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Register a new userAttractors in coherent systems of differential equations
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Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.
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We introduce new weak topologies and spaces of Caratheodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Caratheodory vector field $f$ provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of $f$. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.
We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic effective dynamics of solitons and their asymptotic stability. Results of numerical simulation are given. The obtained results allow us to formulate a new general conjecture on attractors of $G$ -invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohrs transitions between quantum stationary states, wave-particle duality and probabilistic interpretation.
We show that any two trajectories of solutions of a one-dimensional fractional differential equation (FDE) either coincide or do not intersect each other. In contrary, in the higher dimensional case, two different trajectories can meet. Furthermore, one-dimensional FDEs and triangular systems of FDEs generate nonlocal fractional dynamical systems, whereas a higher dimensional FDE does, in general, not generate a nonlocal dynamical system.
In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Holder space which is separable.
This paper shows that the celebrated Embedding Theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.
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