Dynamics of the stochastic Lorenz chaotic system with long memory effects

Area of Science:

• Dynamical Systems
• Stochastic Processes
• Chaos Theory

Background:

• Ergodic dynamics of stochastic systems with fractional noise are not well understood.
• Fractional noise introduces long memory effects, complicating system analysis.

Purpose of the Study:

• To analyze the long-time dynamics of the stochastic Lorenz chaotic system (SLCS) with fractional noise.
• To determine the existence and uniqueness of invariant measures for the system.

Main Methods:

• Utilized a truncation technique to transform the SLCS into a continuous stochastic dynamical system (Λ).
• Applied the Krylov-Bogoliubov criterion to establish a Lyapunov function, ensuring the existence of an invariant measure.
• Employed the strong Feller property and irreducibility arguments to prove the uniqueness of the invariant measure.

Main Results:

• The stochastic Lorenz chaotic system (SLCS) was shown to generate a continuous stochastic dynamical system (Λ).
• The existence of an invariant measure for Λ was confirmed.
• The uniqueness of the invariant measure for Λ was proven.

Conclusions:

• The stochastic Lorenz chaotic system (SLCS) possesses exactly one adapted stationary solution.
• This research provides crucial insights into the ergodic dynamics of stochastic systems with long memory effects.