Revisiting linear augmentation for stabilizing stationary solutions: potential pitfalls and their application

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the highlight of this scheme but at the same time questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Appropriate examples from conservative as well as dissipative systems are presented in this regard and potential applications for relevant observations in dissipative predator--prey systems are also discussed.Comment: updated version with title change, additional figures, text and explanation

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.Comment: submitted to EP

Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Many real-world processes tend to be chaotic and also do not lead to satisfactory analytical modelling. It has been shown here that for such chaotic processes represented through short chaotic noisy time-series, a multi-input and multi-output recurrent neural networks model can be built which is capable of capturing the process trends and predicting the future values from any given starting condition. It is further shown that this capability can be achieved by the Recurrent Neural Network model when it is trained to very low value of mean squared error. Such a model can then be used for constructing the Bifurcation Diagram of the process leading to determination of desirable operating conditions. Further, this multi-input and multi-output model makes the process accessible for control using open-loop/closed-loop approaches or bifurcation control etc. All these studies have been carried out using a low dimensional discrete chaotic system of Hénon Map as a representative of some real-world processes