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Identification of quasiperiodic processes in the vicinity of the resonance

Fischer, Cyril, Náprstek, Jiří (2023)

Programs and Algorithms of Numerical Mathematics

In nonlinear dynamical systems, strong quasiperiodic beating effects appear due to combination of self-excited and forced vibration. The presence of symmetric or asymmetric beatings indicates an exchange of energy between individual degrees of freedom of the model or by multiple close dominant frequencies. This effect is illustrated by the case of the van der Pol equation in the vicinity of resonance. The approximate analysis of these nonlinear effects uses the harmonic balance method and the multiple...

Index filtrations and Morse decompositions for discrete dynamical systems

P. Bartłomiejczyk, Z. Dzedzej (1999)

Annales Polonici Mathematici

On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.

Intertwined mappings

Jean Ecalle, Bruno Vallet (2004)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Invariant measures for position dependent random maps with continuous random parameters

Tomoki Inoue (2012)

Studia Mathematica

We consider a family of transformations with a random parameter and study a random dynamical system in which one transformation is randomly selected from the family and applied on each iteration. The parameter space may be of cardinality continuum. Further, the selection of the transformation need not be independent of the position in the state space. We show the existence of absolutely continuous invariant measures for random maps on an interval under some conditions.

Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz (2002)

Annales Polonici Mathematici

We consider the stochastic differential equation (1) d u ( t ) = a ( u ( t ) , ξ ( t ) ) d t + Θ σ ( u ( t ) , θ ) p ( d t , d θ ) for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup P t t 0 corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup P t t 0 describing the evolution of measures along trajectories and vice versa.

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