76,628 research outputs found
Chaos synchronization of the master-slave generalized Lorenz systems via linear state error feedback control
This paper provides a unified method for analyzing chaos synchronization of
the generalized Lorenz systems. The considered synchronization scheme consists
of identical master and slave generalized Lorenz systems coupled by linear
state error variables. A sufficient synchronization criterion for a general
linear state error feedback controller is rigorously proven by means of
linearization and Lyapunov's direct methods. When a simple linear controller is
used in the scheme, some easily implemented algebraic synchronization
conditions are derived based on the upper and lower bounds of the master
chaotic system. These criteria are further optimized to improve their
sharpness. The optimized criteria are then applied to four typical generalized
Lorenz systems, i.e. the classical Lorenz system, the Chen system, the Lv
system and a unified chaotic system, obtaining precise corresponding
synchronization conditions. The advantages of the new criteria are revealed by
analytically and numerically comparing their sharpness with that of the known
criteria existing in the literature.Comment: 61 pages, 15 figures, 1 tabl
Synthesizing the L\"{u} attractor by parameter-switching
In this letter we synthesize numerically the L\"{u} attractor starting from
the generalized Lorenz and Chen systems, by switching the control parameter
inside a chosen finite set of values on every successive adjacent finite time
intervals. A numerical method with fixed step size for ODEs is used to
integrate the underlying initial value problem. As numerically and
computationally proved in this work, the utilized attractors synthesis
algorithm introduced by the present author before, allows to synthesize the
L\"{u} attractor starting from any finite set of parameter values.Comment: accepted IJBC, 15 pages, 5 figure
OGY Control of Haken Like Systems on Different Poincare Sections
The Chua system, the Lorenz system, the Chen system and The L\"u system are
chaotic systems that their state space equations is very similar to Haken
system which is a nonlinear model of a optical slow-fast system. These
Haken-Like Sys-tems have very similar properties. All have two slow but
unstable eigenvalues and one fastest but stable eigenvalue. This lets that an
approximation of slow manifold be equivalent with unstable manifold of the
system. In other hand, control of discreet model of the system on a defined
manifold (Poincare map) is main essence of some important control methods of
chaotic systems for example OGY method. Here, by using different methods of
defining slow manifold of the H-L systems the efficiency of the OGY control for
stabilizing problem investigated.Comment: 4 page
Adaptive Hybrid Function Projective Synchronization of Chaotic Systems with Time-Varying Parameters
The adaptive hybrid function projective synchronization (AHFPS) of different chaotic systems with unknown time-varying parameters is investigated. Based on the Lyapunov stability theory and adaptive bounding technique, the robust adaptive control law and the parameters update law are derived to make the states of two different chaotic systems asymptotically synchronized. In the control strategy, the parameters need not be known throughly if the time-varying parameters are bounded by the product of a known function of t and an unknown constant. In order to avoid the switching in the control signal, a modified robust adaptive synchronization approach with the leakage-like adaptation law is also proposed to guarantee the ultimately uni-formly boundedness (UUB) of synchronization errors. The schemes are successfully applied to the hybrid function projective synchronization between the Chen system and the Lorenz system and between hyperchaotic Chen system and generalized Lorenz system. Moreover, numerical simulation results are presented to verify the effectiveness of the proposed scheme
A novel chaotic system and its topological horseshoe
Based on the construction pattern of Chen, Liu and Qi chaotic systems, a new threedimensional (3D) chaotic system is proposed by developing Lorenz chaotic system. It’s found that when parameter e varies, the Lyapunov exponent spectrum keeps invariable, and the signal amplitude can be controlled by adjusting e. Moreover, the horseshoe chaos in this system is investigated based on the topological horseshoe theory
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems
PublishedA one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3 using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.The first author is partially supported by a
MINECO/FEDER grant MTM2008-03437 and
MTM2013-40998-P, an AGAUR grant number
2014SGR-568, an ICREA Academia, the grants
FP7-PEOPLE-2012-IRSES 318999 and 316338,
and UNAB 13-4E-1604
Adaptive Hybrid Function Projective Synchronization of Chaotic Systems with Time-Varying Parameters
The adaptive hybrid function projective synchronization AHFPS of different chaotic systems with unknown time-varying parameters is investigated. Based on the Lyapunov stability theory and adaptive bounding technique, the robust adaptive control law and the parameters update law are derived to make the states of two different chaotic systems asymptotically synchronized. In the control strategy, the parameters need not be known throughly if the time-varying parameters are bounded by the product of a known function of t and an unknown constant. In order to avoid the switching in the control signal, a modified robust adaptive synchronization approach with the leakage-like adaptation law is also proposed to guarantee the ultimately uni-formly boundedness UUB of synchronization errors. The schemes are successfully applied to the hybrid function projective synchronization between the Chen system and the Lorenz system and between hyperchaotic Chen system and generalized Lorenz system. Moreover, numerical simulation results are presented to verify the effectiveness of the proposed scheme
Chaos synchronization between linearly coupled chaotic systems
Abstract This paper investigates the chaos synchronization between two linearly coupled chaotic systems. Some sufficient conditions of global asymptotic synchronization are attained from rigorously mathematical theory. Also, a new method for analyzing the stability of synchronization solution is presented. Using this method, some sufficient conditions of linear stability of the synchronization chaotic solution are gained. The influence of coupling coefficients on chaos synchronization is further studied for three typical chaotic systems: Lorenz system, Chen system, and newly found L€ u u system.
Chaotic motions in the real fuzzy electronic circuits
Fuzzy electronic circuit (FEC) is firstly introduced, which is implementing Takagi-Sugeno (T-S) fuzzy chaotic systems on electronic circuit. In the research field of secure communications, the original source should be blended with other complex signals. Chaotic signals are one of the good sources to be applied to encrypt high confidential signals, because of its high complexity, sensitiveness of initial conditions, and unpredictability. Consequently, generating chaotic signals on electronic circuit to produce real electrical signals applied to secure communications is an exceedingly important issue. However, nonlinear systems are always composed of many complex equations and are hard to realize on electronic circuits. Takagi-Sugeno (T-S) fuzzy model is a powerful tool, which is described by fuzzy IF-THEN rules to express the local dynamics of each fuzzy rule by a linear system model. Accordingly, in this paper, we produce the chaotic signals via electronic circuits through T-S fuzzy model and the numerical simulation results provided by MATLAB are also proposed for comparison. T-S fuzzy chaotic Lorenz and Chen-Lee systems are used for examples and are given to demonstrate the effectiveness of the proposed electronic circuit. © 2013 Shih-Yu Li et al
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