SINGLE LINEAR-BIVARIATE CUBIC SYSTEMS

This book is about the nonlinear dynamics of single linear-bivariate cubic dynamical systems. For such cubic dynamical systems, the inflection-flows and third-order parabola flows exist for appearing bifurcations. The inflection-flows are for appearing bifurcations of two parabola flows on the same direction. The third-order parabola flows are for the appearing bifurcation of inflection and parabola flows, and for the appearing bifurcations of up, down and up-parabola flows or down, up and down-parabola flows. Third-order parabola flow are for the appearing bifurcation among the up and down-parabola flows. There are four types of infinite-equilibriums: (i) The inflection-source and sink infinite-equilibriums are for the switching bifurcations of two parabola flows on the two-directions. (ii) The parabola-source and sink infinite-equilibriums are for the switching bifurcations of parabola and inflection flows on the two-directions. (iii) The inflection-saddle infinite-equilibriums are for the switching bifurcation of two inflection flows in two directions. (iv) The parabola-saddle infinite-equilibriums are for the switching bifurcations of parabola and inflection flows on the two-directions are the switching bifurcations for parabola and inflection flows. Such switching bifurcations for 1-dimensional flow are based on the infinite-equilibriums, which will help one understand global dynamics in nonlinear cubic systems.

Contents:

• Constant and Single-Linear-Bivariate Cubic Systems:
  • Constant and Single-Linear-Bivariate Cubic Vector Fields
  • Proof of Theorem 1.1
  • Inflection and Parabola Flows
  • Parabola Flows Series
• Single-Linear-Bivariate Linear and Cubic Systems:
  • Single-Linear-Bivariate Linear and Cubic Vector Fields
  • Theorem Proof
  • Third-Order Parabola Flows with Bifurcations
  • Second-Order Singular Single-Linear-Bivariate Cubic Systems
  • Parabola Flows' Series and Switching Bifurcations
• Single-Linear-Bivariate Quadratic and Cubic Systems:
  • Single-Linear-Bivariate Quadratic and Cubic Vector Fields
  • Proof of Theorem 3.1
  • Second-Order Singular Quadratic and Cubic Systems
  • Singular Parabola Flows and Infinite-Equilibriums
• Single-Linear-Bivariate Double-Cubic Systems:
  • Two Single-Linear-Bivariate Cubic Vector Fields
  • Proof of Theorem 4.1
  • Singular Single-Linear Bivariate Cubic Systems
  • Singular Parabola Flows and Infinite-Equilibriums

Readership: Advanced undergraduates, graduate students and researchers in mathematics, physics, mechanical and control engineering; as well as those studying dynamical systems and control in these disciplines.