Abstract
Many problems in the natural and engineering sciences can be modeled as evolution processes. Mathematically this leads to either discrete or continuous dynamical systems, i.e. to either difference or differential equations. Usually such dynamical systems are nonlinear or even discontinuous and depend on parameters. Consequently the study of qualitative behaviour of their solutions is very difficult. Rather effective method for handling dynamical systems is the bifurcation theory, when the original problem is a perturbation of a solvable problem, and we are interested in qualitative changes of properties of solutions for small parameter variations. Nowadays the bifurcation and perturbation theories are well developed and methods applied by these theories are rather broad including functional-analytical tools and numerical simulations as well [1–13].
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References
C. CHICONE: Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer, New York, 2006.
S. ELAYDI: An Introduction to Difference Equations, Springer, New York, 2005.
M. FARKAS: Periodic Motions, Springer, New York, 1994.
J. GUCKENHEIMER & P. HOLMES: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
J. K. HALE & H. KOÇAK: Dynamics and Bifurcations, Springer, New York, 1991.
M.C. IRWIN: Smooth Dynamical Systems, Academic Press, London, 1980.
Y.A. KUZNETSOV: Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112, Springer, New York, 2004.
M. MEDVEĎ: Fundamentals of Dynamical Systems and Bifurcation Theory, Adam Hilger, Bristol, 1992.
C. ROBINSON: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, 1998.
M. RONTO & A.M. SAOMOILENKO: Numerical-Analytic Methods in the Theory of Boundary-Value Problems, World Scientific Publishing Co., Singapore, 2001.
J.J. STOKER: Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience, New York, 1950.
A. STUART & A.R. HUMPHRIES: Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1999.
M. TABOR: Chaos and Integrability in Nonlinear Dynamics: An Itroduction, Willey-Interscience, New York, 1989.
J. AWREJCEWICZ & M.M. HOLICKE: Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods, World Scientific Publishing Co., Singapore, 2007.
M. CENCINI, F. CECCONI & A. VULPIANI: Chaos: from Simple Models to Complex Systems, World Scientific Publishing Co., Singapore, 2009.
R. CHACÓN: Control of Homoclinic Chaos by Weak Periodic Perturbations, World Scientific Publishing Co., Singapore, 2005.
S.N. CHOW & J.K. HALE: Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.
R. DEVANEY: An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings, Menlo Park, CA, 1986.
S. WIGGINS: Chaotic Transport in Dynamical Systems, Springer, New York, 1992.
S. WIGGINS: Global Bifurcations and Chaos, Analytical Methods, Applied Mathematical Sciences, 73, Springer-Verlag, New York, NY, 1988.
P. HOLMES & J. MARSDEN: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Rational Mech. Anal., 76 (1981), 135–165.
A. FIDLIN: Nonlinear Oscillations in Mechanical Engineering, Springer, Berlin, 2006.
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© 2011 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
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Fečkan, M. (2011). Introduction. In: Bifurcation and Chaos in Discontinuous and Continuous Systems. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18269-3_1
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DOI: https://doi.org/10.1007/978-3-642-18269-3_1
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