Conclusive Remarks

Abstract

In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos is presented through a 1-dimensional, time-delayed, nonlinear dynamical system. The sequential periodic motions in such a 1-dimensional time-delayed system is achieved semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential order of bifurcation trees of periodic motions to chaos is given by $${G_{1\left( S \right)}} \triangleleft {G_{1\left( A \right)}} \triangleleft {G_{3\left( S \right)}} \triangleleft {G_{2\left( A \right)}} \cdots \triangleleft {G_{m\left( A \right)}} \triangleleft {G_{2m + 1\left( S \right)}} \triangleleft \cdots \left( {m = 1,2, \ldots } \right),$$ where Gm(A) represents a global bifurcation tree of an asymmetric period-m motion to chaos and G2m+1(S) is for a global bifurcation tree of a symmetric period-(2m + 1) motion to chaos. Each bifurcation tree of a specific periodic motion to chaos are presented in detail. The bifurcation tree appearance and vanishing are determined by the saddle-node bifurcation, and the cascaded period-doubled periodic solutions are determined by the period-doubling bifurcation. From finite Fourier series, harmonic amplitude and harmonic phases for periodic motions on the global bifurcation tree are obtained for frequency analysis. Numerical illustrations of periodic motions are given for complex periodic motions in global bifurcation trees. The rich dynamics of the 1-D, delayed, nonlinear dynamical system is presented. Such global sequential periodic motions to chaos also exist in nonlinear dynamical systems.