A Lorenz-type attractor in a piecewise-smooth system: Rigorous results

Area of Science:

• Dynamical Systems and Chaos Theory
• Mathematical Physics
• Nonlinear Dynamics

Background:

• Chaotic attractors are prevalent in physical and biological systems, but rigorous proofs of their existence and bifurcations are scarce.
• The Lorenz attractor is a celebrated example, yet analytical characterization of its dynamics remains challenging.

Purpose of the Study:

• To construct a novel piecewise-smooth dynamical system that rigorously proves the existence and characterizes bifurcations of a chaotic attractor.
• To develop a method for synthesizing and analyzing hyperbolic attractors with predefined properties.

Main Methods:

• Construction of a simple piecewise-smooth model switching between three linear systems.
• Derivation of a Poincaré return map utilizing the integrability of the linear components.
• Analytical calculation of bifurcation curves and identification of key parameter regimes.

Main Results:

• Rigorous proof for the existence of a Lorenz-type singular hyperbolic attractor.
• Explicit characterization of bifurcations, including the formation of a 'homoclinic butterfly' and the birth of the chaotic attractor from heteroclinic orbits.
• Analytical tractability of bifurcations, surpassing limitations of the original nonintegrable Lorenz system.

Conclusions:

• The developed piecewise-smooth model provides a framework for the rigorous synthesis and analysis of hyperbolic chaotic attractors.
• This approach offers exact solutions and analytical insights into complex dynamical phenomena, applicable to various scientific models.