MA 543: Ordinary Differential Equations and Dynamical Systems
Spring 2018, Purdue University

http://www.math.purdue.edu/~yip/543

Course Description:

This is a beginning graduate level course on ordinary differential equations. It covers basic results for linear systems, local theory for nonlinear systems (existence and uniqueness, dependence on parameters, flows and linearization, stable manifold theorem) and their global theory (global existence, limit sets and periodic orbits, Poincare maps). Some further topics include bifurcations, averaging techniques and applications to mechanics and mathematical biology.

Prerequisites:

One undergraduate course in each of the following topics:
linear algebra (for example, MA 265, 351),
differential equation (for example, MA 266, 366),
analysis (for example, MA 341, 440, 504).

Textbooks and References: (All are available online through Purdue library page.)

Main text:
[M] Differentiable Dynamical Systems, J. D. Meiss

References:
[HSD] Differential Equations, Dynamical Systems: An Introduction to Chaos, M. W. Hirsch, S. Smale, R. L. Devaney

[T] Ordinary Differential Equations and Dynamical System, G. Teschl, available online

[S] Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering, S. H. Strogatz

[GH] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, J. Guckenheimer, P. Holmes

[A] Mathematical Methods of Classical Mechanics, V. I. Arnold

Some texts at the undergraduate level:
[BP] Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima (textbook of Purdue MA266, on reserve in libray)

[B] Differential Equations and Their Applications, M. Braun

Course Announcement:

Your grade is based on class participation (though I do not intentionally take attendance), typing up of class notes (shared among the whole class), a short (say 5 to 6 pages) paper, and a short (30 min) presentation about the paper toward the end of the semester, probably during the final exam week or earlier.

Please submit a title and abstract (with just regular text, no mathematical formula) of your paper by e-mail, by Mar 30.


Week 1:

Tue Jan 9:
[M, Chapter 1]
introduction, notations,
conversion between higher order equation and first order system,
examples;



Week 2:

Tue Jan 16:
introduction:
explicit solution, integrating factor, finite time blow-up of nonlinear equations, non-uniqueness of solutions.

Thur Jan 18:
``invariant domain'', estimation of long time behavior of solutions,
matrix exponential, solution of linear, inhomogeneous system with constant matrix, dX/dt = AX + h(t).


Week 3:

Tue Jan 23:
norm(s) of vector and matrices;
uniqueness of solution for linear inhomogeneous system, dX/dt = A(t)X + h(t), Gronwall's inequality,
successive approximation, Picard's iteration.

Thur Jan 25:
existence of solution for linear inhomogeneous system, dX/dt = A(t)X + h(t), Picard's iteration,
fundamental solution, dPhi(t)/dt = A(t)Phi(t), Phi(0)=I,
invertibility of fundamental solution,
variation of parameters and solution formula for linear inhomogeneous systems, dX/dt=A(t)X + h(t)


Week 4:

Thur Feb 1: (M, Chapter 2)
Abel Theorem for det(Phi(t));
(computation of) exponential of a matrix, diagonalization, eigenvectors and generalized eigenvectors for repeated eigenvalues, Jordan canonical form,
invariant subspaces, stable, unstable, and center subspaces.


Week 5:

Tue Feb 6: (M, Chapter 3)
Existence and uniqueness of nonlinear differential equations,
continuity and Lipschitz condition for functions,
existence and uniqueness of dX/dt=F(X) when F is Lipschitz, by Picard's iteration and Gronwall's lemma.

Thur Feb 8: (M, Chapter 3)
Existence and uniqueness of nonlinear differential equations (cont'), space of continuous functions, compactness (Arzela-Ascoli Theorem),
contraction mapping (Banach Fixed Point Theorem), explicit Euler scheme, Schauder Fixed Point Theorem,
Gronwall's Inequality (Lemma 3.13), Lipshitz dependence of solution on initial data (Theorem 3.14).


