Introduction to Infinite-Equilibriums in Dynamical Systems

This book is the fourth of 15 related monographs presents systematically a theory of crossing-cubic nonlinear systems. In this treatment, at least one vector field is crossing-cubic, and the other vector field can be constant, crossing-linear, crossing-quadratic, and crossing-cubic. For constant vector fields, the dynamical systems possess 1-dimensional flows, such as parabola and inflection flows plus third-order parabola flows. For crossing-linear and crossing-cubic systems, the dynamical systems possess saddle and center equilibriums, parabola-saddles, third-order centers and saddles (i.e, (3rd UP+:UP+)-saddle and (3rdUP-:UP-)-saddle) and third-order centers (i.e., (3rd DP+:DP-)-center, (3rd DP-, DP+)-center) . For crossing-quadratic and crossing-cubic systems, in addition to the first and third-order saddles and centers plus parabola-saddles, there are (3:2)parabola-saddle and double-inflection saddles, and for the two crossing-cubic systems, (3:3)-saddles and centers exist. Finally,the homoclinic orbits with centers can be formed, and the corresponding homoclinic networks of centers and saddles exist.

Readers will learn new concepts, theory, phenomena, and analytic techniques, including

· Constant and crossing-cubic systems

· Crossing-linear and crossing-cubic systems

· Crossing-quadratic and crossing-cubic systems

· Crossing-cubic and crossing-cubic systems

· Appearing and switching bifurcations

· Third-order centers and saddles

· Parabola-saddles and inflection-saddles

· Homoclinic-orbit network with centers

· Appearing bifurcations

Quadratic and Cubic Product Systems.- Inflection Singularity and Bifurcation Dynamics.- Saddle-node and hyperbolic-flow singular dynamics.

Self and product cubic systems.- Second and third order equibriliums.- Equilibrium series and switching dynamics.-  Saddle nodes and hyperbolic flow series.- Simple equilibrium series and switching dynamics.

Constant and Product Polynomial Systems.- Proof of Theorem 1.1.- Singular flows bifurcaions and networks.

Introduction.- Homoclinic Networks without Centers.- Bifurcations for Homoclinic Networks without Centers.- Homoclinic Networks with Centers.- Bifurcations for Homoclinic Networks with Centers.

Self and Product Polynomial Systems.- Proof of Theorem 1.1.- Singular Equilibria and Flows, and Simple Networks.