Narrow Results

In this paper, the problem of the global asymptotic stability (GAS) of a class of delayed neural network is investigated. Under the generalization of dropping the boundedness and differentiability hypotheses for activation functions, using some existing results for the existence and uniqueness of the equilibrium point, we obtain a couple of general results concerning GAS by means of Lyapunov functional method without the assumption of symmetry of interconnection matrix. Our results improve and extend some previous works of other researchers. Moreover, our conditions are presented in terms of system parameters, which have leading significance in designs and applications of GAS for Hopfield neural network (HNNs) and delayed cellular neural network (DCNNs).

We consider the following system of delay differential equations

This paper addresses bifurcation properties of equilibria in lumped electrical circuits. The goal is to tackle these properties in circuit-theoretic terms, characterizing the bifurcation conditions in terms of the underlying network digraph and the electrical features of the circuit devices. The attention is mainly focused on so-called singular bifurcations, resulting from the semistate (differential-algebraic) nature of circuit models, but the scope of our approach seems to extend to other types of bifurcations. The bifurcation analysis combines different tools coming from graph theory (such as proper trees in circuit digraphs, Maxwell's determinantal expansions or the colored branch theorem) with several results from linear algebra (matrix pencils, the Cauchy–Binet formula, Schur complements). Several examples illustrate the results.

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

In this letter we investigate the role of complex fixed-points in finding hidden attractors in chaotic flows with no equilibria. If these attractors could be found by starting the trajectory in the neighborhood of complex fixed-points, maybe it would be better not to call them hidden.

Discovering unknown features of no-equilibrium systems with hidden strange attractors is an attractive research topic. This paper presents a novel no-equilibrium chaotic system that is constructed by using a state feedback controller. Interestingly, the new system can exhibit multiwing butterfly attractors. Moreover, a new chaotic system with an infinite number of equilibrium points, which can generate multiscroll attractors, is also proposed by applying the introduced methodology.

Chua's circuit, an archetypal example of nonsmooth dynamical systems, exhibits mostly discontinuous bifurcations. More complex dynamical phenomena of Chua's circuit are presented here due to discontinuity-induced bifurcations. Some new kinds of classical bifurcations are revealed and analyzed, including the coexistence of two classical bifurcations and bifurcations of equilibrium manifolds. The local dynamical behavior of the boundary equilibrium points located on switch boundaries is found to be determined jointly by the Jacobian matrices evaluated before and after switching. Some new discontinuous bifurcations are also observed, such as the coexistence of two discontinuous and one classical bifurcation.

The nonlinear dynamical behavior of an atmospheric circulation in a beta-plane channel is examined on a five-spectral mode model, truncated from the Charney and DeVore quasi-geostrophic equation. Bifurcation and chaos are observed when subjected to a topographic driving disturbance and a thermally driving zonal source. An equilibrium state undergoes supercritical Hopf bifurcation and becomes a stable periodic state with respect to the magnitude of the thermally driving source, whereas the periodic state undergoes a subcritical Hopf bifurcation and transforms into a low-index equilibrium state with respect to the increasing topographic driving disturbance. The stable periodic state further develops into a pair of stable periodic states when increasing the thermally driving source. The first one with the period of 4.3 days exhibits an oscillation of strong and weak zonal flow patterns, whereas the second one with the period of 6.8 days demonstrates a fluctuation amongst weak zonal disturbance flow patterns. Moreover, the two periodic states transform respectively into chaos through separate period-doubling cascades with the further development of the thermally driving source.

Although many chaotic systems have been introduced in the literature, a few of them possess uncountably infinite equilibrium points. The aim of our short work is to widen the current knowledge of the chaotic systems with an infinite number of equilibria. A three-dimensional system with special properties, for example, exhibiting chaotic attractor with circular equilibrium, chaotic attractor with ellipse equilibrium, chaotic attractor with square-shaped equilibrium, and chaotic attractor with rectangle-shaped equilibrium, is proposed.

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

The chaotic systems with hidden attractors, such as chaotic systems with a stable equilibrium, chaotic systems with infinite equilibria or chaotic systems with no equilibrium have been investigated recently. However, the relationships between them still need to be discovered. This work explains how to transform a system with one stable equilibrium into a new system with an infinite number of equilibrium points by using a memristive device. Furthermore, some other new systems with infinite equilibria are also constructed to illustrate the introduced methodology.

