theorem, the sufficient conditions guaranteeing the almost sure and mean square exponential stability of its equilibrium solution are respectively obtained. For illustration, an example is given to show the feasibility of results.

In this paper, the global exponential stability of Chua's reaction–diffusion CNN system is investigated. For this system, some sufficient conditions ensuring the existence and global exponential stability of the equilibrium point is derived by using homeomorphism mapping, the property of coefficient matrix and analytical techniques. Finally, three illustrative examples are given to show the effectiveness of our results.

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chua's robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

In this paper, a modulated optical injection system is proposed. It mainly consists of a semiconductor laser subject to a modulated optical injection. It is known that optical injection will force a laser intensity into high-speed periodic oscillation without modulation, while the oscillation will be locked with weak external modulation. However, the route to chaos in such a system is observed and reported for the first time. It is shown that the generated oscillation frequency shifts away from the external frequency when the modulation amplitude becomes sufficiently large. Due to the nonlinearities of the laser, the frequencies are mixed inside the laser cavity, which causes the dynamics to follow a quasi-periodic route to chaos. In addition, the resultant chaotic signal possesses a larger bandwidth than that attainable with constant optical injection alone. Since the system is capable of generating both narrowband photonic microwave signals with low phase variances and chaotic signals with large bandwidths, it is practically important for Doppler and chaotic lidar applications.

In this paper, we study the localization problem of compact invariant sets of the system modeling the dynamics of acoustic gravity waves constructed by Stenflo (the Lorenz–Stenflo system). We discuss relations between compact localization domains and trapping domains with the help of extended invariance principle due to Rodrigues et al. Based on this analysis, we compute the trapping domain for the system modeling the dynamics of acoustic gravity waves. By using the first order extremum conditions and a comparison with localization results obtained earlier for the Lorenz system we find that all compact invariant sets of the Lorenz–Stenflo system are located in the intersection of one-parameter set of ellipsoids with a few domains bounded by some quadratic surfaces. Further, we derive polytopic bounds for the locus of compact invariant sets. Finally, we present the general formulae validating the application of rational localizing functions and use these formulae for the Lorenz–Stenflo system and for the Lorenz system.

During the winter 1679, R. Hooke challenged I. Newton to predict the dynamics of an object submitted to a constant radial force. This correspondence made a strong impact on I. Newton, who wrote four years later "De Motu", the real ancestor of "The Principia", published in 1687. R. Hooke's problem can be physically linked to the dynamics of a sphere sliding on an inverted cone due to gravitational effects. If the symmetry axis of the cone is parallel to the gravitational field, the ball executes stable precessions. Breaking this symmetry induces the appearance of chaotic motions. After having derived the equations related to the position of the sphere, we analyze its dynamics, and we perform an approximated Floquet analysis that is compared to our numerical results.

A short review is given of recent papers on the relaxation to (incompressible) absolute equilibrium. A new algorithm to construct absolute equilibrium of spectrally truncated compressible flows is described. The algorithm uses stochastic processes based on the Clebsch representation of the velocity field to generate density and velocity fields that follow by construction the absolute equilibrium stationary probability. The new method is shown to reproduce the well-known Gaussian results in the incompressible limit. The irrotational compressible absolute equilibrium case is characterized and the distribution is shown to be non-Gaussian. The high-temperature compressible spectra are found not to obey k² scaling. Finally, oscillating behavior in constant-pressure variable-temperature relaxation is obtained, suggesting the presence of second sound.

We study the effect of external noise on the catalytic oxidation of CO on an Iridium(111) single crystal under ultrahigh vacuum conditions. This reaction can be considered as a model of catalysis used in industry. In the absence of noise the reaction exhibits one or two stable stationary states, depending on control parameters such as temperature and partial pressures. When noise is added, for instance, by randomly varying the quality of the influx mixture, the system exhibits stochastic reaction rate and switching. In this work, we present two approaches: one for the monostable regime, and another for the bistable situation that relies on a white noise approximation. Both approaches rest on the assumption that spatial patterns of coverage on the Iridium plate can be neglected on a first approximation. Using mathematical models, it is possible to reconstruct stationary probability distribution functions that match experimental observations and provide support for the existence of a thermodynamic potential.

We describe how, departing from the Shannon entropy, it is possible to deal with semiquantum time-independent nonlinear Hamiltonians. The interplay between the quantal and classical degrees of freedom can be easily seen, and the set of differential equations that govern the temporal evolution of the quantal mean values and the classical variables is obtained. We find invariants of motion and, particularly, we describe under which conditions, the uncertainty principle remains as an invariant of motion too. Through the analysis of these invariants, it is possible to follow the transition of the system from quantum to classical regime and to conclude that the uncertainty principle behaves as an indicator telling us whether the system is in regular or irregular regime. A simple example is shown.

