A geometrical setting for Lax equations associated to dynamical systems

We develop a geometrical framework for dealing with Lax equations associated to dynamical systems over a manifold M. We also show that this theory reproduces the global versions of Lax equations given before as well as the usual theory of reduced systems obtained from systems defined on Lie groups and with such group as a symmetry group.     

On unified field theories,dynamical torsion and geometrical models

A new model of a nondualistic unified theory is proposed. This model is absolutely consistent from the mathematical and geometrical points of view and is based on a manifold equipped with an underlying hypercomplex structure and zero nonmetricity. Also we showed that interesting wormhole solutions, similar to the non-Abelian Born-Infeld theory of our previous work [14] can be obtained. The solution of this model is explicitly compared with our previous one and we find that the torsion plays in this unified theory a role similar to that of Yang-Mills type strength field coming from the non-Abelian Born-Infeld energy momentum tensor. The meaning of the Hosoya-Ogura ansatz (namely, the alignment of the isospin with the frame geometry of the space-time) is completely elucidated.     

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.     

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.     

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...     

Anomalous diffusion and dynamical localization in polygonal billiards

We study numerically classical and quantum dynamics of a piecewise parabolic area preserving map on a cylinder which emerges from the bounce map of elongated triangular billiards. The classical map exhibits anomalous diffusion. Quantization of the same map results in a system with dynamical localization and pure point spectrum.     

Reversible destruction of dynamical localization

Dynamical localization is a localization phenomenon taking place, for example, in the quantum periodically driven kicked rotor. It is due to subtle quantum destructive interferences and is thus of intrinsic quantum origin. It has been shown that deviation from strict periodicity in the driving rapidly destroys dynamical localization. We report experimental results showing that this destruction is partially reversible when the deterministic perturbation that destroyed it is slowly reversed. We also provide an explanation for the partial character of the reversibility.     

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.     

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.     

On unified field theories,dynamical torsion and geometrical models: II

We analyze in this letter the same space-time structure as that presented in our previous reference (Part. Nucl., Lett. 2010. V. 7, No. 5, P. 299–307), but relaxing now the condition a priori of the existence of a potential for the torsion. We show through exact cosmological solutions from this model, where the geometry is Euclidean R ⊗ O ₃ ∼ R ⊗ SU(2), the relation between the space-time geometry and the structure of the gauge group. Precisely this relation is directly connected with the relation between the spin and torsion fields. The solution of this model is explicitly compared with our previous ones and we find that: (i) the torsion is not identified directly with the Yang-Mills type strength field, (ii) there exists a compatibility condition connected with the identification of the gauge group with the geometric structure of the space-time: this fact leads to the identification between derivatives of the scale factor with the components of the torsion in order to allow the Hosoya-Ogura ansatz (namely, the alignment of the isospin with the frame geometry of the space-time), and (iii) of two possible structures of the torsion the “tratorial” form (the only one studied here) forbids wormhole configurations, leading only to cosmological space-time solution in eternal expansion.     

New solvable dynamical systems

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