Influence of dynamical noise on time series generated by nonlinear maps

We consider periodic and chaotic dynamics of discrete nonlinear maps in the presence of dynamical noise. We show that dynamical noise corrupting dynamics of a nonlinear map may be considered as a measurement “pseudonoise” with the distribution determined by the Jacobian of the map. The formula for the distribution of the measurement “pseudonoise” for one-dimensional quadratic maps has also been obtained in an explicit form. We expect that our results apply to an arbitrary distribution of low-level dynamical noise and hope that these results could help to find a universal method of discriminating dynamical from measurement noise.     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

On ergodic and mixing properties of the triangle map

In this paper, we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both relying on the fact that it maps polygons to polygons: in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons, in the second we consider the standard binary partition of the torus induced by the discontinuity set. These encoding schemes are studied in detail and shown to be compatible, although not equivalent. The ergodic properties of the triangle map are then investigated in terms of the Markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the Koopman operator in L²(T²). Finally, a stochastic version of the triangle map is introduced and studied. A simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map.     

Quantum NOT operation and integrability in two-level systems

We demonstrate the surprising integrability of the classical Hamiltonian associated to a spin 1/2 system under periodic external fields. The one-qubit rotations generated by the dynamical evolution is, on the one hand, close to that of the rotating wave approximation (RWA), on the other hand to two different “average” systems, according to whether a certain parameter is small or large. Of particular independent interest is the fact that both the RWA and the averaging theorem are seen to hold well beyond their expected region of validity. Finally, we determine conditions for the realization of the quantum NOT operation by means of classical stroboscopic maps.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Local and global statistical properties of deterministic chaos

Summary Locla and global statistical properties of a class of one-dimensional dissipative chaotic maps and a class of 2-dimensional conservative hyperbolic maps are investigated. This is achieved by considering the spectral properties of the Perron-Frobenius operator (the evolution operator for probability densities) acting on two different types of function space. In the first case, the function space is piecewise analytic, and includes functions having support over local regions of phase space. In the second case, the function space essentially consists of functions which are “globally? analytic,i.e. analytic over the given systems entire phase space. Each function space defines a space of measurable functions or observables, whose statistical moments and corresponding characteristic times can be exactly determined. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.     

BEC from a time-dependent variational point of view

We use the time-dependent variational principle of Balian and Vénéroni to derive a set of equations governing the dynamics of a trapped Bose gas at finite temperature. We show that this dynamics generalizes the Gross-Pitaevskii equations in that it introduces a consistent dynamical coupling between the evolution of the condensate density, the thermal cloud, and the “anomalous” density.     

Invariants for the topological characterization of the iteration of differentiable functions – the bimodal case

We consider dynamical systems defined by a particular class of differentiable functions, as fixed state space. The dynamics is given by the iteration of an operator induced by a polynomial map which belongs to an appropriate family of isentropic bimodal interval maps. We characterize topologically these dynamical systems, in particular using the invariants defined for the iteration of the bimodal interval maps.     

Dynamics and schedule of shuttle bus controlled by traffic signal

We study the dynamical behavior of a shuttle bus moving through a traffic signal. The dynamics of the bus is expressed in terms of the nonlinear maps. The bus dynamics is controlled by varying the loading parameter, the cycle time of signal, and the degree of speedup. We show the dependence of the tour time on both loading parameter and cycle time. The fluctuation of boarding passengers is highly reduced by varying the cycle time. When the bus speeds up to retrieve the delay induced by loading the passengers, the bus behavior also changes highly. The shuttle bus schedule is connected with the complex motion of the shuttle bus. The region map (phase diagram) is shown to control the complex motion of the bus.     

Character of the map for switched dynamical systems for observations on the switching manifold

It has been proposed to obtain the discrete-time models of switching dynamical systems by observing the states at the switching instants. Apart from the lowering of dimension, such switching maps or impact maps offer advantage in modeling systems that exhibit chattering. In this Letter we derive the nature of the switching map for the special case of grazing orbits. We show that the map is discontinuous in the neighborhood of a grazing orbit, and that it has a square root slope singularity on one side of the discontinuity. We illustrate the above by obtaining the switching maps for two example systems: the Colpitt's oscillator in the electrical domain and the soft impact oscillator in the mechanical domain.     

Relaxation properties of chaotic dynamical systems: Operator approach

A method of analytically determining eigenvalues and the piecewise-continuous eigenfunction systems of the Perron-Frobenius operator for Rényi chaotic map x _n+1 = βx _n mod 1, 1 < β < 2 based on introducing the generating function for the eigenfunctions is described. These characteristics define the relaxation properties and decay of correlations in discrete dynamical systems.     

Quantum Ergodicity for Graphs Related to Interval Maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L ² functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.     

Entropy-driven phase transitions with influence of the field-dependent diffusion coefficient

We present a comprehensive study of phase transitions in a single-field reaction-diffusion stochastic systems with a field-dependent mobility of a power-law form and internal fluctuations. Using variational principles and mean-field theory we have shown that the noise can sustain spatial patterns and leads to phase transitions type of “order-disorder”. These phase transitions can be critical and non-critical in character. Our theoretical results are verified by computer simulations.     

What is the role of chaotic scattering in irreversible processes?

We study kinetic properties of simple mechanical models of deterministic diffusion like the scattering of a point particle in a billiard of Lorentz type and the multibaker area-preserving map. We show how dynamical chaos and, in particular, chaotic scattering are related to the transport property of diffusion. Moreover, we show that the Liouvillian dynamics of the multibaker map can be decomposed into the eigenmodes of diffusive relaxation associated with the Ruelle resonances of the Perron-Frobenius operator.     

Collective behavior in coupled chaotic map lattices with random perturbations

Numerical simulations of coupled map lattices with non-local interactions (i.e., the coupling of a given map occurs with all lattice sites) often involve a large computer time if the lattice size is too large. In order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution. This emulates a “thermal bath”, where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude. We demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied. We considered one-dimensional lattices of chaotic logistic maps at outer crisis x→4x(1−x).     

Expanding direction of the period doubling operator

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a Perron-Frobenius type operator, to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.     

Ulam method for the Chirikov standard map

We introduce a generalized Ulam method and apply it to symplectic dynamical maps with a divided phase space. Our extensive numerical studies based on the Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator on a chaotic component converges to a continuous limit. Typically, in this regime the spectrum of relaxation modes is characterized by a power law decay for small relaxation rates. Our numerical data show that the exponent of this decay is approximately equal to the exponent of Poincaré recurrences in such systems. The eigenmodes show links with trajectories sticking around stability islands.     

From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.