Albert C. J. Luo Livres

Focusing on nonlinear dynamics, this monograph explores two-dimensional quadratic nonlinear systems through bivariate vector fields. It offers insights into the dynamics and bifurcations of these systems on both linear and nonlinear bivariate manifolds. Detailed discussions include singular dynamics, equilibrium behaviors, and one-dimensional flows. The text also covers saddle-focus bifurcations and switching bifurcations involving infinite equilibriums, making it a valuable reference for researchers and students in mathematics and engineering fields.

Focusing on the theoretical framework of product-cubic nonlinear systems, this monograph delves into the dynamics of systems characterized by constant and single-variable linear vector fields. It explores hyperbolic flows and their interactions with cubic product systems, detailing bifurcations and equilibrium points. The text examines both connected and separated hyperbolic flows, highlighting the behavior of inflection-source and sink equilibria in relation to switching bifurcations, providing a comprehensive analysis of complex flow dynamics in these mathematical systems.

Focusing on periodic and chaotic behaviors, this book delves into the intricacies of discontinuous dynamical systems, such as impulsive, discontinuous, and switching systems. It introduces innovative concepts aimed at understanding discontinuous systems and addresses the challenges posed by discontinuities that can mislead dynamical behaviors, providing insights into their smoothening.

The book delves into the theory of crossing-cubic nonlinear systems, examining various vector fields such as constant, crossing-linear, crossing-quadratic, and crossing-cubic. It details the dynamics of these systems, including 1-dimensional flows like parabola and inflection flows, as well as more complex equilibriums like saddle and center points. It also explores higher-order dynamics, including third-order saddles and centers, and discusses the formation of homoclinic orbits and networks, providing a comprehensive framework for understanding these nonlinear systems.

Focusing on cubic nonlinear systems, this monograph introduces a systematic theory centered around single-variable vector fields. It delves into 1-dimensional flow singularities and bifurcations, showcasing previously unexplored bifurcations in 2-dimensional cubic systems. The text covers third-order source and sink flows, as well as parabola flows, highlighting the significance of infinite-equilibriums in switching bifurcations. Additionally, it details various bifurcations, including saddle flows and inflection flows, providing a comprehensive analysis of these complex systems.

Focusing on product-cubic nonlinear systems, this volume delves into the dynamics of two crossing-linear and self-quadratic product vector fields. It explores equilibrium and flow singularities, emphasizing bifurcations, including appearing and switching types. The text details double-saddle equilibria related to saddle source and saddle-sink bifurcations, along with a network of saddles. Additionally, it presents infinite-equilibriums associated with switching bifurcations, offering insights into complex dynamic behaviors and their implications.

Focusing on discontinuous dynamical systems within time-varying domains, this monograph introduces a novel perspective distinct from traditional theories that emphasize time-invariant domains. It explores the switchability of flows to time-varying boundaries and presents principles for system interactions without physical connections. Detailed analyses of various discontinuous systems illustrate practical applications of the theory. This work serves as a valuable reference for researchers and advanced students in mathematics, physics, and mechanics.

Focusing on crossing and product cubic systems, this monograph delves into self-linear and crossing-quadratic product vector fields. Dr. Luo explores singular equilibrium series characterized by inflection-source and parabola-source flows, detailing the dynamics of networks with hyperbolic flows. The study emphasizes the nonlinear dynamics and singularities of these systems, highlighting the bifurcations that arise within them. This work is part of a larger series on Cubic Dynamical Systems, contributing to the understanding of complex mathematical behaviors in this field.

Focusing on cubic nonlinear systems, this monograph delves into the intricacies of single-variable vector fields, particularly those with crossing variables. It explores 1-dimensional flow singularities and bifurcations, presenting novel insights into the switching bifurcations within 2-dimensional cubic systems. The text details third-order parabola flows and saddle flows, highlighting the significance of infinite equilibria and various flow types, including inflection flows and saddle flows, in understanding the dynamics of these systems.

Focusing on the intricate dynamics of polynomial systems, this monograph explores limit cycles and homoclinic networks, addressing Hilbert's sixteenth problem. It examines the equilibrium properties in planar polynomial systems, determining first integral manifolds and developing bifurcation theory related to homoclinic networks. The work identifies the maximum numbers of centers, saddles, sinks, and sources, contributing to a deeper understanding of global dynamics. This resource is invaluable for graduate students and researchers in mathematics and engineering fields.