Applying some transformation to an autonomous system, we obtain a new system, which might keep the dynamical behavior of the original system or generate different dynamics. But this is often accompanied by the appearance of discontinuous points, where the vector field for the new system is not continuous at these points. We discuss the effects of the discontinuous points, and provide two methods to construct systems with any preassigned number of chaotic attractors via some transformation. The first one does not change the geometric structure of the attractors, since the discontinuous points are out of the basin of attraction. The second one might make the new systems have different dynamics, like multiscroll chaotic attractors, or infinitely many chaotic attractors. These results illustrate that both the equilibria and the discontinuous points affect the global dynamics.

We present the development of a new theory of the pitchfork bifurcation, which removes the perspective of the third derivative and a requirement of symmetry.

A new chaotic system having variable equilibrium is introduced in this paper. The presence of an infinite number of equilibrium points, a stable equilibrium, and no-equilibrium is observed in the system. Interestingly, this system is classified as a rare system with hidden attractors from the view point of computation. Complex dynamical behavior and a circuital implementation of the new system have been investigated in our work.

Designing chaotic systems with specific features is a very interesting topic in nonlinear dynamics. However most of the efforts in this area are about features in the structure of the equations, while there is less attention to features in the topology of strange attractors. In this paper, we introduce a new chaotic system with unique property. It has been designed in such a way that a specific property has been injected to it. This new system is analyzed carefully and its real circuit implementation is presented.

In the chaotic polynomial Lorenz-type systems (including Lorenz, Chen, Lü and Yang systems) and Rössler system, their equilibria are unstable and the number of the hyperbolic equilibria are no more than three. This paper shows how to construct a simple analytic (nonpolynomial) chaotic system that can have any preassigned number of equilibria. A special 3D chaotic system with no equilibrium is first presented and discussed. Using a methodology of adding a constant controller to the third equation of such a chaotic system, it is shown that a chaotic system with any preassigned number of equilibria can be generated. Two complete mathematical characterizations for the number and stability of their equilibria are further rigorously derived and studied. This system is very interesting in the sense that some complex dynamics are found, revealing many amazing properties: (i) a hidden chaotic attractor exists with no equilibria or only one stable equilibrium; (ii) the chaotic attractor coexists with unstable equilibria, including two/five unstable equilibria; (iii) the chaotic attractor coexists with stable equilibria and unstable equilibria, including one stable and two unstable equilibria/94 stable and 93 unstable equilibria; (iv) the chaotic attractor coexists with infinitely many nonhyperbolic isolated equilibria. These results reveal an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of some unusual chaotic systems.

This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some $3$D piecewise linear systems.

This paper proposes a novel three-dimensional autonomous chaotic system. Interestingly, when the system has infinitely many stable equilibria, it is found that the system also has infinitely many hidden chaotic attractors. We show that the period-doubling bifurcations are the routes to chaos. Moreover, the Hopf bifurcations at all equilibria are investigated and it is also found that all the Hopf bifurcations simultaneously occur. Furthermore, the approximate expressions and stabilities of bifurcating limit cycles are obtained by using normal form theory and bifurcation theory.

This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, which are not in the sense of Shil’nikov criteria. First, under some conditions, the system has infinitely many isolated equilibria. Moreover, all equilibria are nonhyperbolic and give the first Lyapunov coefficient. Furthermore, when all equilibria are weak saddle-foci, the system also has infinitely many chaotic attractors. Besides, the Lyapunov exponents spectrum and bifurcation diagram are given. Second, under another condition, all the equilibria constitute a curve and there exist infinitely many singular degenerated heteroclinic orbits. At the same time, the system can show infinitely many chaotic attractors.

Social insect colonies’ robust and efficient collective behaviors without any central control contribute greatly to their ecological success. Colony migration is a leading subject for studying collective decision-making in migration. In this paper, a general colony migration model with Hill functions in recruitment is proposed to investigate the underlying decision making mechanism and the related dynamical behaviors. Our analysis provides the existence and stability of equilibrium, and the global dynamical behavior of the system. To understand how piecewise functions and Hill functions in recruitment impact colony migration dynamics, the comparisons are performed in both analytic results and bifurcation analysis. Our theoretical results show that the dynamics of the migration system with Hill functions in recruitment differs from that of the migration system with piecewise functions in the following three aspects: (1) all population components in our colony migration model with Hill functions in recruitment are persistent; (2) the colony migration model with Hill functions in recruitment has saddle and saddle-node bifurcations, while the migration system with piecewise functions does not; (3) the system with Hill functions has only equilibrium dynamics, i.e. either has a global stability at one interior equilibrium or has bistablity among two locally stable interior equilibria. Bifurcation analysis shows that the geometrical shape of the Hill functions greatly impacts the dynamics: (1) the system with flatter Hill functions is less likely to exhibit bistability; (2) the system with steeper functions is prone to exhibit bistability, and the steady state of total active workers is closer to that of active workers in the system with piecewise function.