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Linear time-invariant Systems01:23

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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Inverse synchronizations in coupled time-delay systems with inhibitory coupling.

D V Senthilkumar1, J Kurths, M Lakshmanan

  • 1Centre for Dynamics of Complex Systems, 14469 Potsdam, Germany. skumar@cnld.bdu.ac.in

Chaos (Woodbury, N.Y.)
|July 2, 2009
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Summary

Coupling delay in inhibitory time-delay systems drives transitions between inverse anticipatory, complete, and lag synchronizations. Stability conditions are consistent across various coefficient dependencies, confirmed by Lyapunov exponents.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Complex Systems

Background:

  • Time-delay systems are crucial in modeling phenomena with inherent delays.
  • Synchronization in coupled systems is fundamental to understanding emergent behaviors.
  • Inhibitory coupling introduces unique dynamics not seen in excitatory systems.

Purpose of the Study:

  • To investigate transitions between different types of inverse synchronization.
  • To analyze the impact of coupling delay on synchronization patterns.
  • To establish general asymptotic stability conditions for these systems.

Main Methods:

  • Analysis of unidirectionally coupled time-delay systems with inhibitory coupling.
  • Application of Krasovskii-Lyapunov functional theory for stability analysis.
  • Utilizing similarity functions, probability of synchronization, and Lyapunov exponents for validation.

Main Results:

  • Synchronization transitions (inverse anticipatory, complete, lag) are observed as a function of coupling delay.
  • A unified asymptotic stability condition is derived, applicable to time-independent and mixed time-dependent coefficients.
  • Validation of synchronization existence through multiple quantitative measures.

Conclusions:

  • Coupling delay is a critical parameter governing synchronization regimes in inhibitory time-delay systems.
  • The Krasovskii-Lyapunov method provides robust stability analysis for diverse system parameterizations.
  • Multiple metrics confirm the occurrence and characteristics of various inverse synchronization states.