667 research outputs found
Invariant cones for strange attractors of Lozi, H\'{e}non and Belykh type maps
We consider strange attractors of two dimensional generalized map with one
nonlinearity such that Lozi, H\'{e}non and Belykh maps are particular cases of
it. We describe technique of invariant expanding and contracting cones creation
for study of hyperbolic properties. Theorems of singular hyperbolic attractors
for new modifications of Lozi, H\'{e}non and Belykh-type maps are presented
Critical Lattice Size Limit for Synchronized Chaotic State in 1-D and 2-D Diffusively Coupled Map Lattices
We consider diffusively coupled map lattices with neighbors (where is
arbitrary) and study the stability of synchronized state. We show that there
exists a critical lattice size beyond which the synchronized state is unstable.
This generalizes earlier results for nearest neighbor coupling. We confirm the
analytical results by performing numerical simulations on coupled map lattices
with logistic map at each node. The above analysis is also extended to
2-dimensional -neighbor diffusively coupled map lattices.Comment: 4 pages, 2 figure
Generalized Turing Patterns and Their Selective Realization in Spatiotemporal Systems
We consider the pattern formation problem in coupled identical systems after
the global synchronized state becomes unstable. Based on analytical results
relating the coupling strengths and the instability of each spatial mode
(pattern) we show that these spatial patterns can be selectively realized by
varying the coupling strengths along different paths in the parameter space.
Furthermore, we discuss the important role of the synchronized state (fixed
point versus chaotic attractor) in modulating the temporal dynamics of the
spatial patterns.Comment: 9 pages, 3 figure
General Stability Analysis of Synchronized Dynamics in Coupled Systems
We consider the stability of synchronized states (including equilibrium
point, periodic orbit or chaotic attractor) in arbitrarily coupled dynamical
systems (maps or ordinary differential equations). We develop a general
approach, based on the master stability function and Gershgorin disc theory, to
yield constraints on the coupling strengths to ensure the stability of
synchronized dynamics. Systems with specific coupling schemes are used as
examples to illustrate our general method.Comment: 8 pages, 1 figur
Stability analysis of coupled map lattices at locally unstable fixed points
Numerical simulations of coupled map lattices (CMLs) and other complex model
systems show an enormous phenomenological variety that is difficult to classify
and understand. It is therefore desirable to establish analytical tools for
exploring fundamental features of CMLs, such as their stability properties.
Since CMLs can be considered as graphs, we apply methods of spectral graph
theory to analyze their stability at locally unstable fixed points for
different updating rules, different coupling scenarios, and different types of
neighborhoods. Numerical studies are found to be in excellent agreement with
our theoretical results.Comment: 22 pages, 6 figures, accepted for publication in European Physical
Journal
Stability of Synchronized Chaos in Coupled Dynamical Systems
We consider the stability of synchronized chaos in coupled map lattices and
in coupled ordinary differential equations. Applying the theory of Hermitian
and positive semidefinite matrices we prove two results that give simple bounds
on coupling strengths which ensure the stability of synchronized chaos.
Previous results in this area involving particular coupling schemes (e.g.
global coupling and nearest neighbor diffusive coupling) are included as
special cases of the present work.Comment: 9 page
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