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Exploring and establishing artificial neural networks with electrophysiological characteristics and high computational efficiency is a popular topic in the field of computer vision. Inspired by the working mechanism of primary visual cortex, pulse-coupled neural network (PCNN) can exhibit the characteristics of synchronous oscillation, refractory period, and exponential decay. However, electrophysiological evidence shows that the neurons exhibit highly complex non-linear dynamics when stimulated by external periodic signals. This chaos phenomenon, also known as the " butterfly effect", cannot be explained by all PCNN models. In this work, we analyze the main obstacle preventing PCNN models from imitating real primary visual cortex. We consider neuronal excitation as a stochastic process. We then propose a novel neural network, called continuous-coupled neural network (CCNN). Theoretical analysis indicates that the dynamic behavior of CCNN is distinct from PCNN. Numerical results show that the CCNN model exhibits periodic behavior under DC stimulus, and exhibits chaotic behavior under AC stimulus, which is consistent with the results of real neurons. Furthermore, the image and video processing mechanisms of the CCNN model are analyzed. Experimental results on image segmentation indicate that the CCNN model has better performance than the state-of-the-art of visual cortex neural network models., Comment: This manuscript has published in IEEE Transactions on Computers, 2022

In this letter we present a method of constructing dynamical systems with any preassigned number of equilibria by adding symmetry to another system with at least one equilibrium point. If the resulting system is chaotic, we call this procedure chaotification by symmetry since the resulting system is chaotic with many symmetrical equilibrium points., Comment: 6 pages, 0 figures

Using the predictor-corrector scheme, the fractional order diffusionless Lorenz system is investigated numerically. The effective chaotic range of the fractional order diffusionless system for variation of the single control parameter is determined. The route to chaos is by period-doubling bifurcation in this fractional order system, and some typical bifurcations are observed, such as the flip bifurcation, the tangent bifurcation, an interior crisis bifurcation, and transient chaos. The results show that the fractional-order diffusionless Lorenz system has complex dynamics with interesting characteristics., Comment: 10 pages, 11 figures, 33 references

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance., Comment: 21 pages, 19 figures

An extensive statistical survey of universal approximators shows that as the dimension of a typical dissipative dynamical system is increased, the number of positive Lyapunov exponents increases monotonically and the number of parameter windows with periodic behavior decreases. A subset of parameter space remains in which topological change induced by small parameter variation is very common. It turns out, however, that if the system's dimension is sufficiently high, this inevitable, and expected, topological change is never catastrophic, in the sense chaotic behavior is preserved. One concludes that deterministic chaos is persistent in high dimensions., Comment: 4 pages, 3 figures; Changes in response to referee comments

This paper examines the most probable route to chaos in high-dimensional dynamical systems in a very general computational setting. The most probable route to chaos in high-dimensional, discrete-time maps is observed to be a sequence of Neimark-Sacker bifurcations into chaos. A means for determining and understanding the degree to which the Landau-Hopf route to turbulence is non-generic in the space of $C^r$ mappings is outlined. The results comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss., Comment: 17 pages, 8 figures

This report investigates the dynamical stability conjectures of Palis and Smale, and Pugh and Shub from the standpoint of numerical observation and lays the foundation for a stability conjecture. As the dimension of a dissipative dynamical system is increased, it is observed that the number of positive Lyapunov exponents increases monotonically, the Lyapunov exponents tend towards continuous change with respect to parameter variation, the number of observable periodic windows decreases (at least below numerical precision), and a subset of parameter space exists such that topological change is very common with small parameter perturbation. However, this seemingly inevitable topological variation is never catastrophic (the dynamic type is preserved) if the dimension of the system is high enough., Comment: 31 pages, 19 figures