Week 6:

Tue Feb 13: (M, Chapter 3)
Differentiability of vector valued functions, linear approximation (D_X F(X)),
smooth (C^1) dependence of solution on initial data (Theorem 3.15), smooth (C^1) dependence of solution on initial data and parameter (Theorem 3.16),
local Lipschitz condition, local in time existence of solution (Lemma 3.12), maximal interval of existence (Theorem 3.17).

Thur Feb 15:
[M 3.5] limit of solution near the end of the maximal interval of existence,
[M 4.3] global existence of solutions with linear growth of F(X),
(Bellman, Chapter 2) exponential decay of solutions and eigenvalues, perturbation results for stability of solutions.


Week 7:

Tue Feb 20: (M, Chapter 4)
flow map, phi(t,x), Lyapunov stability, asymptotic stability,
asymptotic stability for nonlinear system when the linearization has eigenvalues with strictly negative real parts.

Tue Feb 22: (M, Chapters 4, 5)
example of a system which are Lyapunov not asymptotically stable,
example of a system which is not structually stable,
(linear, invariant) stable, unstable, and center subspaces,
(Theorem 5.8, nonlinear, invariant) stable, unstable, and center manifolds,
example of nonuniqueness of center manifold.


Week 8:

Tue Feb 27: (M, Chapter 4, 5)
homeomorphism and diffeomorphism between topological spaces,
topological conjugacy, equivalence, and diffeomophism between dynamical systems,
reparametrization of time.

Thur Mar 1: (M, Chapter 4, 5)
[M, Theorem 4.11] diffeomorphism between linear systems and similaritiy between matrices,
[M, Theorem 4.12] topological conjugacy between linear systems and equality between dimensions of stable and unstable subspaces,
[M, Theorem 4.13] Hartman-Grobman Theorem: topological conjugacy between nonlinear system and its linearization for hyperbolic system,
setting up of proof of existence and uniqueness of local stable manifold theorem [M Theorem 5.3].


Week 9:

Tue Mar 6: (M, Chapter 5)
Proof of existence and uniqueness of local stable manifold theorem [M Theorem 5.3]:
existence and uniqueness of bounded solution of linear system with bounded inhomogeneous term [M Lemma 5.2].
setting up of fixed point theorem for nonlinear system.

Tue Mar 8: (M, Chapter 5)
Proof of existence and uniqueness of local stable manifold theorem [M Theorem 5.3],
existence and uniqueness of a fixed point in the space of bounded continuous functions,
existence and uniqueness of a fixed point in exponentially weighted space.


Week 10:

Tue Mar 20:
(M, Chapter 5.4) analytical computation by iteration of (local) stable manifolds,
(M, Chapter 5.6) center manifold theorem [M, Theorem 5.8], non-hyperbolic Hartman-Grobman Theorem [M, 5.9],
(M, Chapter 5.6) approximation of center manifold using Taylor expansion,
Two excellent references (both available online through Purdue library page):
Applications of Centre Manifold Theory, by J. Carr,
Hopf Bifurcation and Its Applications, by J.E. Marsden and M. McCracken.

Thur Mar 22:
(M, Chapter 5.6) approximation of center manifold using Taylor expansion,
importance of nonlinear terms in the dynamics on center manifolds.
(M, 6.1, 6.2) Behavior near a non-hyperbolic critical points in R^2, use of polar-coordinates.


Week 11:

Tue Mar 27:
(M, 6.1, 6.2) Behavior near a non-hyperbolic critical points in R^2, use of polar-coordinates,
(M, 6.3, Lemma 6.1, 6.2, 6.3) classification of behaviors near a linear center:
topological center (periodic orbits), non-hyperbolic focus (spiral in or out), center-focus (nested limit cycles shrinking to the origin).
Note on examples of behaviors near non-hyperbolic critical point.
(M, 4.9) omega- and alpha-limit sets.

Thur Mar 29:
(M, 4.9, 4.10) omega- and alpha-limit sets, attractors,
Note on Limit Sets and Attractors.
(M, 4.6) Dynamical system from an energetic point of view:
gradient flow, Hamiltonian system.


