Although continuous systems such as the Chua circuit are known as systems with hidden attractors, hidden attractors also exist in classical discrete maps, such as a generalized Hénon map. A hidden attractor is an attractor that does not overlap with its own attracting region in its vicinity, which makes it difficult to visualize. In this paper, a local bifurcation analysis method for discrete maps is described, and the bifurcation analysis of the generalized Hénon map is performed using the method. The bifurcation structure, as the parameters are changed, shows a certain law, and the interesting Neimark–Sacker bifurcation and period‐doubling bifurcation are confirmed to occur simultaneously. It was also found that the hidden attractors exist in the rectangular characteristic chaotic regions, and they appear relatively frequently near the window of chaos. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.
This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finite-dimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.
In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.
The paper studies the stabilization problem for a dynamic neural network disturbed by additive noise. The stabilization is achieved from the inverse optimal control approach, introduced in nonlinear control theory, using a quadratic Lyapunov function. A simple feedback control law is derived, which ensures that the neural network state is globally asymptotically stable in probability.
No abstract is provided for this article.
This paper studies the problem of making an arbitrary discrete system chaotic, or enhancing its existing chaotic behaviors, by designing a universal controller. The only assumption is that the arbitrarily given system has a bounded first derivative in a (small) region of interest.
Look around—Is there anything that is not networked to something else out there?
This technical note considers consensus or flocking of coupled multiple double-integrator agents, in which the velocity coupling and position coupling (VCPC) between agents, respectively, are generally non-equal, differing from the equal VCPC setting in the vast literature. This technical note addresses the problem in two aspects: the convergence condition, and particularly the designated convergence rate (or designated convergence margin) condition that was rarely investigated for either flocking, formation, or consensus. Correspondingly, this technical note has two contributions: 1) some necessary and sufficient convergence conditions for flocking or consensus are established, which extend the existing results in the field; and particularly 2) some necessary and sufficient conditions are derived, to guarantee the designated convergence rate of consensus or flocking, which are more valuable for systems design than just convergence analysis performed by most other works on multi-agent systems.
A new method is introduced for controlling chaos in continuous systems, and stabilizing one of the unstable periodic orbits embedded in the chaotic attractor. The stabilization of the orbit is obtained by applying a discontinuous perturbation to one parameter of the system in a neighborhood of the orbit. The analysis is carried out by means of Poincare surfaces, which makes possible to develop the method based on previous results applicable to discrete systems. The discrete nature of the method allows to stabilize three-dimensional systems applying only two changes to the parameter, although in principle more changes may be applied for each period of the orbit. The method is easily generalized to n-dimensional continuous systems of higher order.
This paper proposes a method of generating multi-scroll chaos using second-order linear systems with a hysteresis series. It shows that multi-scroll chaos can be produced in any direction in the phase plane. Furthermore, two-dimensional multi-scroll chaos can be generated as well. Both computer simulations and circuitry implementation have verified the multi-scroll chaos generation scheme.
In this paper, we derive a sharp condition on the equivalence of topological transitivity among an interval autonomous dynamical system, its induced set-valued system and induced normal fuzzified system. We also prove that their sensitivity (resp., total transitivity) are equivalent. For a general non-autonomous dynamical system, we show the equivalence of topological mixing (resp., mild mixing, cofinite sensitivity, multi-sensitivity and syndetic sensitivity) among the non-autonomous system and its two induced systems. In contrast, we construct a non-autonomous system that is weakly mixing but neither of its two induced systems is weakly mixing. We extend the topological equi-conjugacy between two non-autonomous systems to their two induced systems. Finally, we verify some basic properties of topological entropy among a non-autonomous system and its two induced systems, and establish some sufficient conditions for the topological equi-conjugacy between the fuzzification of a non-autonomous system and a subshift of finite type.
In October each year, Nobel Prizes are doled out. My excitement this time came not only from the numbers--5 out of 13 names on the list of the 2009 Nobel Laureates are women, the largest number ever to receive these most prestigious awards in a single year in the history, but also from the broad scope of the awarded contributions, ranging from protecting and deciphering DNA to international diplomacy and to optical fibers.
This paper offers some new observer-based criteria for discrete-time generalized chaos synchronization, including synchronization conditions on construction of a synchronizing system via a suitable non-singular coordinate transform, using two scalar driving signals, and by using an extended observer.
