Diffeomorphical equivalence vs topological equivalence among Sprott systems

Area of Science:

• Nonlinear dynamics
• Chaos theory
• Dynamical systems analysis

Background:

• Sprott proposed 19 simple dynamical systems producing chaotic attractors in 1994.
• 14 of these systems feature a single nonlinear term.
• Diffeomorphical and topological equivalence of these systems' chaotic attractors remain under-investigated.

Purpose of the Study:

• To systematically investigate the diffeomorphical and topological equivalence of Sprott's 14 simple dynamical systems.
• To classify these systems based on their unique characteristics.

Main Methods:

• Rewriting systems in terms of one variable and its first two derivatives to obtain jerk functions.
• Analyzing the functional form and coefficients of jerk functions to determine diffeomorphical equivalence.
• Characterizing chaotic attractors using branched manifolds.
• Comparing attractors to known systems like the Lorenz system.

Main Results:

• Established criteria for diffeomorphical equivalence based on identical jerk functions (form and coefficients).
• Characterized chaotic attractors as branched manifolds.
• Identified systems with chaotic attractors similar to the Lorenz system.
• Classified systems according to their diffeomorphical and topological equivalence.

Conclusions:

• The study provides a systematic classification of Sprott's dynamical systems.
• Diffeomorphical equivalence is determined by the jerk function.
• Topological equivalence of chaotic attractors is analyzed in relation to branched manifolds and known systems.