Chaotic Attractor in a Novel Time-Delayed System with a Saturation Function

From the viewpoint of engineering applications, time delay is useful for constructing a chaotic signal generator, which is the major part of diverse potential applications. Although different mathematical models of time-delay systems have been known, few models can exhibit chaotic behaviors. Motivated by attractive features and potential applications of time-delay models, a new chaotic system with a single scalar time delay and a nonlinearity described by a saturation function is proposed in this chapter. Nonlinear behavior of the system is discovered through bifurcation diagrams and the maximum Lyapunov exponent with the variance of system parameters. Interestingly, the system shows double-scroll chaotic attractors for some suitable chosen system parameters. In order to confirm the correction and feasibility of the theoretical model, the system is also implemented with analog electronic circuit. Finally, a practical application of such circuit is discussed at the end of this chapter.

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A Chaotic Time-Delay System with Saturation Nonlinearity

International Journal of System Dynamics Applications ◽

10.4018/ijsda.2017070106 ◽

2017 ◽

Vol 6 (3) ◽

pp. 111-129 ◽

Cited By ~ 3

Author(s):

Viet-Thanh Pham ◽

Christos Volos ◽

Sundarapandian Vaidyanathan

Keyword(s):

Time Delay ◽

Complex Dynamics ◽

Chaotic Signal ◽

Delay System ◽

Delay Systems ◽

Maximum Lyapunov Exponent ◽

Time Delay Systems ◽

Time Delay System ◽

System Parameters ◽

Saturation Nonlinearity

Complex dynamics are observed in time-delay systems because the presence of time delay could induce unexpected oscillations. Therefore, time-delay systems are effective for constructing chaotic signal generators which have used in various engineering applications. In this paper, a new system with a single scalar time delay and a saturation nonlinearity is introduced. Dynamics of such time-delay system are investigated by using phase planes, bifurcation diagrams and the maximum Lyapunov exponent with the variance of system parameters. It is interesting that the time-delay system can generate double-scroll chaotic attractors despite its elegant model. Circuitry of the system is also presented to show the feasibility of the theoretical model.

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THEORY AND EXPERIMENT OF A FIRST-ORDER CHAOTIC DELAY DYNAMICAL SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413300206 ◽

2013 ◽

Vol 23 (06) ◽

pp. 1330020 ◽

Cited By ~ 16

Author(s):

TANMOY BANERJEE ◽

DEBABRATA BISWAS

Keyword(s):

Time Delay ◽

Electronic Circuit ◽

Stable Limit Cycle ◽

Nonlinear Behavior ◽

Mathematical Function ◽

Hyperchaotic System ◽

Stable Limit ◽

System Parameters ◽

Chaotic Electronic Circuit ◽

Theory And Experiment

We report the theory and experiment of a new time-delayed chaotic (hyperchaotic) system with a single scalar time delay and a nonlinearity described by a closed form mathematical function. Detailed stability and bifurcation analyses establish that with the suitable delay and system parameters, the system shows a stable limit cycle through a supercritical Hopf bifurcation. Numerical simulations exemplify that the system depicts mono-scroll and double-scroll chaos and hyperchaos for a range of delay and other system parameters. Nonlinear behavior of the system is characterized by Lyapunov exponents and Kaplan–Yorke dimension. It is established that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate time delay. Finally, the system is implemented in an analogue electronic circuit using off-the-shelf circuit elements. It is shown that the behavior of the time delay chaotic electronic circuit qualitatively agrees well with our analytical and numerical results.

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The Stability Analysis and Planning for Multivariable Linear Time-Delay Systems and its Application in Supervision Oilfield

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.709.727 ◽

2013 ◽

Vol 709 ◽

pp. 727-730

Author(s):

Ren Wang ◽

Xue Kun Qi ◽

Long Xing

Keyword(s):

Time Delay ◽

Exponential Stability ◽

Linear Time ◽

Control Process ◽

Delay Systems ◽

Sufficient Condition ◽

Water Separation ◽

Delayed Systems ◽

Delayed System ◽

The Stability

Based on Lyapunovs theory of stability, analyze on a control process of oil-water separation with time-delay, coupling and uncertainty in the unite station. a equation of sufficient condition for multi-variable Linear Delayed Systems with delayed independent stabilization is derived and another forms are also given. Based on this, several simple criterions for judging independent stabilization from linear delayed system are presented. We also discuss the exponential stability for delayed system and also provide the sufficient condition of exponential stability with any appointed convergent rate and its corresponding deductions. Using these conditions, we can choose a set of suitable parameters to reduce the conservation. By calculating and being compared with methods of the literature, the results show that our methods have less conservation.

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The Parameters Identification of Time - Delay Models of Heat Transfer Systems From Input-Output Measurements

Volume 3: Thermal Hydraulics; Current Advanced Reactors: Plant Design, Construction, Workforce and Public Acceptance ◽

10.1115/icone17-75361 ◽

2009 ◽

Author(s):

Goran Simeunovic´ ◽

Pavel Zitek ◽

Jan Chysky´

Keyword(s):

Heat Transfer ◽

Time Delay ◽

Time Delays ◽

Parameters Estimation ◽

Parameters Identification ◽

Delay Systems ◽

Parameter Estimates ◽

Identification Algorithm ◽

System Parameters ◽

Variable Projection

The paper presents the identification issues and proposes a parameter identification algorithm that separates the system parameters from the time-delays for a class of single input single output (SISO) linear time delay systems (LTDS). The presence of the unknown time delay greatly complicates the parameters estimation problem, because the parameters of the model are not linear with respect to the time-delay. However, once the time delay is determined, the model becomes linear for the other parameters and hence the common least square method can be utilized directly. Motivated by the nonlinear least squares problem developed in the paper Golub and Pereyra (1973), a novel modification of the so-called variable projection functional is worked out for identification of time delays. In this way, the parameters estimation is separated from the estimation of time delays and the large errors in the parameter estimates, in the case of presence of errors in the time-delay identified values, are avoided. Namely, the small error in the time-delay identified values may often cause a large error in the system parameters identification. A hybrid optimalization method combining a Genetic Algorithm and Nelder-Mead technique is used for minimization of variable projection functional for the identification of time delays. This approach is illustrated by a particular application in the field of heat transfer, concretely on the time-delay model of the recuperative heat exchanger.

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WEAK RESONANT DOUBLE HOPF BIFURCATIONS IN AN INERTIAL FOUR-NEURON MODEL WITH TIME DELAY

International Journal of Neural Systems ◽

10.1142/s0129065712002980 ◽

2012 ◽

Vol 22 (01) ◽

pp. 63-75 ◽

Cited By ~ 31

Author(s):

JUHONG GE ◽

JIAN XU

Keyword(s):

Time Delay ◽

Neuron Model ◽

Hopf Bifurcations ◽

Bidirectional Associative Memory ◽

System Parameters ◽

Dynamical Behaviors ◽

Delayed System ◽

Incremental Scheme ◽

The Relationship ◽

Theoretical Results

In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

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