Differential Bivariate Regression Interpolation for Chaos Modeling and Feature Analysis with the Lorenz and Rossler Systems

This paper proposes a Differential Bivariate Regression (DBR) interpolation approach for chaos modeling and dynamic analysis with applications to the Lorenz and Rossler chaotic systems. Specifically, chaos models using the proposed approach are developed, whose dynamic features in terms of equilibria, bifurcation, Lyapunov exponents, and curl/divergence/gradient are analyzed geometrically and algebraically. As case studies, the Lorenz and Rossler systems are scrutinized regarding these features for the first time, and then DBR chaos models are formed by preserving these features. It is revealed that chaotic trajectories of the Lorenz and Rossler systems, when displayed, always swirl around their unique curving axes that are chaotic, independent of initial states and in the form of 1D lines or 2D/3D curves, together with chaotic gradient ramifications. In this way, infinitely many triplets in terms of Lorenz (or Rossler) chaos, curving axis and gradient ramification can be constructed synchronically. The results are significant and inspiring for chaos analysis, construction, reduction, and generalization.