Jaime Rössler

Random chains and complex transfer matrix attractors

Abstract The physical properties of one-dimensional disordered systems are re-obtained with a method which is a generalization of work by Schmidt. This generalization is carried out reformulating Schmidts procedure in the language of non-linear processes and extending it to the complex plane. The average Greens functions are evaluated in this way and from them the density of eigenstates for each atomic species. The coherence length of these eigenstates is also obtained and related to the Lyapunov exponent of the non-linear process mentioned above. Illustrative numerical results are provided.

Populations under Periodically and Randomly Varying Growth Conditions

A large number of natural populations result from single generations that do not overlap, so that population growth occurs in discrete steps. The growth of a single species can then be described by an equation of the type (1) where we consider the time lapse between two generations as time unity. Examples of this type of poulation are many temperate zone arthropod species with one short-lived adult generation per year [15], bivoltine insects (i.e. insects having a summer and a winter generation [81) and cicadas with adults emerging every 13 years [13].

Lineair Chain with Free End Boundary Conditions

The problem of energy propagation through a one-dimensional open-ended finite chain is critically re-examined. Free end boundary conditions are imposed and quantum-field-theory operators, consistent with these boundary conditions, are derived for both the local and total energy current operators, avoiding the shortcomings of previous work reported on the subject. The formal treatment is illustrated through an example, and a comparison with results obtained with cyclic boundary conditions is given.

Ecosystems Under Varying Ambient Conditions

Ecosystems governed by random dynamic non-linear equations are studied in detail. The process is precisely characterized, both for the non-markovian and the markovian case, including short range order. A new feature, the appearance of early chaos, is introduced and analyzed. The Lyapunov exponent is defined and evaluated. Its precise meaning is discussed and its significance to characterize the dynamics of a random succession of events is investigated and clarified.

Arnold Tongues In A Periodically Perturbed Logistic Oscillator

A logistic oscillator, perturbed by a periodic force of amplitude D and frequency ω, is studied. We obtain the condition for phase locking between the nonlinear oscillator, of frequency ω 0, and the external perturbation. The locked—unlocked boundary, D = D L (ω), has a remarkable structure, with “V-shaped” minima appearing for every rational value of ω/ω 0.

Electron-lattice interaction in Sm1-xYxS-like systems

Abstract An Anderson-like Hamiltonian, which describes a cluster of a rare earth (Sm or Y) cation surrounded by six S anions, is used to model the electron-lattice interaction in mixed-valence systems. Coupling between the electronic and phononic variables is introduced, and two different phonon modes are considered: a breathing and an asymmetric one. The first, related to the ionic radius, is treated exactly. The asymmetric mode, which determines the sd-f hybridization, is dealt with in the Born-Oppenheimer approximation. Experimental results are adequately accounted for by this simple model.

EFFECT OF A RESONANT PERIODIC PERTURBATION ON NONLINEAR PROCESSES

The effect of a periodic modulation on the bifurcation parameter of the logistic map is investigated. Attention is limited to the case where the modulation parameter A t always lies within the stability range of a P -furcation of the unperturbed logistic map. In particular, we focus our interest in the near resonance behavior, i. e. when the external modulation frequency ω is close to the “natural” logistic oscillator frequency, ω 0 = 1/ P. The study of the Lyapunov exponent shows that, above a critical modulation amplitude, a sharply delimited frequency interval exists around ω 0 in which the external modulation has a resonant effect, inducing a qualitative modification of the dynamics, even when the modulation amplitude is relatively small. It consists in locking the phase of the external perturbation to our nonlinear logistic oscillator.

Periodic Modulation of a Logistic Oscillator

We study the effect of a periodic modulation on the bifurcation parameter of the logistic oscillator. Remarkable results are obtained, including early chaos, resonance phenomena and the disappearance of period—doubling bifurcations. Also, long term unpredictable dynamics appears fornegativevalues of the Lyapunov exponent; the latter occurs when a resonant modulation is applied within the period-3 window.

Structural instability in a boson system

Abstract We have found a boson system (libron interacting with phonons in a lattice) which is unstable in such a way that at high temperature (H.T.) it becomes deformed while at low Temperatures is homogeneous. The stable periodicity at H.T. occurs at a finite wavelength. This a first order phase transition.

Periodically and Randomly Modulated Non Linear Processes

Many natural phenomena are governed by nonlinear recursive relations of the type x t +1 = f(x t ), where f does depend on t. We focus our interest on the particularly simple case x t +1 = r t x t (1-x t ), where r t adopts either periodically or at random, the values A and B. Graphical representations of the Lyapunov exponent on the AB-plane show unexpected features, like self-similarity and early chaos (i.e. chaos for very low parameter values). The latter constitutes a novel mechanism to induce chaotic behavior. The meaning of the Lyapunov exponent for random processes is examined.