A novel double-convection chaotic attractor, its adaptive control and circuit simulation

A 3-D novel double-convection chaotic system with three nonlinearities is proposed in this research work. The dynamical properties of the new chaotic system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, stability analysis of equilibria, etc. Adaptive control and synchronization of the new chaotic system with unknown parameters are achieved via nonlinear controllers and the results are established using Lyapunov stability theory. Furthermore, an electronic circuit realization of the new 3-D novel chaotic system is presented in detail. Finally, the circuit experimental results of the 3-D novel chaotic attractor show agreement with the numerical simulations

Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force

We identify a novel route to the birth of a strange nonchaotic attractor (SNA) in a quasiperiodically forced electronic circuit with a nonsinusoidal (square wave) force as one of the quasiperiodic forces through numerical and experimental studies. We find that bubbles appear in the strands of the quasiperiodic attractor due to the instability induced by the additional square wave type force. The bubbles then enlarge and get increasingly wrinkled as a function of the control parameter. Finally, the bubbles get extremely wrinkled (while the remaining parts of the strands of the torus remain largely unaffected) resulting in the birth of the SNA which we term as the \emph{bubbling route to SNA}. We characterize and confirm this birth from both experimental and numerical data by maximal Lyapunov exponents and their variance, Poincar\'e maps, Fourier amplitude spectra and spectral distribution function. We also strongly confirm the birth of SNA via the bubbling route by the distribution of the finite-time Lyapunov exponents.Comment: 11 pages. 11 figures, Accepted for publication in Phys. Rev.

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

Design of time delayed chaotic circuit with threshold controller

A novel time delayed chaotic oscillator exhibiting mono- and double scroll complex chaotic attractors is designed. This circuit consists of only a few operational amplifiers and diodes and employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of piecewise linear function facilitates controlling the shape of the attractors. This is demonstrated by constructing the phase portraits of the attractors through numerical simulations and hardware experiments. Based on these studies, we find that this circuit can produce multi-scroll chaotic attractors by just introducing more number of threshold values.Comment: 21 pages, 12 figures; Submitted to IJB

Optimal Piecewise-Linear Approximation of the Quadratic Chaotic Dynamics

This paper shows the influence of piecewise-linear approximation on the global dynamics associated with autonomous third-order dynamical systems with the quadratic vector fields. The novel method for optimal nonlinear function approximation preserving the system behavior is proposed and experimentally verified. This approach is based on the calculation of the state attractor metric dimension inside a stochastic optimization routine. The approximated systems are compared to the original by means of the numerical integration. Real electronic circuits representing individual dynamical systems are derived using classical as well as integrator-based synthesis and verified by time-domain analysis in Orcad Pspice simulator. The universality of the proposed method is briefly discussed, especially from the viewpoint of the higher-order dynamical systems. Future topics and perspectives are also provide

Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. Firstly we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the R\"ossler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures -- thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length and network diameter are highly sensitive to the interior crisis captured in this particular data set