×

Time-delayed chaotic dynamical systems. From theory to electronic experiment. (English) Zbl 1397.34001

SpringerBriefs in Applied Sciences and Technology. Nonlinear Circuits. Cham: Springer (ISBN 978-3-319-70992-5/pbk; 978-3-319-70993-2/ebook). xi, 114 p. (2018).
In the “Preface”, the authors write “Delays arise in various natural and man-made systems due to the finite speed of signal propagation, finite response time, and switching speed. The presence of time delay in a dynamical system makes it infinite dimensional and if the system is nonlinear it may give rise to many interesting phenomena, like bifurcation, chaos, and multistability”. The book aims at studying the application of Delay Differential Equations, in particular the above mentioned phenomena arising in these, to several fields of Electronics with a special focus on numerical studies and electronic experiments.
In a short introduction some examples of relevant classes of delay differential equations are given. The second chapter, “First-Order Time-Delayed Chaotic Systems: Design and Experiment”, contains analysis of systems with bimodal and unimodal nonlinearities, both of which are described as representable “by closed-form mathematical functions”. The third chapter is titled “Chaotic Time-Delayed System with Hard Nonlinearity: Design and Characterization”. Chapters 4 and 5 are devoted to “Collective Behavior” which refers to the coupling of several systems. Chapter 4 is titled “Collective Behavior-I: Synchronization in Hyperchaotic Time-Delayed Oscillators Coupled Through a Common Environment” and Chapter 5 “Collective Behavior-II: Amplitude Death and the Corresponding Transitions in Coupled Chaotic Time-Delayed Systems”. The chapters of the main body contain essentially stability and bifurcation analysis with numerical and experimental studies. Hopf bifurcations, Lyapunov exponents, Krasovskii-Lyapunov Theory and Synchronization are among the discussed topics. The Epilogue outlines some “Future directions”. The book closes with “Appendix A: A brief tutorial on XPPAUT and LabVIEW”, which are, respectively, a numerical package and one for the acquisition of experimental data.

MSC:

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
94C05 Analytic circuit theory
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K35 Control problems for functional-differential equations

Software:

LabVIEW; XPPAUT
PDFBibTeX XMLCite
Full Text: DOI