Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Mathematical Physics

Background:

• The stability of parabolic equilibria and existence of heteroclinic orbits in Lorenz-like systems with high-order nonlinearities remain underexplored.
• Existing generalized Lorenz systems do not encompass all complex dynamic behaviors.

Purpose of the Study:

• To introduce and analyze a novel 3D cubic Lorenz-like system with added nonlinear terms.
• To rigorously prove the global exponential asymptotic stability of its parabolic type equilibria.
• To establish the existence of symmetrical heteroclinic orbits.

Main Methods:

• Modification of the standard Lorenz system by incorporating nonlinear terms yz and x^2z.
• Mathematical analysis to investigate bifurcations (pitchfork, Hopf) and attractor types (hidden Lorenz-like).
• Rigorous proof techniques to demonstrate stability and orbit existence.

Main Results:

• Introduction of a new 3D cubic Lorenz-like system not belonging to the generalized Lorenz family.
• Demonstration of various bifurcations and hidden chaotic attractors.
• Rigorous proof of global exponential asymptotic stability for parabolic type equilibria.
• Confirmation of a pair of symmetrical heteroclinic orbits.

Conclusions:

• The novel Lorenz-like system exhibits rich dynamics, including bifurcations and chaotic attractors.
• The parabolic equilibria are proven to be globally exponentially asymptotically stable.
• The existence of heteroclinic orbits is confirmed, contributing to understanding Lorenz-like system families.