Family Of Dynamical Systems Research Articles

Normal linear stability of quasi-periodic tori

We consider families of dynamical systems having invariant tori that carry quasi-periodic motions. Our interest is the persistence of such tori under small, nearly-integrable perturbations. This persistence problem is studied in the dissipative, the Hamiltonian and the reversible setting, as part of a more general kam theory for classes of structure preserving dynamical systems. This concerns the parametrized kam theory as initiated by Moser [J.K. Moser, On the theory of quasiperiodic motions, SIAM Rev. 8 (2) (1966)145–172; J.K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136–176] and further developed in [G.B. Huitema, Unfoldings of quasi-periodic tori, PhD thesis, University of Groningen, 1988; H.W. Broer, G.B. Huitema, F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (421) (1990) 1–82; H.W. Broer, G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1) (1995) 191–212]. The corresponding nondegeneracy condition involves certain (trans-)versality conditions on the normal linear, leading, part at the invariant tori. We show that as a consequence, a Cantor family of Diophantine tori with positive Hausdorff measure is persistent under nearly-integrable perturbations. This result extends the above references since presently the case of multiple Floquet exponents is included. Our leading example is the normal 1 : − 1 resonance, which occurs a lot in applications, both Hamiltonian and reversible. As an illustration of this we briefly describe the Lagrange top coupled to an oscillator.

On the qualitative analysis of the solutions of a mathematical model of social dynamics

This work deals with a family of dynamical systems which were introduced in [M.L. Bertotti, M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Models Methods Appl. Sci. 7 (2004) 1061–1084], modelling the evolution of a population of interacting individuals, distinguished by their social state. The existence of certain uniform distribution equilibria is proved and the asymptotic trend is investigated.

Generalized radix representations and dynamical systems II

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

Generalized radix representations and dynamical systems. I

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

On the Stability of Families of Dynamical Systems

On the Stability of Families of Dynamical Systems

A topological analysis of a family of dynamical systems with non-standard chaotic and periodic behaviour

We examine a family of maps, which are motivated by a specific flow model for a manufacturing system, that act on polygons inscribed within an equilateral triangle. The family of maps is interesting because it provides an example of chaotic behaviour in one extreme and periodic behaviour in the other, with a non-standard transition between these extremes. We characterize the behaviour of this family of maps and describe the interesting phenomena which occur between the two extremes of chaos and periodicity. We also use symbolic dynamics to relate one of the maps to a particular subshift of finite type.

The dynamies of rolling disks and sliding disks

The dynamics of disks which move with one point in contact with a fixed horizontal surface are investigated. Two cases are considered: one where the disk rolls without slipping and a second where the disk slides. A two-parameter family of integrable second order differential equations is established for both of these systems by exploiting classical integrability results. Using these families of dynamical systems, complete descriptions of the bifurcations and stability of the steady motions of the disks are presented. One of these families is also used to establish the existence of motions which result in the rolling disk colliding with the horizontal surface. These motions occur for arbitrarily large angular velocities and in the absence of dissipation. The paper closes with some comments on non-integrability.

Universal canonical forms for time-continuous dynamical systems.

A natural embedding for most time-continuous systems is presented. A set of nonlinear transformations is shown to split the whole family of dynamical systems into equivalence classes. The degree of nonlinearity is not a relevant characteristic of these classes of ordinary differential equations (ODEs). In each class there exist two particular simple canonical forms into which any such ODE can be cast. We show on an example how to use these canonical forms to find nontrivial integrability conditions. © 1989 The American Physical Society.

A remark about topological and measure entropy for a class of maps of the interval [0,1

In this paper we consider the family of dynamical systems on [0,1] in which the evolution is described by the class of mapsSa(x)=ax(1−x),x∈[0,1], for the countable setA={ap}p≥2 of values ofa∈[0,4] satisfying the conditions of Pianigiani. For such values ofa we evaluate the topological entropyht(Sa) and show a lower bound for the measure entropyh(Sa,μa), where μa is the a.c.Sa-invariant measure. Moreover, we show that the sequence {ht(Sap} is increasing withp, and that for a subsequence{pk}⊂{p}, asapk→4, one has:\(h_t (S_{a_n } ) \to h_t (S_4 ,\mu _4 )\).

Well-posedness of dynamical system models

Abstract The problem of well-posedness has been discussed by several authors, G. Zanies and J. C. Willems in particular. Wo propose a new conceptual framework for the well-posedness of system models. In this framework well-posedness is viewed as the continuity of a specific property of a family of dynamical systems with'respect to a parameter change. This parameter change mny represent either errors due to modelling procedures, manufacturing tolerance, or tolerances involved in measuring the input data of the system. Several examples from different subfields of system theory are exhibited to illustrate the generality and flexibility of this new conceptual framework.

Derived stable matrices

Given a stable linear time-invariant dynamical system x·= Ax, it is shown that there exists a related stable family of dynamical systems x·=A(k)x. The Lyapunov stability criterion remains valid for all linear operators A(k)=DkA of this family.

General relations concerning multiply-periodic excitation of nonlinear dynamical systems

If a nonlinear dynamic system is in a state of steady multiply-periodic excitation with a number of incommensurable fundamental frequencies, there exist the same number of relations governing the transfer of energy in this system. General forms for these relations may be established by modifications of the theorem of the Poincaré invariant. These theorems enable one to associate a conservation relation with a closed (cyclic) family of dynamical systems; such a family may be formed by varying any one of the phase parameters which one may associate with the fundamental frequencies. The relations are first set up in a form appropriate to systems with a finite number of degrees of freedom. When applied to a purely reactive electrical network, these relations reduce to those derived by Manley and Rowe. The relations are next established in a form appropriate to the study of nonlinear fields. The general relations are specialized for the following examples: nonlinear electromagnetic media; one-dimensional electron beam with electrostatic field; and unrestricted electron flow with electromagnetic field. These examples reproduce and extend relations established by Haus and Grau.