A new approach for controlling chaotic dynamical systems

1998 ◽  
Vol 238 (6) ◽  
pp. 349-357 ◽  
Author(s):  
Kazuyuki Yagasaki ◽  
Tomotsugu Uozumi
Keyword(s):  
Dynamical Systems ◽  
New Approach ◽  
Chaotic Dynamical Systems

Pseudorandom Number Generators Based on Chaotic Dynamical Systems

2001 ◽  
Vol 08 (02) ◽  
pp. 137-146 ◽  
Author(s):  
Janusz Szczepański ◽  
Zbigniew Kotulski
Keyword(s):  
Dynamical Systems ◽  
Communication Systems ◽  
Initial Conditions ◽  
Pseudorandom Number ◽  
New Approach ◽  
Chaotic Dynamical Systems ◽  
Pseudorandom Number Generators ◽  
Transmission Of Information ◽  
Extreme Sensitivity ◽  
Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

scholarly journals DIFFERENTIAL GEOMETRY AND MECHANICS: APPLICATIONS TO CHAOTIC DYNAMICAL SYSTEMS

2006 ◽  
Vol 16 (04) ◽  
pp. 887-910 ◽  
Author(s):  
JEAN-MARC GINOUX ◽  
BRUNO ROSSETTO
Keyword(s):  
Differential Geometry ◽  
Dynamical Systems ◽  
Slow Manifold ◽  
Van Der Pol ◽  
New Approach ◽  
Metric Properties ◽  
Chaotic Dynamical Systems ◽  
Mechanics Concepts ◽  
The One ◽  
Curvature And Torsion

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk). The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra–Gause are proposed.

The plausibility of a chaotic brain theory

2001 ◽  
Vol 24 (5) ◽  
pp. 829-840 ◽  
Author(s):  
Ichiro Tsuda
Keyword(s):  
Dynamical Systems ◽  
Dynamic Behavior ◽  
Predictive Power ◽  
High Dimensional ◽  
New Approach ◽  
Dynamic Memory ◽  
Chaotic Dynamical Systems ◽  
Brain Theory ◽  
The Brain ◽  
Cantor Coding

We consider the significance of high-dimensional transitory dynamics in the brain and mind. In particular, we highlight the roles of high-dimensional chaotic dynamical systems as an “adequate language” (Gelfand 1989), which should possess both explanatory and predictive power of description. We discuss the methods of description of dynamic behavior of the brain. These methods have been adopted to capture the averaged or deterministic complexity, and further to allow for discussion of a new approach to capture the complexity of the deviation from such an averaged complexity and also the complexity of interactive modes. We also give arguments in defense of our models for dynamic memory with chaotic itinerancy and Cantor coding. In addition, we discuss the reality that a model of the brain and mind should reflect.

CHARACTERIZING LOSS OF MEMORY IN CHAOTIC DYNAMICAL SYSTEMS

1991 ◽  
Vol 05 (14) ◽  
pp. 2323-2345 ◽  
Author(s):  
R.E. AMRITKAR ◽  
P.M. GADE
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Chaotic Dynamical Systems

We discuss different methods of characterizing the loss of memory of initial conditions in chaotic dynamical systems.

FREQUENCY SYNCHRONIZATION IN NETWORKS OF COUPLED OSCILLATORS, A MONOTONE DYNAMICAL SYSTEMS APPROACH

2009 ◽  
Vol 19 (12) ◽  
pp. 4107-4116 ◽  
Author(s):  
WEN-XIN QIN
Keyword(s):  
Dynamical Systems ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Systems Approach ◽  
System Size ◽  
Coupling Scheme ◽  
Frequency Synchronization ◽  
Nonlinear Coupling ◽  
New Approach ◽  
Monotone Dynamical Systems

We propose a new approach to investigate the frequency synchronization in networks of coupled oscillators. By making use of the theory of monotone dynamical systems, we show that frequency synchronization occurs in networks of coupled oscillators, provided the coupling scheme is symmetric, connected, and strongly cooperative. Our criterion is independent of the system size, the coupling strength and the details of the connections, and applies also to nonlinear coupling schemes.

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