International Journal of Bifurcation and Chaos

We first summarize features of free, forced and stochastic harmonic oscillations and, following an idea first proposed by Lord Rayleigh in 1883, we discuss the possibility of maintaining them in the presence of dissipation. We describe how phonons appear in a harmonic (linear) lattice and then use the Toda exponential interaction to illustrate solitonic excitations (cnoidal waves) in a one-dimensional nonlinear lattice. We discuss properties such as specific heat (at constant length/volume) and the dynamic structure factor, both over a broad range of temperature values. By considering the interacting Toda particles to be Brownian units capable of pumping energy from a surrounding heat bath taken as a reservoir we show that solitons can be excited and sustained in the presence of dissipation. Thus the original Toda lattice is converted into an active lattice using Lord Rayleigh's method. Finally, by endowing the Toda–Brownian particles with electric charge (i.e. making them positive ions) and adding free electrons to the system we study the electric currents that arise. We show that, following instability of the base linear Ohm(Drude) conduction state, the active electric Toda lattice is able to maintain a form of high-T supercurrent, whose characteristics we then discuss.

This paper reviews the important role of resonances in the structure of planetary systems. After a short introduction to the basics of orbital dynamics of motion in resonances we describe the dynamics of our planetary systems and also of extrasolar planetary systems, where up to now more than 100 are known. In our planetary system the planets move in quite regular orbits with small eccentricities although it was found that the motion of the inner planets is "slightly" chaotic on time scales of tenths of millions of years. The quasi regularity (close to so-called quasi-periodic motion on a torus) is not true for the small bodies: the main belt of asteroids between Mars and Jupiter with gaps for special values of semimajor axes on one hand and with families of many small bodies on the other, is sculpted due to the presence of first mean motion resonances with Jupiter and second secular resonances with long-periodic motions of the nodes and perihelia of Jupiter and Saturn. In extrasolar systems the planets — rather surprisingly — are found to move sometimes in very high eccentric orbits when they are at distances comparable to the size of our planets. Because of our still limited observational techniques using indirect methods we have only discovered massive planets comparable to the size of Jupiter. When these planets orbit alone around their host star our research aims at the possibility of additional terrestrial planets moving in such a system. Because of mostly large eccentricities here the resonances are, in contrary to our planets, essential for the stability of orbits, and may protect or destroy an orbit. On the other hand, in multiple planetary systems we concentrate on the stability of their orbits as they are observed: a very interesting new result is that most of these multiple planetary systems with high eccentric orbits move in resonances with a special configuration which protects them from close encounters although these orbits are crossing.

This work provides a description of the main computational tools for the study of discrete breathers. It starts with the observation of breathers through simple numerical runs, the study uses targeted initial conditions, and discrete breather impact on transient processes and thermal equilibrium. We briefly describe a set of numerical methods to obtain breathers up to machine precision. In the final part of this work we apply the discussed methods to study the competing length scales for breathers with purely anharmonic interactions — favoring superexponential localization — and long range interactions, which favor algebraic decay in space. As a result, we observe and explain the presence of three different spatial tail characteristics of the considered localized excitations.

The imaging techniques of magnetoencephalography (MEG), computed tomography (CT), positron emission tomography (PET) and single photon emission computed tomography (SPECT) are reviewed, and the analytical solutions of the relevant inverse problems are presented. The numerical implementation of the exact formulas yield accurate reconstructions for both realistic phantoms as well as real data.

The relevance of self-organization, nonlinear phenomena and nonequilibrium behavior in a wide range of problems in space science and microgravity research, calls for a concerted approach using the tools of statistical physics, thermodynamics, nonlinear dynamics, mathematical modeling and numerical simulation, in synergy with experimentally oriented work.

The reason behind the ubiquity of such concepts is that in many instances of relevance in space science one witnesses an interplay between molecular and macroscopic-level entities and processes, such as nucleation in phase transitions or aggregation and self-assembly in materials science and biology. These phenomena are fundamentally dynamic in nature. We present two minimal models one dealing with issues of complex behavior at the microscopic level and the second referring to the strong nonlinear nature of macroscopic evolution, relevant to soft-condensed matter under observation both on the ground and in microgravity conditions.

These investigations open the way to a whole class of unexpected situations whose proper characterization and control can only be expected in the framework of nonequilibrium physics and complexity theory.

