Two-dimensional Crossing-Variable Cubic Nonlinear Systems - Hardcover

Luo, Albert C. J.

Hardcover

ISBN 10: 3031628098 ISBN 13: 9783031628092

Publisher: Springer, 2025

Synopsis

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

 

About this title

About the Author

Dr. Albert C. J. Luo is a Distinguished Research Professor at the Southern Illinois University Edwardsville, in Edwardsville, IL, USA. Dr. Luo worked on Nonlinear Mechanics, Nonlinear Dynamics, and Applied Mathematics. He proposed and systematically developed: (i) the discontinuous dynamical system theory, (ii) analytical solutions for periodic motions in nonlinear dynamical systems, (iii) the theory of dynamical system synchronization, (iv) the accurate theory of nonlinear deformable-body

dynamics, (v) new theories for stability and bifurcations of nonlinear dynamical systems. He discovered new phenomena in nonlinear dynamical systems. His methods and theories can help understanding and solving the Hilbert sixteenth problems and other nonlinear physics problems. The main results were scattered in 45 monographs in Springer, Wiley, Elsevier, and World Scientific, over 200 prestigious journal papers, and over 150 peer-reviewed conference papers.

From the Back Cover

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

• ​Develops equilibrium singularity and bifurcations in 2-dimensional self-cubic systems;
• Presents (1,3) and (3,3)-sink, source, and saddles; (1,2) and (3,2)-saddle-sink and saddle-source; (2,2)-double-saddles;
• Develops homoclinic networks of source, sink and saddles.

Edition details

• Publisher: Springer
• Publication date: 2025
• Language: English
• ISBN 10: 3031628098
• ISBN 13: 9783031628092
• Binding: Hardcover
• Number of pages: 396