Dynamics of Nonlinear Time-Delay Systems

Lakshmanan, M.; Senthilkumar, D. V.

doi:10.1007/978-3-642-14938-2

Cited by 284 publications

(270 citation statements)

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“…Often, the transmission time is longer than the internal time scales of these units; the coupling has a long delay time. When the evolution of a deterministic system depends not only on its present state but also on its past state, the mathematical space of its solution becomes infinite dimensional, and the system is flexible enough to develop instabilities which lead to highdimensional chaos [2,3]. For example, if the beam of a semiconductor laser is reflected by an external mirror back to the laser cavity, then a chaotic intensity on the time scale of picoseconds is observed [4].…”

Section: Introductionmentioning

confidence: 99%

Chaos in networks with time-delayed couplings

Kinzel

2013

Phil. Trans. R. Soc. A.

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Networks of nonlinear units coupled by time-delayed signals can show chaos. In the limit of long delay times, chaos appears in two ways: strong and weak, depending on how the maximal Lyapunov exponent scales with the delay time. Only for weak chaos, a network can synchronize completely, without time shift. The conditions for strong and weak chaos and synchronization in networks with multiple delay times are investigated.

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Section: Introductionmentioning

confidence: 99%

Chaos in networks with time-delayed couplings

Kinzel

2013

Phil. Trans. R. Soc. A.

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“…Some articles [10,11] and approaches [7] have already been studied on this subject. The goal is to find out weather this model admits real human reaction times and simultaneously ensures solution stability.…”

Section: Discussionmentioning

confidence: 99%

Traffic and Granular Flow '13

Flow¹,

Chraibi²,

Boltes³

et al. 2015

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The Intelligent Driver Model (IDM) is studied and several drawbacks with respect to driving simulators are defined. We present two modifications of the IDM. The first one gives any predefined distance to the leading vehicle in a steady state. The second modification is a combination of the first one and the optimal velocity model. It takes into account driver's reaction time explicitly and is described by delay differential equation. This model always results in realistic vehicles accelerations what allows simulating real traffic collisions.Necessary and sufficient conditions are obtained, that guarantee a non-oscillating solution near the equilibrium for the vehicle platoon. We suggest the calibrating framework based on a numerical solution of the constrained optimization problem. Nonlinear constraints are generated by the numerical integration scheme. The suggested procedure incorporates the local stability conditions obtained and takes into account vehicle dynamics, drivers' behavior and weather conditions.

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“…(3). To quantify anticipatory synchronization, we use the following similarity function [27] defined with respect to one dynamical variable, say x1 of the Van der pol oscillator, Synchronization S(τ )=0, τ=0. Numerically as the coupling strength is increased, one observes the transition from anticipatory synchronization to complete synchronization.…”

Section: Figmentioning

confidence: 99%

The dynamics of the two mutually coupled autonomous Van der Pol Oscillator

Thamilmaran¹

2017

IOSR JAP

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In this work, the dynamical behavior of the two mutually coupled Autonomous Van der Pol oscillator studied via numerical and experimental observations. We show, when the appropriate coupling co-efficient, the transition from phase synchronization to anticipatory synchronization.The transition confirmed through similarity function. The rich nonlinear dynamical phenomena of chaos identified and confirmed through bifurcation and Lyapunov exponent spectrum analysis. The well known period doubling route to chaos are identified. The above mentioned dynamical behavior are observed in the real time hardware experimental circuit simulations.

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Dynamics of Nonlinear Time-Delay Systems

Cited by 284 publications

References 0 publications

Chaos in networks with time-delayed couplings

Chaos in networks with time-delayed couplings

Traffic and Granular Flow '13

The dynamics of the two mutually coupled autonomous Van der Pol Oscillator

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