We study by means of the Landau–Lifshitz equation both the dynamical and the thermodynamical equilibrium properties of the interacting anisotropic magnetic particles on a chain in the presence of an applied magnetic field. The interaction between the magnetic particles is produced by an exchange coupling between first neighbors with periodic boundary conditions. We find that the total magnetization modulus can be a fluctuating function of time, since the system under study contains anisotropic terms. We obtain an analytical expression for the configurational temperature and an approximate formula for the case of a large number of particles. When the direction of the external field is different to the easy axes, the configurational temperature presents an oscillatory behavior as a function of time. Finally, by using the time average of the temperature, we calculate both the energy and the magnetization along the external field as a function of temperature, recovering the equilibrium thermodynamics results.

The purpose of this paper is to analyze in some detail the arguably simplest case of diversity-induced resonance: that of a system of globally-coupled linear oscillators subjected to a periodic forcing. Diversity appears as the parameters characterizing each oscillator, namely its mass, internal frequency and damping coefficient are drawn from a probability distribution. The main ingredients for the diversity-induced-resonance phenomenon are present in this system as the oscillators display a variability in the individual responses but are induced, by the coupling, to synchronize their responses. A steady-state solution for this model is obtained. We also determine the conditions under which it is possible to find a resonance effect.

A particular type of localized structure in a prototypical model for population dynamics interaction is studied. The model considers cooperative and competitive interaction among the individuals. Interaction at distance (or nonlocal interaction) and a simple random walk for the motion of the individuals are included. The system exhibits the formation of a periodic cellular pattern in some region of its parameter space. Inside this parameter region, it is possible to observe the localization of a single cell from the cellular pattern into an unpopulated background. The stability of this localized structure is discussed, as well as the destabilization process that gives rise to its own self-replication, inducing the propagation of the cellular pattern. The long distance interaction between these localized structures is also studied which results in a mutual repulsion.

Intermittent search processes alternate between two different stochastic motions in order to reach a given target. If the faster motion has a lower probability to detect the target, a question arises concerning the efficiency of both processes, and it may be possible to minimize the search time by a convenient choice of the parameters. This argument has been used to interpret observations in molecular biology or to explain the behavior of animals when searching for food. It can also have interesting consequences for the kinetics of reactions in heterogeneous media. In particular, the reaction kinetics in a biological cell can be enhanced when the active molecules occasionally bind to molecular motors that inactivate their reactivity and carry them far away. Here, we present a synthesis of the recent results obtained on these topics, with new perspectives and possible applications of intermittent behavior in reaction kinetics to be soon developed.

Parametric instability of quasi-reversal system — i.e. time reversible systems perturbed with injection and dissipation of energy — is studied in a unified manner. We infer and characterize an adequate amplitude equation, which is the parametrically driven damped nonlinear Schrödinger equation, corrected with higher order terms. This model exhibits rich dynamical behavior which are lost in the parametrically driven damped nonlinear Schrödinger equation such as: uniform states, fronts and coherent states. The dynamical behavior of a simple parametrically driven system, the vertically driven chain of pendula, exhibits quite good agreement with the amended amplitude equation.

Underground mining has become a viable alternative method for ore extraction when the operation cost of an open pit mine becomes too high. Its success is based on a good characterization of the rocks flow inside the cave. Kinematic and plasticity models, used in the granular flow description in hoppers, have been adapted to describe the experimental situations aimed at sketching the mine operation. The local packing changes that occur at the early stages of rock motion that significantly affect the flow are considered by introducing a dilation front propagating upward. The dynamics of the dilation front limits the extent of the flow generated by the drawing process allowing the assessment of the loosening zone. Based on the linear character of the kinematic model, it is shown that the loosening zone resulting from the simultaneous extraction at two adjacent draw points is accurately described by the dilation front approach.

We use a family of stationary solutions of the Cahn–Hilliard dynamics with the aim of describing the process of coarsening during a first order phase transition. With this analytical ansatz, we can compute the characteristic time for one step of period doubling in Langer's self-similar scenario for Ostwald ripening. As an application, the same ansatz is also used to compute the thermodynamically stable period of a 1D modulated phase pattern, described by a modified Cahn–Hilliard dynamics with a long range interaction term.

We present an experimental study on the local dynamics of parametrically excited waves at an air–water interface when defects are present in the wave pattern. The probability density function (PDF) of the local wave amplitude displays an exponential part for values close to the average amplitude and decreases sharply to zero for large amplitudes. The power spectral density (PSD) of the local amplitude fluctuations shows a power-law behavior over one decade which we relate to a regime of defect-mediated turbulence.

Dislocations in a material will, when present in enough numbers, change the speed of propagation of elastic waves. Consequently, two material samples, differing only in dislocation density, will have different elastic constants, a quantity that can be measured using Resonant Ultrasound Spectroscopy. Measurements of this effect on aluminum samples are reported. They compare well with the predictions of the theory.