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Networked dynamical systems with linear coupling: synchronisation patterns, coherence and other behaviours.

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  • 1School of Mathematics and Statistics, University of Western Australia, Perth, Western Australia, 6009.

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Network structure significantly impacts chaotic dynamics in physical and biochemical systems. Small networks exhibit exact synchronization, while large networks show partial synchronization, explained by symmetry and shadowing methods.

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Area of Science:

  • Complex systems
  • Non-linear dynamics
  • Network science

Background:

  • Physical and biochemical systems often comprise identical non-linear dynamical elements with linear coupling.
  • Understanding how network topology influences emergent behaviors like synchronization and coherence is crucial.

Purpose of the Study:

  • To investigate the relationship between network structure and chaotic dynamics.
  • To characterize synchronization patterns in networks of varying sizes.

Main Methods:

  • Analysis of synchronization patterns in small and large networks.
  • Application of symmetry groups and invariance for explaining exact synchronization.
  • Utilizing finite-time shadowing of exact synchronization modes for partial synchronization.

Main Results:

  • Small networks demonstrate precise patterns of exact synchronization.
  • Large networks exhibit partial synchronization at both local and global scales.
  • Exact synchronization modes are mathematically explained via symmetry and invariance.
  • Partial synchronization is elucidated through the concept of finite-time shadowing.

Conclusions:

  • Network size is a critical determinant of synchronization behavior in non-linear dynamical systems.
  • Distinct theoretical frameworks explain exact and partial synchronization modes.
  • The findings provide insights into the collective dynamics of complex networks.