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The book is about the global stability and bifurcation of equilibriums in polynomial functional systems. Appearing and switching bifurcations of simple and higher-order equilibriums in the polynomial functional systems are discussed, and such bifurcations of equilibriums are not only for simple equilibriums but for higher-order equilibriums. The third-order sink and source bifurcations for simple equilibriums are presented in the polynomial functional systems. The third-order sink and source switching bifurcations for saddle and nodes are also presented, and the fourth-order upper-saddle and lower-saddle switching and appearing bifurcations are presented for two second-order upper-saddles and two second-order lower-saddles, respectively. In general, the (2l + 1)th-order sink and source switching bifurcations for (2li)th-order saddles and (2lj +1)-order nodes are also presented, and the (2l)th-order upper-saddle and lower-saddle switching and appearing bifurcations are presented for (2li)th-order upper-saddles and (2lj)th-order lower-saddles (i, j = 1,2,...). The vector fields in nonlinear dynamical systems are polynomial functional. Complex dynamical systems can be constructed with polynomial algebraic structures, and the corresponding singularity and motion complexity can be easily determined.
This book is a monograph about limit cycles and homoclinic networks in polynomial systems. The study of dynamical behaviors of polynomial dynamical systems was stimulated by Hilbert’s sixteenth problem in 1900. Many scientists have tried to work on Hilbert's sixteenth problem, but no significant results have been achieved yet. In this book, the properties of equilibriums in planar polynomial dynamical systems are studied. The corresponding first integral manifolds are determined. The homoclinic networks of saddles and centers (or limit cycles) in crossing-univariate polynomial systems are discussed, and the corresponding bifurcation theory is developed. The corresponding first integral manifolds are polynomial functions. The maximum numbers of centers and saddles in homoclinic networks are obtained, and the maximum numbers of sinks, sources, and saddles in homoclinic networks without centers are obtained as well. Such studies are to achieve global dynamics of planar polynomial dynamical systems, which can help one study global behaviors in nonlinear dynamical systems in physics, chemical reaction dynamics, engineering dynamics, and so on. This book is a reference for graduate students and researchers in the field of dynamical systems and control in mathematics, mechanical, and electrical engineering.
This book introduces to 1-dimensional flow arrays and bifurcations in planar polynomial systems. The 1-dimensional source, sink and saddle flows are discussed, as well as the 1-dimensional parabola and inflection flows. The singular source, sink and saddle flows are the appearing and switching bifurcations for simple sink and source flow arrays and for lower-order singular source, sink and saddle flow arrays. The singular parabola and inflection flows are the appearing and switching bifurcations for simple parabola arrays and also for lower-order singular parabola and inflection flow arrays. The infinite-equilibriums in single-variable polynomial systems are also discussed, which are the appearing and switching bifurcations of hybrid arrays of source, sink, and saddle flows with parabola and inflections. This book helps readers understand the global dynamics of planar polynomial systems and the Hilbert sixteen problem.
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).