Narrow Results

Using an approach developed by one of the authors, approximate solutions of the null-Dirichlet problem for the forced Nagumo equation are obtained. It is shown that these solutions are chaotic whatever the values of the control parameter in this equation. It is also shown that in the absence of the external force the solutions are chaotic for negative values of the control parameter. However, these solutions are quasi-periodic for large time scales.

Generally, it is difficult to obtain convergent chaotic solution in an arbitrarily given finite interval of time. Some researchers even believe that all chaotic responses are simply numerical noise and have nothing to do with solutions of differential equations. However, using 1200 CPUs of the National Supercomputer TH-A1 at Tianjin and a parallel integration algorithm of the so-called "Clean Numerical Simulation" (CNS) based on the 3500th-order Taylor expansion and data in 4180-digit multiple precision, one can obtain reliable convergent chaotic solution of the Lorenz equation in a rather long time interval [0,10 000]. This supports Lorenz's optimistic viewpoint [Lorenz 2008] that "numerical approximations can converge to a chaotic true solution throughout any finite range of time". It also supports Tucker's proof [Tucker 1999, 2002] for the famous Smale's 14th problem that the strange attractor of the Lorenz equation indeed exists.

The Chen circuit system, a three-dimensional autonomous system with intriguing dynamics, has gained considerable attention due to its potential applications in diverse scientific fields. In this article, a Chen-like system is defined and a comprehensive dynamics studied. The system exhibits chaotic dynamics, limit cycles, and bifurcations, making it a captivating subject of study. By employing numerical simulations and analytical techniques, we explore the system’s stability and identify critical parameter values leading to qualitative changes. Notably, we delve into Hopf bifurcations, which give rise to stability changes and the emergence of limit cycles. Furthermore, we analyze the fractal dimension of the system’s attractor, providing insights into its complexity and self-similarity. Through a systematic examination of the Chen-like system, we deepen our understanding of its intricate dynamics and offer valuable insights into the underlying mechanisms. The findings contribute to the field of dynamical systems and hold potential implications in areas such as chaos-based secure communications, signal processing, and nonlinear control. This work serves as a valuable reference for researchers and practitioners interested in the dynamics and bifurcation analysis of nonlinear systems.

We study an impulsive delay differential predator–prey model with Holling type II functional response. The stability of the trivial equilibrium is analyzed by means of impulsive Floquet theory providing a sufficient condition for extinction. Using coincidence degree theory we show the existence of positive periodic solutions. The system is then analyzed numerically, revealing that the presence of delays and impulses may lead to chaotic solutions, quasi-periodic solutions, or multiple periodic solutions. Several simulations and examples are presented.