Identifying chaotic dynamics in nonuniformly dissipative and conservative dynamical systems from the varying divergence

As well-known, chaotic dynamics can appear in both living and artificial systems, resulting in several applications in science and engineering. Thus, the persistent question is whether a determined time series represents truly chaotic behavior. Although many familiar tools exist to respond to that question, new efficient algorithms in time and complexity must be developed to cope with the striking characteristics of novel chaotic systems, e.g., systems with non-uniform divergence.

In this framework, a new metric to evaluate the chaotic behavior in nonuniformly dynamical systems using the divergence operator of the vector field is proposed. Such systems are characterized by presenting a non-constant divergence, i.e., the divergence changes as time evolves since it depends on the system states. The proposed metric can be applied in dissipative and conservative systems with non-uniform divergence. By expanding the dynamical system with the time derivative of the divergence operator, the proposed approach can identify periodic and chaotic regions from the averages of the derivative of the divergence for the first orbit (ADDFO) and second orbit (ADDSO). We evaluate the performance of the algorithm in various systems: the typical Rossler system, a three-neuron Hopfield neural network, the Pernarowski model of pancreatic beta-cells, and the Sprott D conservative system. From the numerical results, we explicitly demonstrate that the proposed metric provides an efficient algorithm regarding simulation time and complexity with the best performance compared to Lyapunov exponents and bifurcation diagrams.