Riddled basins of chaotic synchronization and unstable dimension variability in coupled Lorenz-like systems

Unstable dimension variability is an extreme form of non-hyperbolic behavior that causes a severe shadowing breakdown of chaotic trajectories. This phenomenon can occur in coupled chaotic systems possessing symmetries, leading to an invariant attractor with riddled basins of attraction. We consider the coupling of two Lorenz-like systems, which exhibits chaotic synchronized and anti-synchronized states, with their respective basins of attraction. We demonstrate that these basins are riddled, in the sense that they verify both the mathematical conditions for their existence, as well as the characteristic scaling laws indicating power-law dependence of parameters. Our simulations have shown that a biased random-walk model for the log-distances to the synchronized manifold can accurately predict the scaling exponents near blowout bifurcations in this high-dimensional coupled system. The behavior of the finite-time Lyapunov exponents in directions transversal to the invariant subspace has been used as numerical evidence of unstable dimension variability.

Generation of multi-scrolls in corona virus disease 2019 (COVID-19) chaotic system and its impact on the zero-covid policy

In this paper, we discussed the impossibility of achieving zero-covid cases per day for all time with the help of fuzzy theory, while how a single case can trigger chaotic situation in the nearby city is elaborated using multi-scrolls. To accomplish this goal, we consider the number of new cases per day; [Formula: see text] to be the preferred state variable by restricting its value to the interval (0, 1). One can need to think of [Formula: see text] as a member of a fuzzy set and provide that set with appropriate membership functions. Moreover, how a single incident in one city can spread chaos to other cities is also addressed at length, using multi-scroll attractors and the signal excitation function. In addition, a bifurcation diagram of daily new instances vs the parameter [Formula: see text] is shown, elaborating that daily new cases may show a decrease under strict rules and regulations, but can again lead to chaos. Apart from biologist, this paper can play vital role for engineers as well in a sense that, a signal function can be embedded in non-symmetric systems for the creation of multi-scroll attractors in all directions using a generalized algorithm that has been designed in the current work. Finally, it is our future target to show that the covid is leading towards influenza and will be no more dangerous as was in the past.

Reducing network size and improving prediction stability of reservoir computing

Reservoir computing is a very promising approach for the prediction of complex nonlinear dynamical systems. Besides capturing the exact short-term trajectories of nonlinear systems, it has also proved to reproduce its characteristic long-term properties very accurately. However, predictions do not always work equivalently well. It has been shown that both short- and long-term predictions vary significantly among different random realizations of the reservoir. In order to gain an understanding on when reservoir computing works best, we investigate some differential properties of the respective realization of the reservoir in a systematic way. We find that removing nodes that correspond to the largest weights in the output regression matrix reduces outliers and improves overall prediction quality. Moreover, this allows to effectively reduce the network size and, therefore, increase computational efficiency. In addition, we use a nonlinear scaling factor in the hyperbolic tangent of the activation function. This adjusts the response of the activation function to the range of values of the input variables of the nodes. As a consequence, this reduces the number of outliers significantly and increases both the short- and long-term prediction quality for the nonlinear systems investigated in this study. Our results demonstrate that a large optimization potential lies in the systematical refinement of the differential reservoir properties for a given dataset.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results

Chaotic attractors appear in various physical and biological models; however, rigorous proofs of their existence and bifurcations are rare. In this paper, we construct a simple piecewise-smooth model which switches between three three-dimensional linear systems that yield a singular hyperbolic attractor whose structure and bifurcations are similar to those of the celebrated Lorenz attractor. Due to integrability of the linear systems composing the model, we derive a Poincaré return map to rigorously prove the existence of the Lorenz-type attractor and explicitly characterize bifurcations that lead to its birth, structural changes, and disappearance. In particular, we analytically calculate a bifurcation curve explicit in the model's parameters that corresponds to the formation of homoclinic orbits of a saddle, often referred to as a "homoclinic butterfly." We explicitly indicate the system's parameters that yield a bifurcation of two heteroclinic orbits connecting the saddle fixed point and two symmetrical saddle periodic orbits that gives birth to the chaotic attractor as in the Lorenz system. These analytical tasks are out of reach for the original nonintegrable Lorenz system. Our approach to designing piecewise-smooth dynamical systems with a predefined chaotic attractor and exact solutions may open the door to the synthesis and rigorous analysis of hyperbolic attractors.

Multistability in Chua's circuit with two stable node-foci

Only using one-stage op-amp based negative impedance converter realization, a simplified Chua's diode with positive outer segment slope is introduced, based on which an improved Chua's circuit realization with more simpler circuit structure is designed. The improved Chua's circuit has identical mathematical model but completely different nonlinearity to the classical Chua's circuit, from which multiple attractors including coexisting point attractors, limit cycle, double-scroll chaotic attractor, or coexisting chaotic spiral attractors are numerically simulated and experimentally captured. Furthermore, with dimensionless Chua's equations, the dynamical properties of the Chua's system are studied including equilibrium and stability, phase portrait, bifurcation diagram, Lyapunov exponent spectrum, and attraction basin. The results indicate that the system has two symmetric stable nonzero node-foci in global adjusting parameter regions and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.

A Novel Method for Constructing Grid Multi-Wing Butterfly Chaotic Attractors via Nonlinear Coupling Control

A new method is presented to construct grid multi-wing butterfly chaotic attractors. Based on the three-dimensional Lorenz system, two first-order differential equations are added along with one linear coupling controller, respectively. And a piecewise linear function, which is taken into the linear coupling controller, is designed to form a nonlinear coupling controller; thus a five-dimensional chaotic system is produced, which is able to generate gird multi-wing butterfly chaotic attractors. Through the analysis of the equilibrium points, Lyapunov exponent spectrums, bifurcation diagrams, and Poincaré mapping in this system, the chaotic characteristic of the system is verified. Apart from the research above, an electronic circuit is designed to implement the system. The circuit experimental results are in accordance with the results of numerical simulation, which verify the availability and feasibility of this method.