14.3 Stochastic dynamical systems

Stochastic dynamical systems add randomness to the mix, shaking up our understanding of predictable behavior. These systems incorporate noise and uncertainty, making them more realistic models of real-world phenomena like stock markets, weather patterns, and biological processes.

By using tools like stochastic differential equations and Itô calculus, we can analyze how random fluctuations affect system behavior. This approach reveals fascinating phenomena like noise-induced transitions and stochastic resonance, where randomness can actually enhance signal detection.

Stochastic Processes and Equations

Brownian Motion and Stochastic Differential Equations

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• Brownian motion describes the random motion of particles suspended in a fluid (liquid or gas) resulting from collisions with molecules of the fluid
• Mathematically modeled using stochastic processes, which are random processes that evolve over time
• Stochastic differential equations (SDEs) extend ordinary differential equations by incorporating random terms to model systems subject to noise or uncertainty
• SDEs consist of a deterministic term describing the system's average behavior and a stochastic term representing random fluctuations (Wiener process or Brownian motion)

Langevin Equation and Markov Processes

• Langevin equation is a stochastic differential equation that describes the motion of a particle subject to random forces and friction
• Models the velocity of a particle as a combination of a deterministic term (friction) and a random term (noise)
• Markov processes are stochastic processes where the future state depends only on the current state, not on the past states
• Markov processes have the "memoryless" property, meaning that the probability distribution of the next state depends only on the current state (transition probabilities)

Mathematical Tools for Stochastic Systems

Itô Calculus

• Itô calculus is a mathematical framework for dealing with stochastic differential equations and stochastic integrals
• Extends the rules of ordinary calculus to handle stochastic processes and random variables
• Itô's lemma provides a formula for computing the differential of a function of a stochastic process, analogous to the chain rule in ordinary calculus
• Itô integrals define the integration of stochastic processes with respect to Brownian motion or other stochastic processes

Fokker-Planck Equation

• Fokker-Planck equation, also known as the Kolmogorov forward equation, is a partial differential equation that describes the time evolution of the probability density function of a stochastic process
• Provides a way to determine the probability distribution of a system's state at a given time, given an initial probability distribution and the system's dynamics
• Useful for analyzing the behavior of stochastic systems and understanding how the probability distribution evolves over time
• Can be derived from the Langevin equation or other stochastic differential equations using Itô calculus

Stochastic Phenomena

Noise-Induced Transitions

• Noise-induced transitions refer to the phenomenon where noise or random fluctuations can cause a system to transition between different states or regimes
• In the presence of noise, a system can overcome potential barriers and switch between stable states (bistability) or exhibit new behaviors not present in the deterministic case
• Examples include:
  • Stochastic switching in gene expression, where noise can cause cells to switch between different gene expression states
  • Noise-induced transitions in climate systems, such as the transition between ice ages and warm periods

Stochastic Resonance

• Stochastic resonance is a phenomenon where the presence of noise can enhance the detection or transmission of weak signals in nonlinear systems
• Occurs when the addition of an optimal level of noise to a system increases its sensitivity to weak input signals, improving the signal-to-noise ratio
• The noise helps the system overcome potential barriers and synchronize with the weak input signal, resulting in amplified output
• Examples include:
  • Enhanced sensitivity of sensory neurons in the presence of background noise, improving the detection of weak stimuli
  • Improved climate signal detection in paleoclimatic records by adding noise to the data analysis process