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Polynomial Functional Dynamical Systems
The book is about the global stability and bifurcation of equilibriums in polynomial functional systems. Appearing and switching bifurcations of simple and higher-order equilibriums in the polynomial functional systems are discussed, and such bifurcations of equilibriums are not only for simple equilibriums but for higher-order equilibriums. The third-order sink and source bifurcations for simple equilibriums are presented in the polynomial functional systems. The third-order sink and source switching bifurcations for saddle and nodes are also presented, and the fourth-order upper-saddle and lower-saddle switching and appearing bifurcations are presented for two second-order upper-saddles an...
Discretization and Implicit Mapping Dynamics
This unique book presents the discretization of continuous systems and implicit mapping dynamics of periodic motions to chaos in continuous nonlinear systems. The stability and bifurcation theory of fixed points in discrete nonlinear dynamical systems is reviewed, and the explicit and implicit maps of continuous dynamical systems are developed through the single-step and multi-step discretizations. The implicit dynamics of period-m solutions in discrete nonlinear systems are discussed. The book also offers a generalized approach to finding analytical and numerical solutions of stable and unstable periodic flows to chaos in nonlinear systems with/without time-delay. The bifurcation trees of periodic motions to chaos in the Duffing oscillator are shown as a sample problem, while the discrete Fourier series of periodic motions and chaos are also presented. The book offers a valuable resource for university students, professors, researchers and engineers in the fields of applied mathematics, physics, mechanics, control systems, and engineering.
Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems
In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is presented for a better understanding of global behaviors and motion transitions for one periodic motion to another one. A 1-dimensional (1-D), time-delayed, nonlinear dynamical system is considered as an example to show how to determine the global sequential scenarios of the bifurcation trees of periodic motions to chaos. All stable and unstable periodic motions on the bifurcation trees can be determined. Especially, the unstable periodic motions on the bifurcation trees cannot be achieved from the traditional analytical methods, and such unstable periodic motions ...
Toward Analytical Chaos in Nonlinear Systems
Exact analytical solutions to periodic motions in nonlinear dynamical systems are almost not possible. Since the 18th century, one has extensively used techniques such as perturbation methods to obtain approximate analytical solutions of periodic motions in nonlinear systems. However, the perturbation methods cannot provide the enough accuracy of analytical solutions of periodic motions in nonlinear dynamical systems. So the bifurcation trees of periodic motions to chaos cannot be achieved analytically. The author has developed an analytical technique that is more effective to achieve periodic motions and corresponding bifurcation trees to chaos analytically. Toward Analytical Chaos in Nonli...
Discontinuous Dynamical Systems
“Discontinuous Dynamical Systems” presents a theory of dynamics and flow switchability in discontinuous dynamical systems, which can be as the mathematical foundation for a new dynamics of dynamical system networks. The book includes a theory for flow barriers and passability to boundaries in discontinuous dynamical systems that will completely change traditional concepts and ideas in the field of dynamical systems. Edge dynamics and switching complexity of flows in discontinuous dynamical systems are explored in the book and provide the mathematical basis for developing the attractive network channels in dynamical systems. The theory of bouncing flows to boundaries, edges and vertexes i...
Two-dimensional Self and Product Cubic Systems, Vol. I
This book, the twelfth of 15 related monographs on Cubic Systems, discusses self and product cubic systems with a self-linear and crossing-quadratic product vector field. Equilibrium series with flow singularity are presented and the corresponding switching bifurcations are discussed. The volume explains how the equilibrium series with connected hyperbolic and hyperbolic-secant flows exist in such cubic systems, and that the corresponding switching bifurcations are obtained through the inflection-source and sink infinite-equilibriums. Finally, the author illustrates how, in such cubic systems, the appearing bifurcations include saddle-source (sink) for equilibriums and inflection-source and sink flows for the connected hyperbolic flows, and the third-order saddle, sink and source are the appearing and switching bifurcations for saddle-source (sink) with saddles, source and sink, and also for saddle, sink and source.
Vibro-impact Dynamics
Presents a systematic view of vibro-impact dynamics based onthe nonlinear dynamics analysis Comprehensive understanding of any vibro-impact system iscritically impeded by the lack of analytical tools viable forproperly characterizing grazing bifurcation. The authors establishvibro-impact dynamics as a subset of the theory of discontinuoussystems, thus enabling all vibro-impact systems to be explored andcharacterized for applications. Vibro-impact Dynamics presents an original theoreticalway of analyzing the behavior of vibro-impact dynamics that can beextended to discontinuous dynamics. All topics are logicallyintegrated to allow for vibro-impact dynamics, the central theme,to be presented. ...
Analytical Routes to Chaos in Nonlinear Engineering
Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations. This works well up to some accuracy and some range for the input values, but some interesting phenomena such as chaos and singularities are hidden by linearization and perturbation analysis. It follows that some aspects of the behavior of a nonlinear system appear commonly to be chaotic, unpredictable or counterintuitive. Although such a chaotic behavior may resemble a random behavior, it is absolutely deterministic. Analy...
Bifurcation Dynamics in Polynomial Discrete Systems
This is the first book focusing on bifurcation dynamics in 1-dimensional polynomial nonlinear discrete systems. It comprehensively discusses the general mathematical conditions of bifurcations in polynomial nonlinear discrete systems, as well as appearing and switching bifurcations for simple and higher-order singularity period-1 fixed-points in the 1-dimensional polynomial discrete systems. Further, it analyzes the bifurcation trees of period-1 to chaos generated by period-doubling, and monotonic saddle-node bifurcations. Lastly, the book presents methods for period-2 and period-doubling renormalization for polynomial discrete systems, and describes the appearing mechanism and period-doublization of period-n fixed-points on bifurcation trees for the first time, offering readers fascinating insights into recent research results in nonlinear discrete systems.
Periodic Flows to Chaos in Time-delay Systems
This book for the first time examines periodic motions to chaos in time-delay systems, which exist extensively in engineering. For a long time, the stability of time-delay systems at equilibrium has been of great interest from the Lyapunov theory-based methods, where one cannot achieve the ideal results. Thus, time-delay discretization in time-delay systems was used for the stability of these systems. In this volume, Dr. Luo presents an accurate method based on the finite Fourier series to determine periodic motions in nonlinear time-delay systems. The stability and bifurcation of periodic motions are determined by the time-delayed system of coefficients in the Fourier series and the method for nonlinear time-delay systems is equivalent to the Laplace transformation method for linear time-delay systems.