We consider noise-induced charge density dynamics in a semiconductor superlattice. The parameters are fixed in the regime below the Hopf bifurcation that gives birth to spatio-temporal oscillations, where in the absence of noise of the system rests at a fixed point. It is shown that in this case noise can induce in the superlattice quite coherent oscillations of the current through the device. While the regularity of these oscillations depends on the noise intensity, their dominant frequency remains almost constant with variation of the noise level in the system. Further, we demonstrate that a time-delayed feedback scheme that was previously used to control purely temporal oscillations induced by noise, can not only enhance or deteriorate the regularity of stochastic spatio-temporal patterns but also allow for the manipulation of the system's time scales with varying time delay.

Ensembles of charged particles (plasmas) are a highly complex form of matter, most often modeled as a many-body system characterized by weak inter-particle interactions (electrostatic coupling). However, strongly-coupled plasma configurations have recently been produced in laboratory, either by creating ultra-cold plasmas confined in a trap or by manipulating dusty plasmas in discharge experiments. In this paper, the nonlinear aspects involved in the motion of charged dust grains in a one-dimensional plasma monolayer (crystal) are discussed. Different types of collective excitations are reviewed, and characteristics and conditions for their occurrence in dusty plasma crystals are discussed, in a quasi-continuum approximation. Dust crystals are shown to support nonlinear kink-shaped supersonic solitary longitudinal excitations, as well as modulated envelope localized modes associated with longitudinal and transverse vibrations. Furthermore, the possibility for intrinsic localized modes (ILMs) — Discrete Breathers (DBs) — to occur is investigated, from first principles. The effect of mode-coupling is also briefly considered. The relation to previous results on atomic chains, and also to experimental results on strongly-coupled dust layers in gas discharge plasmas, is briefly discussed.

Many natural and artificial systems whose range of interaction is long enough are known to exhibit (quasi)stationary states that defy the standard, Boltzmann–Gibbs statistical mechanical prescriptions. For handling such anomalous systems (or at least some classes of them), nonextensive statistical mechanics has been proposed based on the entropy , with (Boltzmann–Gibbs entropy). Special collective correlations can be mathematically constructed such that the strictly additive entropy is now S_q for an adequate value of q ≠ 1, whereas Boltzmann–Gibbs entropy is nonadditive. Since important classes of systems exist for which the strict additivity of Boltzmann–Gibbs entropy is replaced by asymptotic additivity (i.e. extensivity), a variety of classes are expected to exist for which the strict additivity of S_q (q ≠ 1) is similarly replaced by asymptotic additivity (i.e. extensivity). All probabilistically well defined systems whose adequate entropy is S₁ are called extensive (or normal). They correspond to a number W^eff of effectively occupied states which grows exponentially with the number N of elements (or subsystems). Those whose adequate entropy is S_q (q ≠ 1) are currently called nonextensive (or anomalous). They correspond to W^eff growing like a power of N. To illustrate this scenario, recently addressed [Tsallis, 2004] we provide in this paper details about systems composed by N = 2, 3 two-state subsystems.

A circular array of optical waveguides collectively coupled with a central core is investigated. Nonlinear losses, both linear and nonlinear coupling as well as energy transfer between neighboring array elements and between the array and the core are allowed. The concept is ideal for the design of high power stable amplifiers as well as of all-optical data processing devices in optical communications. The existence of stable steady-state continuous wave modes as well as of localized solitary and breathing type modes is demonstrated. These properties render the proposed system functionally rich, far more controllable than a planar one and easier to stabilize.

Solitary wave propagation under interaction with continuous waves is studied in the context of the Nonlinear Schrödinger Equation. An analytical approach, based on the conserved quantities of the wave evolution, is used to study transverse velocity variations for the case of nonzero transverse wavenumber difference between the solitary and continuous waves. The method is applicable for any number of transverse dimensions and any kind of nonlinearity. Moreover, the presence of a coherent continuous background is shown to be responsible for the creation of solitary rings and spirals, under interaction with solitary structures with nonzero topological charge. Numerical simulations for specific cases were used to confirm the analytical results.

Duffing-type electrical oscillator is a second-order nonlinear electric circuit driven by a sinusoidal voltage source. The nonlinear element is a nonlinear inductor. We have studied the dynamics of two resistively coupled oscillators of this type in two cases. The first, when the oscillators are identical having chaotic dynamics, and the second, when the oscillators are in different dynamic states (periodic and chaotic, respectively). In the first case, chaotic synchronization is observed, while in the second case control of the chaotic behavior is achieved.

