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Author: Amjad Islam Pitafi Publisher: LAP Lambert Academic Publishing ISBN: 9783659157097 Category : Languages : en Pages : 84
Book Description
In the last twenty years, the work was done on the different problems related to the Qualitative Theory of differential equations. But during the last few years, the interest surrounded around the well-known Hilbert's Sixteenth Problem which he posed at Paris Conference of International Congress of Mathematicians in 1900, together with other twenty-two problems [17]. In this book we are mainly concerned in the second part of Hilbert's sixteenth Problem, which poses the question of maximal number and relative position of limit cycles of the polynomial system of the form: (A) in which P and Q are polynomials in x and y. We write the system A in the form of (B) Where, and, are homogeneous quadratic and cubic polynomials in x and y. Chapter No. 1 comprises the basic concepts for general theory of limit cycles and Hilbert's Sixteenth Problem. Chapter No. 2 contains an Algorithm for determining so called focal basis. This can be implemented on the computer to get the estimate for the number of small-amplitude limit cycles. Chapter No. 3 deals with some classes of system (B) with several small-amplitude limit cycles.
Author: Amjad Islam Pitafi Publisher: LAP Lambert Academic Publishing ISBN: 9783659157097 Category : Languages : en Pages : 84
Book Description
In the last twenty years, the work was done on the different problems related to the Qualitative Theory of differential equations. But during the last few years, the interest surrounded around the well-known Hilbert's Sixteenth Problem which he posed at Paris Conference of International Congress of Mathematicians in 1900, together with other twenty-two problems [17]. In this book we are mainly concerned in the second part of Hilbert's sixteenth Problem, which poses the question of maximal number and relative position of limit cycles of the polynomial system of the form: (A) in which P and Q are polynomials in x and y. We write the system A in the form of (B) Where, and, are homogeneous quadratic and cubic polynomials in x and y. Chapter No. 1 comprises the basic concepts for general theory of limit cycles and Hilbert's Sixteenth Problem. Chapter No. 2 contains an Algorithm for determining so called focal basis. This can be implemented on the computer to get the estimate for the number of small-amplitude limit cycles. Chapter No. 3 deals with some classes of system (B) with several small-amplitude limit cycles.
Author: Colin Christopher Publisher: Springer Science & Business Media ISBN: 3764384107 Category : Mathematics Languages : en Pages : 167
Book Description
This textbook contains the lecture series originally delivered at the "Advanced Course on Limit Cycles of Differential Equations" in the Centre de Rechercha Mathematica Barcelona in 2006. It covers the center-focus problem for polynomial vector fields and the application of abelian integrals to limit cycle bifurcations. Both topics are related to the authors' interests in Hilbert's sixteenth problem, but would also be of interest to those working more generally in the qualitative theory of dynamical systems.
Author: V. Gaiko Publisher: Springer Science & Business Media ISBN: 1441991689 Category : Mathematics Languages : en Pages : 199
Book Description
On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second Interna tional Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathema tics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possi ble complete information on the qualitative behaviour of integral curves defined by this equation (176].
Author: Colin Christopher Publisher: Springer Science & Business Media ISBN: 3764384093 Category : Mathematics Languages : en Pages : 167
Book Description
This textbook contains the lecture series originally delivered at the "Advanced Course on Limit Cycles of Differential Equations" in the Centre de Rechercha Mathematica Barcelona in 2006. It covers the center-focus problem for polynomial vector fields and the application of abelian integrals to limit cycle bifurcations. Both topics are related to the authors' interests in Hilbert's sixteenth problem, but would also be of interest to those working more generally in the qualitative theory of dynamical systems.
Author: Valery Romanovski Publisher: Springer Science & Business Media ISBN: 0817647279 Category : Mathematics Languages : en Pages : 336
Book Description
Using a computational algebra approach, this comprehensive text addresses the center and cyclicity problems as behaviors of dynamical systems and families of polynomial systems. The book gives the main properties of ideals in polynomial rings and their affine varieties followed by a discussion on the theory of normal forms and stability of differential equations. It contains numerous examples, pseudocode displays of all the computational algorithms, historical notes, nearly two hundred exercises, and an extensive bibliography, making it a suitable graduate textbook as well as research reference.
Author: Albert C. J. Luo Publisher: Springer ISBN: 9789819726165 Category : Mathematics Languages : en Pages : 0
Book Description
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.
Author: Maoan Han Publisher: Springer Science & Business Media ISBN: 1447129180 Category : Mathematics Languages : en Pages : 408
Book Description
Dynamical system theory has developed rapidly over the past fifty years. It is a subject upon which the theory of limit cycles has a significant impact for both theoretical advances and practical solutions to problems. Hopf bifurcation from a center or a focus is integral to the theory of bifurcation of limit cycles, for which normal form theory is a central tool. Although Hopf bifurcation has been studied for more than half a century, and normal form theory for over 100 years, efficient computation in this area is still a challenge with implications for Hilbert’s 16th problem. This book introduces the most recent developments in this field and provides major advances in fundamental theory of limit cycles. Split into two parts, the first focuses on the study of limit cycles bifurcating from Hopf singularity using normal form theory with later application to Hilbert’s 16th problem, while the second considers near Hamiltonian systems using Melnikov function as the main mathematical tool. Classic topics with new results are presented in a clear and concise manner and are accompanied by the liberal use of illustrations throughout. Containing a wealth of examples and structured algorithms that are treated in detail, a good balance between theoretical and applied topics is demonstrated. By including complete Maple programs within the text, this book also enables the reader to reconstruct the majority of formulas provided, facilitating the use of concrete models for study. Through the adoption of an elementary and practical approach, this book will be of use to graduate mathematics students wishing to study the theory of limit cycles as well as scientists, across a number of disciplines, with an interest in the applications of periodic behavior.
Author: Amjad Islam Pitafi
Author: Amjad Islam Pitafi
Author: Colin Christopher
Author: B. D. Sleeman
Author: S. Yakovenko
Author: V. Gaiko
Author: Colin Christopher
Author: Valery Romanovski
Author: Courtney S. Coleman
Author: Albert C. J. Luo
Author: Maoan Han