Observing the symmetry of attractors (original) (raw)
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The structure of symmetric attractors
Archive for Rational Mechanics and Analysis, 1993
We consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems. We show, for example, that the symmetry of an attractor cannot, in general, be an arbitrary subgroup of the group of symmetries. In addition, there are group-theoretic restrictions on the symmetry of connected components of a symmetric attractor. The symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time-average of a chaotic dynamical system.
Dynamical symmetry and geometric phase
Physica Scripta, 2007
By considering dynamical symmetry between canonically equivalent systems, we investigate the connection between the geometric phase and dynamical invariants, where the Liouville-von-Neumann equation is directly deduced. Furthermore, we show that an arbitrary shift of the Hamiltonian
Symmetric Attractors for Diffeomorphisms and Flows
Proceedings of the London Mathematical Society, 1996
Let Γ ⊂ O(n) be a finite group acting on R n . In this work we describe the possible symmetry groups that can occur for attractors of smooth (invertible) Γ-equivariant dynamical systems. In case R n contains no reflection planes and n ≥ 3, our results imply there are no restrictions on symmmetry groups. In case n ≥ 4 (diffeomorphisms) and n ≥ 5 (flows), we show that we may construct attractors which are Axiom A. We also give a complete description of what can happen in low dimensions.
Prepotential approach to systems with dynamical symmetries
2012
A prepotential approach to constructing the quantum systems with dynamical symmetry is proposed. As applications, we derive generalizations of the hydrogen atom and harmonic oscillator, which can be regarded as the systems with position-dependent mass. They have the symmetries which are similar to the corresponding ones, and can be solved by using the algebraic method.
We construct polynomial dynamical systems x = P(x) with symmetries present in the local phase portrait. This point of view on symmetry yields the approaches to ODEs construction being amenable to classical methods of the Spectral Analysis.
ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS
International Journal of Modern Physics A, 1995
In this paper we show how the well-known local symmetries of Lagrangean systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian-and Lagrangean formalisms is found. The possible applications of our results are discussed.
Koopman Operator Methods for Global Phase Space Exploration of Equivariant Dynamical Systems
IFAC-PapersOnLine, 2020
In this paper, we develop the Koopman operator theory for dynamical systems with symmetry. In particular, we investigate how the Koopman operator and eigenfunctions behave under the action of the symmetry group of the underlying dynamical system. Further, exploring the underlying symmetry, we propose an algorithm to construct a global Koopman operator from local Koopman operators. In particular, we show, by exploiting the symmetry, data from all the invariant sets are not required for constructing the global Koopman operator; that is, local knowledge of the system is enough to infer the global dynamics.
Symmetry groups for 3D dynamical systems
Journal of Physics A: Mathematical and Theoretical, 2007
We present a systematic way to construct dynamical systems with a specific symmetry group G. Each symmetric strange attractor has a unique image attractor that is locally identical to it but different at the global topological level. Image attractors can be lifted to many inequivalent covering attractors. These are distinguished by an index that has related topological, algebraic and group theoretical interpretations. These methods are used to describe dynamical systems with symmetry groups V 4 , S 4 and S 6 .
Parameter uncertainties in models of equivariant dynamical systems
Physica D: Nonlinear Phenomena, 1997
We examine the advantages of using equivariant versus nonequivariant phenomenological models for chaotic systems that possess an inversion symmetry. Numerical experiments used the minimum description length (MDL) criteria for model selection. They indicated that the selection between equivariant versus nonequivariant models should be made on an a priori basis. We also examine the relationship between the optimal truncation accuracy of the parameters of the model (as predicted by the MDL criteria) and the true uncertainty in the parameters of the model. Numerical experiments indicate that the true uncertainties are poorly approximated by the optimal truncation accuracy. They also indicate that the true uncertainties may be useful for determining the presence of an inversion symmetry in reconstructed attractors.