Discrete dynamical systems: Limit sets Stability of invariant sets and asymptotically smooth maps Examples of asymptotically smooth maps Dissipativeness and global attractors Dependence on parameters Fixed point theorems Stability relative to the global attractor and Morse-Smale maps Dimension of the global attractor Dissipativeness in two spaces Continuous dynamical systems: Limit sets Asymptotically smooth and $\alpha$-contracting semigroups Stability of invariant sets Dissipativeness and global attractors Dependence on parameters Periodic processes Skew product flows Gradient flows Dissipativeness in two spaces Properties of the flow on the attractor Applications: Retarded functional differential equations Sectorial evolutionary equations A scalar parabolic equation The Navier-Stokes equation Neutral functional differential equations Some abstract evolutionary equations A one-dimensional damped wave equation A three-dimensional damped wave equation Remarks on other applications Dependence on parameters and approximation of the attractor.

Asymptotic Behavior of Dissipative Systems

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

/pdf/functional-differential-equations-3kbvbdqaw3.pdf

Functional Differential Equations

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

Preface.- Introduction.- Invariant Sets and Attractors.- Functional Differential Equations on Manifolds.- The Dimension of the Attractor.- Stability and Bifurcation.- Stability of Morse-Smale Maps and Semiflows.- One-to-oneness, Persistence and Hyperbolicity.- Realization of Vector Fields and Normal Forms.- Attractor Sets as C1-Manifolds.- Monotonicity.- The Kupka-Smale Theorem.- Appendix A: Conley Index Theory in Noncompact Spaces.- References.- Index.

Dynamics in Infinite Dimensions

An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory

The paper deals with smooth nonlinear ODE systems in ℝn, ẋ = f(x), such that the derivative f′(x) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f′(x(t))y. Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝn is then explicitly constructed.

Jacobi matrices and transversality

Dynamics of the Attractor in the Lotka–Volterra Equations☆☆☆

Transversality of stable and unstable manifolds of critical elements plays a central role in connection with generic results and structural stability in the theory of Dynamical Systems. That property is true, generically, for vector fields and diffeomorphisms defined on a finite dimensional manifold (see [PM] for references). Also the structural stability of Morse-Smale vector fields and diffeomorphisms on compact manifolds and the stability with respect to compact attractors in the infinite dimensional case depend strongly on transversality (see [PM] for the finite dimensional case and [HMO], [0] for infinite dimensions). In spite of this fact there is no general available method for checking transversality.

Waldyr M. Oliva

Papers

Dynamics in Infinite Dimensions

An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory

Jacobi matrices and transversality

Dynamics of the Attractor in the Lotka–Volterra Equations☆☆☆

Jacobi matrices and transversality