Luo Albert C. J.

This book is a six-volume handbook about two-product polynomial systems. The corresponding hybrid networks of singular and non-singular, 1-dimensional flows and equilibriums are presented. The higher-order singular 1-dimensional flows and singular equilibriums are for the appearing bifurcations of lower-order singular and non-singular 1-dimensional flows and equilibriums. The infinite-equilibriums are the switching bifurcations for two associated networks of singular and non-singular, 1-dimensional flows and equilibriums. Volume I of this book presents a theorem for the bifurcation dynamics of two-product polynomial systems through a theorem. The nonlinear dynamics of ingular flows and equilibriums with the corresponding infinite-equilibriums in two-product polynomial systems in theorem. Volume II of this book presents the methodology to achieve the mathematical conditions for singular equilibriums, singular 1-dimensional flows, two network switching in the theorem through local analysis and the first integral manifolds. Volume III of this book discusses the nonlinear dynamics of two-product polynomial systems with ([m1, 2n11+1], [m2, 2n21+1])-vector fields. Volume IV of this book discusses the nonlinear dynamics of two-product polynomial systems with ([m1, 2n11], [m2, 2n21+1])-vector fields. Volume V of this book discusses the nonlinear dynamics of two-product polynomial systems with ([m1, 2n11+1], [m2, 2n21])-vector fields. Volume VI of this book discusses the nonlinear dynamics of two-product polynomial systems with ([m1, 2n11], [m2, 2n21])-vector fields. In volumes III-VI, the singular equilibriums and 1-dimensional flows in such two-product polynomial systems are presented first, and the singular infinite-equilibriums are presented for the switching bifurcations of two singular/simple hybrid networks of the two-product polynomial system.

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.

This book is a monograph about the analytical dynamics of nonlinear rotor systems. The analytical solutions of periodic motions in nonlinear rotor systems are presented in this book. To help one understand the analytical solutions, the generalized harmonic balance method for periodic motions in polynomial nonlinear systems is briefly reviewed first, and then the semi-analytical method for periodic motions in any nonlinear systems is presented briefly too, which is also called the implicit mapping method. Such two analytical methods are employed to determine the solutions of periodic motions in nonlinear rotor systems. The analytical expressions of periodic motions to chaos for nonlinear rotor systems are presented, and the frequency-amplitude characteristics of nonlinear rotor systems are discussed. In addition, the accurate modeling of nonlinear rotors with oil films is presented, and the periodic motions of such fully nonlinear oil-film rotor systems are developed through the semi-analytical method. This book provides a better understanding of frequency-amplitude characteristics in nonlinear rotor systems. The methodology presented in this book can help one study complicated nonlinear rotor systems.

This book, the 15th of 15 related monographs on Cubic Dynamic Systems, discusses crossing and product cubic systems with a crossing-linear and self-quadratic product vector field. The author discusses series of singular equilibriums and hyperbolic-to-hyperbolic-scant flows that are switched through the hyperbolic upper-to-lower saddles and parabola-saddles and circular and hyperbolic upper-to-lower saddles infinite-equilibriums. Series of simple equilibrium and paralleled hyperbolic flows are also discussed, which are switched through inflection-source (sink) and parabola-saddle infinite-equilibriums. Nonlinear dynamics and singularity for such crossing and product cubic systems are presented. In such cubic systems, the appearing bifurcations are: parabola-saddles, hyperbolic-to-hyperbolic-secant flows, third-order saddles (centers) and parabola-saddles (saddle-center).

This book, the 14th of 15 related monographs on Cubic Dynamical Systems, discusses crossing and product cubic systems with a self-linear and crossing-quadratic product vector field. Dr. Luo discusses singular equilibrium series with inflection-source (sink) flows that are switched with parabola-source (sink) infinite-equilibriums. He further describes networks of simple equilibriums with connected hyperbolic flows are obtained, which are switched with inflection-source (sink) and parabola-saddle infinite-equilibriums, and nonlinear dynamics and singularity for such crossing and product cubic systems. In such cubic systems, the appearing bifurcations are: - double-inflection saddles, - inflection-source (sink) flows, - parabola-saddles (saddle-center), - third-order parabola-saddles, - third-order saddles and centers.