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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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Pathwise synchronization of global coupled system with linear multiplicative rough noise.

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This study proves pathwise synchronization for stochastic differential equations driven by fractional Brownian rough paths. Numerical simulations confirm synchronization results in Hölder space.

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Area of Science:

  • Stochastic Analysis
  • Dynamical Systems Theory

Background:

  • Stochastic differential equations (SDEs) are crucial for modeling complex systems.
  • Synchronization phenomena in SDEs are of significant interest.
  • Rough noise introduces complexities not captured by traditional SDEs.

Purpose of the Study:

  • To investigate pathwise synchronization in SDEs with linear multiplicative rough noises.
  • To analyze the stability and dynamical behavior of these SDEs.
  • To verify synchronization results using theoretical and numerical methods.

Main Methods:

  • Application of rough paths theory for SDE analysis.
  • Transformation of rough SDEs into random differential equations.
  • Analysis of stability and dynamical properties.
  • Verification of synchronization in Hölder space.

Main Results:

  • Pathwise synchronization of solutions to coupled SDE systems with rough noises is proven.
  • Stability and dynamical behavior of solutions are thoroughly discussed.
  • Synchronization results are validated through numerical simulations.

Conclusions:

  • The proposed methods effectively achieve and verify pathwise synchronization for SDEs with rough noises.
  • The findings extend synchronization theory to systems driven by fractional Brownian rough paths.
  • Numerical simulations corroborate the theoretical results, demonstrating practical applicability.