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.

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.

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.

Data-Driven Detection and Classification of Regimes in Chaotic Systems Via Hidden Markov Modeling

2020 ◽  
Vol 1 (2) ◽  
Author(s):  
Chandrachur Bhattacharya ◽  
Asok Ray
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Hidden Markov ◽  
Urban Traffic ◽  
Data Driven ◽  
Bench Mark ◽  
Lorenz Attractor ◽  
Markov Modeling ◽  
Chaotic Dynamical Systems ◽  
Hidden Markov Modeling

Abstract Chaotic dynamical systems are essentially nonlinear and are highly sensitive to variations in initial conditions and process parameters. Chaos may appear both in natural (e.g., heartbeat rhythms and weather fluctuations) and human-engineered (e.g., thermo-fluid, urban traffic, and stock market) systems. For prediction and control of such systems, it is often necessary to be able to distinguish between non-chaotic and chaotic behavior; several methods exist to detect the presence (or absence) of chaos, specially in noisy signals. A dynamical system may exhibit multiple chaotic regimes, and apparently, there exist no methods, reported in open literature, to classify these regimes individually. This paper demonstrates an application of standard hidden Markov modeling (HMM), which is a commonly used supervised method, as a technique to classify multiple regimes from a time series of dynamical systems, where classified regimes could be chaotic or non-chaotic. The proposed HMM-based method of regime classification has been tested using numerical data obtained from several well-known chaotic dynamical systems (e.g., Hénon, forced Duffing, Rössler, and Lorenz attractor). It is apparently well-suited to serve as a bench mark for the development of alternative data-driven methods to enhance the performance (e.g., accuracy and computational speed) of regime classification in chaotic dynamical systems.

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

A NEW APPROACH IN THE SEARCH FOR PERIODIC ORBITS

2013 ◽  
Vol 23 (11) ◽  
pp. 1350186 ◽  
Author(s):  
ROMINA COBIAGA ◽  
WALTER REARTES
Keyword(s):  
Dynamical Systems ◽  
Linear Operator ◽  
Periodic Orbits ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Initial Conditions ◽  
Formal Series ◽  
Analysis Method ◽  
New Approach ◽  
Homotopy Analysis

In this paper, we present a new approach for finding periodic orbits in dynamical systems modeled by differential equations. It is based on the Homotopy Analysis Method (HAM) but it differs from the usual way it is applied. We apply HAM to construct approximations of a formal series solution of the equation. These approximations exist for any value of the frequency. They can be obtained by choosing a suitable linear operator and allowing the initial conditions to vary freely. Then we study the behavior of the obtained expressions as a function of the frequency. This procedure allows us to find those values of the frequency for which the series converges and therefore to find the periodic orbits. We show several examples of application of the proposed method. Mathematica and MATCONT were applied for all the calculations.

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