This book is part of a two volume set which presents the analysis of nonlinear phenomena as a long-standing challenge for research in basic and applied science as well as engineering. It discusses nonlinear differential and differential equations, bifurcation theory for periodic orbits and global connections. The integrability and reversibility of planar vector fields and theoretical analysis of classic physical models are sketched.

This first volume concentrates on the mathematical theory and computational techniques that are essential for the study of nonlinear science, a second volume deals with real-world nonlinear phenomena in condensed matter, biology and optics.

Autorentext

Prof. Victoriano Carmona Centeno obtained his PhD in Mathematics in 2002 at the University of Seville, Spain, under the supervision of Prof. E. Freire and Prof. F. Torres. His PhD Thesis was awarded with the Extraordinary Prize of Town Hall of Sevilla. Currently, he is an Associate Professor at the Department of Applied Mathematics II of the University of Seville.

His research covers different aspects of Dynamical Systems, in particular, those related to piecewise smooth systems of differential equations. He is an active member of the Group of Dynamical Systems in Engineering of the University of Sevilla.

Prof. Victoriano Carmona is author of many scientific publications in international journals, books and conferences proceedings. Some of these works have been highly cited and they are basic references in the field of piecewise linear dynamical systems. For instance, V. Carmona, E. Freire and F. Torres, On simplifying and classifying piecewise-linea r systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49: 609 (2002).

He has reviewed many papers in international journals (for instance, Physica D, Nonlinearity, Proceedings London Mathematical Society,...). He has taken part in the organization of several international conferences, the most recent one in Sevilla (Spain, 2016). Prof. Jesús Cuevas-Maraver obtained his PhD in Physics in 2003 in the Department of Condensed Matter Physics of the University of Seville, Spain, under the supervision of Prof. Francisco R. Romero and Prof. Juan F.R. Archilla. Presently he is as Associate Professor at the Department of Applied Physics I of the same university. His lectures in Elementary Physics (Mechanics, Thermodynamics and Electromagnetism) take place at the Polytechnic School in the degree of Industrial Chemical Engineering.

His research covers many aspects of Nonlinear Science and is based in the the oretical study of nonlinear localized waves as solitons, kinks or breathers that emerge in many physics systems. He has considered a wide range of such systems, which includes crystals, biomolecules, Bose-Einstein condensates, waveguide arrays, nonlinear circuits or transmission lines and metamaterials. He is a frequent referee in international journals, including Nature Physics, Physical Review Letters or Optics Letters.

Prof. Jesús Cuevas-Maraver has written 96 scientific publications covering international journals and books. He has been author of several chapters in Springer books (P.G. Kevrekidis (ed.), The Discrete Nonlinear Schrödinger Equation (2009); B.A. Malomed (ed.), Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations (2013); Machado et al (eds.), Nonlinear Science and Complexity (2011)]. He has been the main editor of Vol. 10 in Springer Series in Nonlinear Systems and Complexity [The sine-Gordon model and its app lication (2014)], where he also wrote a chapter, and coeditor of the Vol. 7 of the same series [Localized Excitations in Nonlinear Complex Systems (2014)].

Prof. Fernando Fernández-Sánchez obtained his PhD in Mathematics in 2002 at the University of Seville, Spain, under the supervision of Prof. E. Freire and Prof. A.J. Rodríguez-Luis. His PhD Thesis was awarded with the Extraordinary Prize of the University of Sevilla. Currently, Prof. Fernández-Sánchez is an Associate Professor at the Department of Applied Mathematics II of the University of Seville.

His research covers different aspects of Dynamical Systems, in particular, those related to global bifurcations (homoclinic and heteroclinic connections) in three dimensional dynamical systems and piecewise smooth systems of differential equations. He is an active member of the Group of Dynamical Systems in Engineering of the University of Sevilla.

Prof. Fernando Fernández-Sánchez is autho r/co-author of more than 40 papers in international journals (Chaos, Nonlinearity, SIAM Journal on Applied Dynamical Systems, etc.). He has also published several chapters in scientific books and attended many conferences, meetings and workshops, presenting several invited talks.

He has reviewed many papers in international journals (for Systems & Control Letters; Physics Letters A.; Chaos, Solitons and Fractals; Journal of Nonlinear Analysis; etc.). He has taken part in the organization of several international conferences and Schools: Workshop on Analysis and Continuation of Bifurcations, Sevilla (Spain), 2004; RTNS 2009. 6th Winter School in Dynamical Systems of the DANCE (Dinámica y Atractores y No linealidad: Caos y Estabilidad) Spanish Network, Carmona (Sevilla, Spain), 2009; RTNS 2016. Sevilla (Spain), 2016; Nolineal. International conference on nonlinear mathematics and physics, Sevilla (Spain), 2016.

Dr. Elisabeth García-Medina obtained h

is PhD in Mathematics in 2011 at the University of Seville, Spain, under the supervision of Prof. V. Carmona and Prof. F. Fernández-Sánchez. Currently, she is an Assistant Professor at the Department of Applied Mathematics II of the University of Seville.

Her research covers different aspects of Dynamical Systems, in particular, those related to global bifurcations and periodic orbits in piecewise linear systems of differential equations. She is an active member of the Group of Dynamical Systems in Engineering of the University of Sevilla.

Prof. Elisabeth García-Medina is a young researcher that has published several works in international journals (Chaos, Nonlinearity, Nonlinear Analysis, etc.), chapters of scientific books and conference proceedings. She has attended many conferences, meeting and workshops related to dynamical systems.

She has also taken part in the organization of several international conferences and Schools, the most recent one in Sevilla (Spain, 2016).

Inhalt

Part 1. BIFURCATION ANALYSIS

· Analytic integrability of some degenerate centers . Checa, Isabel.

· Analysis of the Hopf-zero bifurcation and their degenerations in a quasi-Lorenz system.

Domínguez, Cinta.

· Normal forms for a class of tridimensional vector fields with free-divergence in its first component.

Fuentes, Natalia.

· Takens-Bogdanov bifurcations and resonances of periodic orbits in the Lorenz system.

Rodríguez-Luis, Alejandro J.

Part 2. WAVE EQUATIONS

· Solitons and vortices in the Nonlinear Dirac Equation. Kevrekedis, Panayotis.

· ; Stochastic Korteweg - de Vries type equations. Karczewska, Anna.

· Exact and adiabtic invariants of KdV type equations. Rozmej, Piotr.

· Gravitational waves as nonlinear waves and solitons. Villatoro, Francisco R.

Part 3. OTHER DIFFERENTIAL AND DIFFERENCE EQUATIONS

· On the dynamics of the nonlinear logistic difference equation with two delays.

Balibrea-Gallego, Francisco,

· Simplifying singular perturbation theory in the canard regime using piecewise-linear (PWL) systems.

Desroches, Mathieu.

· Principal solutions and variation of constants formula for a class of functional differential equations.

Marot o, Ismael.

· Diffusion Equations in Inhomogeneous Media from the Master Equation. Salasnich, Luca.

Part 4. COMPUTATIONAL METHODS

· On the numerical approximation to generalized Ostrovsky equations. Durán, Ángel.

Simulation o…