Higher-Dimensional Dynamical Systems

1999 ◽  
pp. 81-90
Author(s):  
Graham Everest ◽  
Thomas Ward
Keyword(s):  
Dynamical Systems ◽  
Higher Dimensional

DYNAMICAL SYSTEMS FROM QUANTUM INTEGRABILITY

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3567-3596 ◽  
Author(s):  
M.P. Bellon ◽  
J-M. Maillard ◽  
C-M. Viallet
Keyword(s):  
Dynamical Systems ◽  
Projective Spaces ◽  
Linear Transformations ◽  
Quantum Integrability ◽  
Higher Dimensional ◽  
Algebraic Invariants ◽  
Non Linear ◽  
Birational Mappings

We describe a class of non-linear transformations acting on many variables. These transformations have their origin in the theory of quantum integrability: they appear in the description of the symmetries of the Yang-Baxter equations and their higher dimensional generalizations. They are generated by involutions and act as birational mappings on various projective spaces. We present numerous figures, showing successive iterations of these mappings. The existence of algebraic invariants explains the aspect of these figures. We also study deformations of our transformations.

AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS

2009 ◽  
Vol 21 (08) ◽  
pp. 949-979 ◽  
Author(s):  
BENOIT SAUSSOL
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism ◽  
Dimension Theory ◽  
Recurrence Rates ◽  
One Dimensional ◽  
Expanding Map ◽  
System A ◽  
Higher Dimensional ◽  
Technical Details ◽  
Hyperbolic Behavior

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

On the Nonlinear Observability of Polynomial Dynamical Systems

2021 ◽  
Vol 1 (1) ◽  
pp. 88-94
Author(s):  
Daniel Gerbet ◽  
Klaus Röbenack
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
State Space ◽  
Polynomial Dynamical Systems ◽  
Higher Dimensional ◽  
Dimensional State Space ◽  
System Properties ◽  
Controllability And Observability ◽  
Important System ◽  
Nonlinear Observability

Controllability and observability are important system properties in control theory. These properties cannot be easily checked for general nonlinear systems. This paper addresses the local and global observability as well as the decomposition with respect to observability of polynomial dynamical systems embedded in a higher-dimensional state-space. These criteria are applied on some example system.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS GOVERNED BY CONTINUOUS MAPS

2005 ◽  
Vol 15 (02) ◽  
pp. 547-555 ◽  
Author(s):  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Relevant Literature ◽  
Discrete Dynamical Systems ◽  
One Dimensional ◽  
Control Techniques ◽  
Controlled Systems ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Continuous Maps

This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

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