Global Bifurcations and Chaos Analytical Methods
Global Bifurcations and Chaos: Analytical Methods is unique in the literature of chaos in that it not only defines the concept of chaos in deterministic systems, but it describes the mechanisms which give rise to chaos (i.e., homoclinic and heteroclinic m
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1. E. Marsden
L. Sirovich
Advisors M. Ghil 1.K. Hale 1. Keller K. Kirchgassner B.l. Matkowsky 1.T. Stuart A. Weinstein
Applied Mathematical Sciences
I. Johll: Partial Differential Equations. 4th ed. 2. SirOl'ich: Techniques of Asymptotic Analysis.
3. 4. 5. 6.
Hale: Theory of Functional Differential Equations. 2nd ed. Percus: Combinatorial Methods. VOII MiseslFriedrichs: Fluid Dynamics. FreibergerlGrenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. II. WolOl'ich : Linear Multivariable Systems. 12. Berkol'itz: Optimal Control Theory. 13. BllimalllCole: Similarity Methods for Differential Equations. 14. Yoshi:awa : Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. 15. Braull: Differential Equations and Their Applications. 3rd cd. 16. LefscltelZ: Applications of Algebraic Topology. 17. ColIlll:IWellerling: Optimization Problems. IS. Grellllllder: Pattern Synthesis: Lectures in Pattern Theory. Vol I. 20. DrilU: Ordinary and Delay Differential Equations. 21. COliralltlFriedrichs: Supersonic Flow and Shock Waves. 22 . RouchelHabelSlLalov: Stability Theory by Liapunov 's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grellallder: Pattern Analysis: Lectures in Pattern Theory. Vol. II. 25. Dm'ies: Integral Transforms and Their Applications. 2nd ed. 26. Klis/1Iler/C/ark: Stochastic Approximation Methods for Constrained and Unconstrained Systems 27. de Boor: A Practical Guide to Splines. 2S. KeilsOl': Markov Chain Models-Rarity and Exponentiality. 29. de Vellbl'ke: A Course in Elasticity. 30. SIIilllycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturm ian Theory for Ordinary Differential Equations. 32. MeislMarkowitz? Numerical Solution of Partial Differential Equations. 33. Grellallder: Regular Structures: Lectures in Pattern Theory. Vol. III. 34. KemrkianlCole: Perturbation methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. BenXr.mmIGltillKiillen: Dynamic Meteorology: Data Assimilation Methods. 37. Saperswlle: Semidynamical Systems in Infinite Dimensional Spaces. 3S. LicirtenberglLiebermlll': Regular and Stochastic Motion. 39. PicciniISramp 0 is constant. The phase space of this equation is the interval
(0,211"] with 0 and 211" identified. Thus, the phase space has the structure of a circle having length 21l" which we denote as 8 1 (note: the superscript one refers to the dimension of the phase space). Cylinder : Mathematically the cylinder is denoted by
R1 x 8 1. Consider the
following equation which describes the dynamics of a free undamped pendulum
v = - sin8.
(1.1.23)
The angular velocity, v, can take on any value in R but, since the motion is rotational, the position, 8, is periodic with period 211". Hence, the phase space of the pendulum is the cylinder R 1 x 8 1. Figure 1.1.4a shows the orbits of the pendulum on R 1 x 8 1 and Figure 1.1.4b gives an a
