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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
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Basin stability in delayed dynamics.

Siyang Leng1,2, Wei Lin1, Jürgen Kurths2,3,4

  • 1School of Mathematical Sciences, LNSM and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China.

Scientific Reports
|February 25, 2016
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Summary
This summary is machine-generated.

This study introduces a new method to calculate basin stability in complex systems with time delays. The technique quantifies the basin of attraction volume, crucial for understanding multi-stability in fields like neuroscience and climate science.

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Area of Science:

  • Complex Systems Dynamics
  • Nonlinear Dynamics and Chaos
  • Computational Neuroscience

Background:

  • Basin stability (BS) is vital for analyzing complex systems, particularly those exhibiting multi-stability.
  • Traditional BS methods struggle with delayed dynamics due to infinite-dimensional initial state spaces.
  • Multi-stability is prevalent in systems like firing neurons, climate processes, and power grids.

Purpose of the Study:

  • To develop a method for defining and calculating the basin of attraction volume in delayed dynamical systems.
  • To address the fundamental challenge posed by the infinite dimensionality of initial conditions in such systems.
  • To provide a practical algorithm for estimating basin volumes in complex, multi-stable, and delayed systems.

Main Methods:

  • Projecting the infinite-dimensional initial state space onto a finite-dimensional Euclidean space using basis expansions.
  • Developing a generalized concept of basin volume applicable to delayed dynamics.
  • Implementing a numerical algorithm with cross-validation for basin volume estimation.

Main Results:

  • A novel technique for estimating the basin of attraction volume in delayed dynamics.
  • Demonstrated applicability to the delayed Hopfield neuronal model and delayed complex networks.
  • Successful numerical estimation of basin volumes in systems with multi-stability and synchronization dynamics.

Conclusions:

  • The proposed method offers a practical approach to quantifying basin stability in complex systems with time delays.
  • This technique enhances the understanding of multi-stability and dynamics in biologically and physically relevant systems.
  • The generalized basin volume concept and algorithm provide valuable tools for future research in delayed dynamics.