Week 12:

Tue Apr 3:
(M, 4.6) Dynamical system from an energetic point of view:
Hamiltonian system with dissipation, Liapunov (Lyapunov) function.
(M 4.11) Stability of periodic orbits, linearization, Floquet Theorem (M, Theorem 2.13)

Thur Apr 5:
(M, 4.11) Stability of periodic orbits, linearization, Floquet Theorem, monodromy matrix,
Poincare map, fixed points of Poincare map and their stability,
Spec(M) = Spec(DP) + {1} (M, Theorem 4.20, Theorem 4.21)
Note on Periodic Orbits and Their Stability.


Week 13:

Tue Apr 10:
Review of Floquet Theory, Poincare map and their relationship,
simplification in R^2, connection to Abel's formula (M, Theorem 2.11, p. 63),
an analytical example (M, p. 153).
Note on Existence of Periodic Orbits (Hartman, ODEs)

Thur Apr 12:
(M 6.6) Poincare Bendixson Theorem (M, Theorem 6.12, with no equilibrium points) and its proof:
(1) definition of a transervsal segment;
(2) (Lemma 6.15) omega limit set and any transversal segment intersect at at most one point (use of Jordan Curve Theorem, Theorem 6.14);
(3) (Lemma 6.16) omega limit set contains a periodic orbit;
(4) (Lemma 6.17) omega limit set is the periodic orbit.


Week 14:

Tue Apr 17:
(M 6.6) Poincare Bendixson Theorem (M, Theorem 6.12) and
its generalization (M Theorem 6.19, with finitely many equilibrium points).
examples, and applications,
Bendixson (and Dulac) criterior for non-existence and uniqueness of periodic orbit,
example of Markus and Yamabe (dX/dt = A(t)X, where for each fixed t, A(t) is stable (i.e. Re(lambda_1(t)) and Re(lambda_2(t)) are negative) and yet there is an exponential growth solution so that the zero solution is unstable;
application of abstract existence theorem to linear harmonic oscillator with friction and external forcing.

Thur Apr 19:
van der Pol oscillator, Duffing oscillator,
[M 6.7] Lienard's system, existence, uniqueness and stability of periodic orbits.


Week 15:

Tue Apr 24:
van der Pol oscillator, Duffing oscillator,
[M 6.7] Lienard's system, existence, uniqueness and stability of periodic orbits.
Note on Lienard System,
[S, Strogatz, 7.5] asymptotics on nonlinear oscillator: large friction (relaxation oscillations), behavior slow/fast system for Lienard system.

Thur Apr 26:
[S, Strogatz, 7.6] asymptotics on nonlinear oscillator: small friction: weakly nonlinear oscillators,
regular versus multiple-time scales expansion, van der Pol oscillator


Note Taking Activity:
(Each week's typed note is due in two Tuesdays afterward.)

Jan 23, 25: Dustin Enyeart
Jan 30, Feb 1: Fouad Khoury
Feb 6, 8: Yang Mo
Feb 13, 15: Vivek Muralidharan
Feb 20, 22: Nik Prljevic
Feb 27, Mar 1: Alex Van Anderlecht and Yongjia Xie
Mar 6, 8: Dustin Enyeart
(Spring Break)
Mar 20, 22: Fouad Khoury
Mar 27, 29: Yang Mo
Apr 3, 5: Vivek Muralidharan
Apr 10, 12: Nik Prljevic
Apr 17, 19: Alex Van Anderlecht and Yongjia Xie
(Last week of class)


Final Exam Presentation Schedule:

Location: Rec 103

Wed, May 2, 2-3pm (Dustin Enyeart)
Wed, May 2, 3-4pm (Yongjia Xie)

Thur, May 3, 2-3pm (Alexandre Van Anderlecht)
Thur, May 3, 3-4pm (Fouad Khoury)
Thur, May 3, 4-5pm (Yang Mo)

Fri, May 4, 2-3pm (Vivek Muralidharan)
Fri, May 4, 3-4pm (Nikola Prljevic)