Strange Attractors and Classical Stability Theory: Stability, Instability, Lyapunov Exponents and Chaos

Secure communication via chaotic synchronization is demonstrated using dynamical systems governed by delay deferential equations. Strange attractors of such systems can have an arbitrarily large number of positive Lyapunov exponents giving rise to very complex time signals. This features can provide high security of masked messages.

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Approximations and Applications of a Reciprocal Fifth Power Mapping

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.a1210.1291s419 ◽

2019 ◽

Vol 9 (1S4) ◽

pp. 983-986

Keyword(s):

Stability Theory ◽

Research Topic ◽

Rational Form ◽

Classical Approximation ◽

Interesting Study ◽

Classical Stability ◽

Power Mapping

The approximation of different rational form of equations involving functions on both sides is an interesting study in the research topic of classical approximation of equations. The intention of this study is to obtain approximate reciprocal fifth power mapping through classical stability theory and to link the equations dealt in this study with various postulations occurring in physics, chemistry and mechanics.

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Megastability, Multistability in a Periodically Forced Conservative and Dissipative System with Signum Nonlinearity

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418300306 ◽

2018 ◽

Vol 28 (09) ◽

pp. 1830030 ◽

Cited By ~ 6

Author(s):

Pankaj Prakash ◽

K. Rajagopal ◽

J. P. Singh ◽

B. K. Roy

Keyword(s):

Lyapunov Exponents ◽

Dissipative System ◽

Strange Attractors ◽

Lyapunov Spectrum ◽

Bifurcation Diagrams

A sinusoidally-driven conservative and dissipative system with signum nonlinearity is presented in this paper. The proposed system exhibits strange attractors, multistability, and megastability. The system exhibits both conservative and dissipative behaviors with the change of its parameters. The Lyapunov exponents, Lyapunov spectrum plots, bifurcation diagrams, and phase plots are used to show the special features of the proposed system.

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Lyapunov Exponents and Strange Attractors

Encyclopedia of Mathematical Physics ◽

10.1016/b0-12-512666-2/00100-0 ◽

2006 ◽

pp. 349-356 ◽

Cited By ~ 1

Author(s):

M. Viana

Keyword(s):

Lyapunov Exponents ◽

Strange Attractors

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Flight Stability and Safety of WIG Craft With and Without Feedback Control

Volume 1: Applied Mechanics; Automotive Systems; Biomedical Biotechnology Engineering; Computational Mechanics; Design; Digital Manufacturing; Education; Marine and Aerospace Applications ◽

10.1115/esda2014-20254 ◽

2014 ◽

Author(s):

Andreas Groß ◽

Nikolai Kornev ◽

Tobias Hahn ◽

Bernhard Lampe ◽

Wolfgang Drewelow

Keyword(s):

Control System ◽

Dynamic Stability ◽

Stability Theory ◽

High Performance ◽

Controller Design ◽

Ground Effect ◽

Weather Conditions ◽

Static Stability ◽

Flight Safety ◽

Classical Stability

The paper presents the motion simulation of a Wing in Ground craft with a Lippisch configuration which is stable in terms of static stability. The influence of wind and wave perturbations on the dynamic stability of the flight are investigated. The analysis of safety and stability used two different approaches. Based on frequency domain simulation, the first approach derived from the classical stability theory utilizes the aerodynamic coefficients and their linearized derivatives. The second approach based on time domain simulation uses the coefficients with full respect to the nonlinear character of aerodynamics in ground effect. The trajectory is used to evaluate the dynamic stability and flight safety of the craft. From the results for typical perturbations, the limitations of the classical stability theory for the flight safety of Wing in Ground craft are shown. The results show the necessity of the automatic control system even for a satisfying stable craft predicted by the linear theory. The demand of high performance together with the hard constraints for states and control inputs are challenging for the controller design. A proportional-integral-derivative controller is studied to secure the flight safety of small craft under different perturbation and a methodology for designing a control system for operation under various weather conditions is illustrated.

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The Fractional Form of the Tinkerbell Map Is Chaotic

Applied Sciences ◽

10.3390/app8122640 ◽

2018 ◽

Vol 8 (12) ◽

pp. 2640 ◽

Cited By ~ 14

Author(s):

Adel Ouannas ◽

Amina-Aicha Khennaoui ◽

Samir Bendoukha ◽

Thoai Vo ◽

Viet-Thanh Pham ◽

...

Keyword(s):

Lyapunov Exponents ◽

Stability Theory ◽

Chaotic Map ◽

Discrete Systems ◽

Numerical Results ◽

Asymptotic Convergence ◽

Bifurcation Diagrams ◽

Difference Form ◽

The Stability

This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings.

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