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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Information Geometry of Nonlinear Stochastic Systems.

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Nonlinear forces shape stochastic processes by altering probability density functions (PDFs). Noise strength dictates relaxation timescales and PDF features, revealing universal power-law behaviors in information geometry.

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Area of Science:

  • Nonlinear dynamics
  • Statistical physics
  • Information geometry

Background:

  • Stochastic processes are fundamental in modeling complex systems.
  • Understanding the influence of nonlinear forces on these processes is crucial.
  • Characterizing the geometric structure of probability density functions (PDFs) under varying conditions is key.

Purpose of the Study:

  • To investigate the impact of deterministic nonlinear forces (-xn, n=3,5,7) on the geometric structure of stochastic processes.
  • To analyze the transient relaxation of initial probability density functions (PDFs) under different noise strengths (D).
  • To quantify information change and determine information length as a measure of evolving statistical states.

Main Methods:

  • Investigating the time-evolution of PDFs under nonlinear forces and stochastic noise.
  • Computing the rate of information change from PDF evolution.
  • Determining information length L(t) and analyzing its scaling behavior.

Main Results:

  • Identified three distinct stages: nondiffusive, quasi-linear Gaussian, and stationary PDF evolution.
  • Established that noise strength (D) critically influences relaxation timescales, PDF peak amplitude, and width.
  • Discovered a robust geodesic in the initial stage and mapped attractor geometry with L(t→∞)∝μm, where scaling exponent m depends on n and D.

Conclusions:

  • Nonlinear interactions lead to ubiquitous power-laws and multi-scalings in information geometry.
  • The study provides insights into the geometric structure of attractors in nonlinear stochastic systems.
  • Results highlight the interplay between deterministic forces, noise, and information evolution.