Nonsteady condensation and evaporation waves

Recent scaling results for the ac conductivity of ionic glasses by Roling et al. [Phys. Rev. Lett. 78, 2160 (1997)] and Sidebottom [Phys. Rev. Lett. 82, 3653 (1999)] are discussed. We prove that Sidebottom's version of scaling is completely general. A new approximation to the universal ac conductivity arising in the extreme disorder limit of the symmetric hopping model, the "diffusion cluster approximation," is presented and compared to computer simulations and experiments.     

Topology in dynamical systems

Previously the torsion number and the relative torsion number have been introduced to describe the topological character of periodic solutions of a three-dimensional ordinary differential equation. In this letter, considering the rewinding mechanism of tangent vectors, we investigate the change of the torsion numbers and the relative torsion numbers as period-doubling bifurcation cascades.     

Dispersion-induced dynamical transition in parametric solitary waves

We investigate the influence of dispersion on parametric solitary waves. We show that dispersion is responsible for a transition towards a new type of dynamical solitary wave characterized by the presence of traveling phase defect arrays within their envelopes. The transition is described analytically through an original extension of the Kolmogorov-Petrovskii-Piskunov approach to front propagation into unstable states.     

Robust feed-back control of travelling waves in a class of reaction-diffusion distributed biological systems

Reaction-Diffusion (RD) mechanisms can describe many biological phenomena such as neuron firing in the brain, the heartbeat, cellular organization activities or even biological disorders such as fibrillation. The FitzHugh-Nagumo (FHN) model is a particular case of RD systems. It is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years. Some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena, related to a travelling plane wave, as well as biological disorders associated with spiral or irregular fronts. The objective of this work is to design a control law that is able to stabilize complex behaviors (travelling plane wave) in biological systems using the FHN model as a case study. Since, in biological systems there usually exists a lack of detailed information on the system structure, our control law will be designed to be robust, i.e., it must be able to reach the predefined reference regardless the presence of structural uncertainties. To this purpose, we will extend some classical results on the finite-dimensional robust control theory to RD systems by means of order reduction techniques, in particular the Proper Orthogonal Decomposition method.     

Non-Hamiltonian dynamical systems

The class of dynamical systems is considered, which are described by several mutually noncommuting Hamiltonian currents, in particular, relativistic bi-Hamiltonian systems, the evolution of which is described by a pair of 4-momenta p and p The examination is conducted in classical and quantum realizations. The evolution equations are derived of relativistic bi-Hamiltonian systems in the Heisenberg and Schrödinger pictures. It is shown that the quantum theory of relativistic bi-Hamiltonian systems is not compatible with the unitary condition and is nonunitary. A physical interpretation is given of nonunitary quantum theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 5–12, October, 1990.     

Resynchronizing dynamical systems

Resynchronizing dynamical systems are important for certain chaotic signal masking methods. We demonstrate the dependence of the resynchronizing property of linear dynamical systems on choices of coordinate systems. Using these insights, we demonstrate how a nonlinear system not previously known to be synchronizable can be used for chaotic signal masking.     

Nonlinear standing waves,resonance phenomena,and frequency characteristics of distributed systems

This review is dedicated to resonator oscillations under conditions of a strongly expressed nonlinearity under which steep shock fronts emerge in the wave profiles. Models and approximated methods for their analysis for quadratic and cubic nonlinear media are examined, as well as for nonlinearity when taking into account the mobility of boundaries. The forms of the profiles are calculated both for a steady-state oscillation regime and during the establishment of the profiles. Dissipative losses and selective losses at specially chosen frequencies are considered. An analysis of nonlinear Q-factor is given. The possibility of increasing the acoustic energy accumulated in the cavity of the resonator is discussed. Special attention is given to various physical phenomena that are exhibited only in nonlinear acoustic fields.     

Recurrent synchronism in dynamical systems

A mechanism which shows recurrent synchronism between some pair(s) of variables of flows and maps is studied. This mechanism is based on the nonuniform influence of bursts on coupled oscillators and the subsequent oscillators' tendency to synchronize. We discuss this mechanism in representative physical models.     

Finite-time blow-up in dynamical systems

A new method to detect finite-time blow-up in systems of ordinary differential equations is presented. This simple algorithmic procedure is based on the analysis of singularities in complex time and amounts to checking the real-valuedness of the leading order term in the asymptotic series describing the behavior of the general solution around movable singularities. Illustrative examples and an application to a magneto-hydrodynamic problem are given.     

Intermittency in dissipative dynamical systems

We show that the intermittency displayed by a differential system proposed by Yamada and Fujisaka can be interpreted in the general framework of intermittent transitions to turbulence studied by Pomeau and the author.     

Exponent inequalities in dynamical systems

In this Letter, we derive exponent inequalities relating the dynamic exponent z to the steady state exponent Γ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating z and Γ. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems, are discussed.     

Time inversion in dynamical systems

We consider the case of a dynamical system when the time evolution is generated by a nonhermitian superoperator on the states of the system. Assuming the left and right eigenvectors of this to provide complete basis sets, we propose a generalized scalar product which can be used to construct a monotonically changing functional of the state, a generalized entropy. Combining the time-dependent state with its time-reversed counterpart we can define the operation of time inversion even in this case of irreversible evolution. We require that both the forward and reversed time evolution can be obtained from a generalized action principle, and this demand serves to define the form of the time-reversed state uniquely. The work thus generalizes the quantum treatment from the unitary case to the irreversible one. We present a discussion of the approach and derive some of the direct consequences of our results.     

Iterated function systems and dynamical systems

We study the relationship between measures invariant for a piecewise expanding transformation tau of a compact metric space endowed with a underlying measure and measures invariant for an iterated function system T(tau), generated by inverse branches of tau. The main result says that the tau-invariant absolutely continuous measure &mgr; is also T(tau) invariant if and only if tau is absolutely continuously conjugated with a piecewise linear transformation. Measures of maximal entropy and general equilibrium states are also discussed. (c) 1995 American Institute of Physics.     

New solvable dynamical systems

New solvable dynamical systems are identified and the properties of their solutions are tersely discussed.     

Punctures and dynamical systems

Letters in Mathematical Physics - With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of...