A New Chaotic System with Only Nonhyperbolic Equilibrium Points: Dynamics and Its Engineering Application

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.

Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model are experimentally investigated via the maximum Lyapunov exponent, phase portraits, and bifurcation diagrams. Numerical simulation demonstrates that the new network has various types of coexisting attractors. Moreover, it is of note that the interesting phenomena of extreme multistability is discovered, i.e., the coexistence of symmetric multiple attractors.

Sensory-motor cortices shape functional connectivity dynamics in the human brain

Large-scale biophysical circuit models provide mechanistic insights into the micro-scale and macro-scale properties of brain organization that shape complex patterns of spontaneous brain activity. We developed a spatially heterogeneous large-scale dynamical circuit model that allowed for variation in local synaptic properties across the human cortex. Here we show that parameterizing local circuit properties with both anatomical and functional gradients generates more realistic static and dynamic resting-state functional connectivity (FC). Furthermore, empirical and simulated FC dynamics demonstrates remarkably similar sharp transitions in FC patterns, suggesting the existence of multiple attractors. Time-varying regional fMRI amplitude may track multi-stability in FC dynamics. Causal manipulation of the large-scale circuit model suggests that sensory-motor regions are a driver of FC dynamics. Finally, the spatial distribution of sensory-motor drivers matches the principal gradient of gene expression that encompasses certain interneuron classes, suggesting that heterogeneity in excitation-inhibition balance might shape multi-stability in FC dynamics.

The Effects of Symmetry Breaking Perturbation on the Dynamics of a Novel Chaotic System with Cyclic Symmetry: Theoretical Analysis and Circuit Realization

Symmetry is an important property shared by a large number of nonlinear dynamical systems. Although the study of nonlinear systems with a symmetry property is very well documented, the literature has no sufficient investigation on the important issues concerning the behavior of such systems when their symmetry is broken or altered. In this work, we introduce a novel autonomous 3D system with cyclic symmetry having a piecewise quadratic nonlinearity [Formula: see text] where parameter [Formula: see text] is fixed and parameter [Formula: see text] controls the symmetry and the nonlinearity of the model. Obviously, for [Formula: see text] the system presents both cyclic and inversion symmetries while the inversion symmetry is explicitly broken for [Formula: see text]. We consider in detail the dynamics of the new system for both two regimes of operation by using classical nonlinear analysis tools (e.g. bifurcation diagrams, plots of largest Lyapunov exponents, phase space trajectory plots, etc.). Several nonlinear patterns are reported such as period doubling, periodic windows, parallel bifurcation branches, hysteresis, transient chaos, and the coexistence of multiple attractors of different topologies as well. One of the most gratifying features of the new system introduced in this work is the existence of several parameter ranges for which up to twelve disconnected periodic and chaotic attractors coexist. This latter feature is rarely reported, at least for a simple system like the one discussed in this work. An electronic analog device of the new cyclic system is designed and implemented in PSpice. A very good agreement is observed between PSpice simulation and the theoretical results.

A Hidden Chaotic System with Multiple Attractors

This paper reports a hidden chaotic system without equilibrium point. The proposed system is studied by the software of MATLAB R2018 through several numerical methods, including Largest Lyapunov exponent, bifurcation diagram, phase diagram, Poincaré map, time-domain waveform, attractive basin and Spectral Entropy. Seven types of attractors are found through altering the system parameters and some interesting characteristics such as coexistence attractors, controllability of chaotic attractor, hyperchaotic behavior and transition behavior are observed. Particularly, the Spectral Entropy algorithm is used to analyze the system and based on the normalized values of Spectral Entropy, the state of the studied system can be identified. Furthermore, the system has been implemented physically to verify the realizability.

A note on symmetry breaking in a non linear marketing model

In this paper, we consider the nonlinear discrete-time dynamic model proposed by Bischi and Baiardi (Chaos Solitons Fractals 79:145-156, 2015a). The model considers players with adaptive adjustment mechanisms towards the best reply and a form of inertia in adopting such mechanism. Moreover, we formulate an extension of the original model, where endogenous market size is considered. Through numerical simulations, we show that multiple attractors may exist in the presence of homogeneous agents and the emergence of non-synchronized trajectories both in the short (on-off intermittency) and long (global riddling) run. Therefore, the article highlights that strategic contexts exist in which the players’ knowledge of the market and the adoption of the best reply do not always allow the use of the representative agent’s rhetoric to describe the dynamics of the model.

Recent Advances in Dimensionality Reduction Modeling and Multistability Reconstitution of Memristive Circuit

Due to the introduction of memristors, the memristor-based nonlinear oscillator circuits readily present the state initial-dependent multistability (or extreme multistability), i.e., coexisting multiple attractors (or coexisting infinitely many attractors). The dimensionality reduction modeling for a memristive circuit is carried out to realize accurate prediction, quantitative analysis, and physical control of its multistability, which has become one of the hottest research topics in the field of information science. Based on these considerations, this paper briefly reviews the specific multistability phenomenon generating from the memristive circuit in the voltage-current domain and expounds the multistability control strategy. Then, this paper introduces the accurate flux-charge constitutive relation of memristors. Afterwards, the dimensionality reduction modeling method of the memristive circuits, i.e., the incremental flux-charge analysis method, is emphatically introduced, whose core idea is to implement the explicit expressions of the initial conditions in the flux-charge model and to discuss the feasibility and effectiveness of the multistability reconstitution of the memristive circuits using their flux-charge models. Furthermore, the incremental integral transformation method for modeling of the memristive system is reviewed by following the idea of the incremental flux-charge analysis method. The theory and application promotion of the dimensionality reduction modeling and multistability reconstitution are proceeded, and the application prospect is prospected by taking the synchronization application of the memristor-coupled system as an example.