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi–Pasta–Ulam (FPU) model and a discretized version of the nonlinear Schrödinger equation related to Bose–Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents L_i, i = 1,…, N - 1, exhibit a transition between two power laws, L_i ∝ E^B_k, B_k > 0, k = 1, 2, occurring at the same value of E. The destabilization energy E_c per particle goes to zero as N → ∞ following a simple power-law, E_c/N ∝ N^-α, with α being 1 or 2 for the cases we studied. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Furthermore, the IPM orbit of the BEC Hamiltonian never destabilizes, having finite-size islands around it, even for very high N and E. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov–Sinai entropies per particle h_KS/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

Normally, conservative systems do not have attractors. However, in a system with escapes, the infinity acts as an attractor. Furthermore, attractors may appear as singularities at a finite distance. We consider the basins of escape in a particular Hamiltonian system with escapes and the rates of escape for various values of the parameters. Then we consider the basins of attraction of a system of two fixed black holes, with particular emphasis on the asymptotic curves of its unstable periodic orbits.

In the planar photogravitational Hill problem (PHP) three equivalent forms of the equations of motion are presented: the physical equations of motion and two more forms of the regularized equations of the problem. Using the Levi–Civita coordinate transformation as well as the corresponding time transformation, we first obtain a simple regularized polynomial Hamiltonian of the dynamical system, and second the regularized equations of motion without using a regularized Hamiltonian. The relation between the physical and the regularized variables are also given. Zero velocity curves and surfaces, Poincaré sections, families of symmetric periodic orbits as well as their stability are given in physical and regularized variables when the larger primary body (the sun) does not radiate and when it does. It is found that for large deviations of its value from the gravitational case, the radiation factor can have a strong effect on the structure of the families. Finally we present the stability regions in both cases with and without radiation using the maximal Lyapunov exponent (LE), working in the regularized equations of motion of the problem since through these the calculation of the stability is more rapid and accurate than using the physical equations of motion.

In the present article we present general linear stability criteria for multibreathers in Klein–Gordon chains at low coupling. Our method is constructive, which means that provided a specific potential the method provides the allowed multibreathers and their characteristic exponent to leading order of approximation.

In the present paper we investigate the nonintegrability of adiabatic one degree of freedom Hamiltonian systems, with the additional assumption that the frozen system possesses an unstable fixed point with two asymmetric homoclinic loops. We prove a criterion for the nonexistence of an integral for such systems, and therefore we prove the nonexistence of a quantity which is conserved in an arbitrarily high order on ε. A specific application is given in the asymmetric quartic oscillator with adiabatic time dependence.

We report the first direct observation of lasing action from a dynamically localized mode in a microdisk resonator with rough boundary. In contrast to microlasers based on stable ray trajectories, the performance of our device is robust with respect to the boundary roughness and corresponding ray chaos, taking advantage of Anderson localization in angular momentum. The resonator design, although demonstrated here in GaAs-InAs microdisk laser, should be applicable to any lasers and sensors based on semiconductor or polymer materials.

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel [1871–1956].

First I derive the power spectrum (of the cumulative or integrated level density) for the regular (Poissonian) energy spectrum which is 1/t², and for the fully chaotic one, which is 1/t, as was observed numerically by Relano et al. [2002], and recently discussed independently by Faleiro et al. [2004]. The statement refers to small values of t ≤ 1, i.e. for times smaller than Heisenberg time. For t ≫ 1 it is always 1/t². Then I analyze the autocorrelation function for spectrum which is a superposition of statistically independent spectral sequences and derive the exact additivity formula for the autocorrelation function, and consequently for the form factor of the density of energy levels and of the cumulative (integrated) density of energy levels (the form factor is the Fourier transform of the corresponding autocorrelation function). Therefore this theory provides prediction for the deep semiclassical regime of sufficiently small effective Planck constant ℏ_eff. However, in not sufficiently deep semiclassical regime (not sufficiently small ℏ_eff) we see a deviation from this behavior, as recently demonstrated numerically by Gomez et al. [2004] in a billiard system [Robnik, 1983] where a power law behavior 1/t^α is found, and the exponent α goes continuously from 2 to 1, as the system's dynamics goes from complete integrability (Poisson) to full chaoticity (GOE and GUE), respectively.

In this paper we formulate a delayed predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other and the growth rate of the predator depends on the prey that was available in the past. If the equilibrium point lies in the Allée effect zone and when the diffusion is present only, we show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, patterns emerge, the spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. When the delay is present only, the increase of delay destabilizes the system and causes the occurrence of periodic oscillations, Andronov–Hopf bifurcation. For the full general model (with both diffusion and delay) if the bifurcation parameters are increased through critical values of diffusion and delay the two new spatially nonconstant stationary solutions lose their stability by Hopf bifurcation.

We present a method that uses distances between nearest neighbors in Takens space to evaluate a level of noise. The method is valid even for high noise levels. The method has been verified by estimation of noise levels in several chaotic systems. We have analyzed the noise level for Dow Jones and DAX indexes and we have found that the noise level ranges from 25 to 80% of the signal variance.