Differential equations and dynamical systems courses

This page collects some information about (ordinary) differential equations and dynamical systems courses offered at Caltech. This page was prepared in preparation for a faculty discussion on integrated ACM 106b, AM 125b and CDS 140a.

Overview of current course sequence

ACM 101b: Methods of Applied Mathematics Catalog listing Analytical methods for the formulation and solution of initial and boundary value problems for ordinary and partial differential equations. Techniques include the use of complex variables, generalized eigenfunction expansions, transform methods and applied spectral theory, linear operators, nonlinear methods, asymptotic and approximate methods, Weiner-Hopf, and integral equations. Taught concurrently with CDS 140 and AM 125. Instructors: Marsden, Murray. Dependent courses: ACM 201 (Advanced PDEs) Partially overlapping courses AM 125b, CDS 140a Topics (Winter 2009) Textbooks Jordan and Smith

AM 125b: Engineering Mathematical Principles Catalog listing Topics include linear spaces, operators and matrices, integral equations, variational principles, ordinary and partial differential equations, stability, perturbation theory. Applications to problems in engineering and science are stressed. Instructor: Beck. Dependent courses AM 125c (partial differential equations) Partially overlapping courses ACM 100b, ACM 101a?? AM 151a (Dynamics and Vibration) CDS 140a (Introductory Concepts for Dynamical Systems) Topics Textbooks Jordan and Smith; Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers

CDS 140a: Catalog listing Basics in topics in dynamics in Euclidean space, including equilibria, stability, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Attractors and structural stability. The Euler-Lagrange equations, mechanical systems, small oscillations, dissipation, energy as a Lyapunov function, conservation laws. Introduction to simple bifurcations and eigenvalue crossing conditions. Discussion of bifurcations in applications, invariant manifolds, the method of averaging, Melnikov’s method, and the Smale horseshoe. Instructors: Marsden, staff. Dependent courses CDS 140b Partially overlapping courses ACM 106b, AM 125b Topics (Fall 2009) Introductory Material Linear Systems Vector Fields and Flows Stability and Linearization Mechanical Systems Invariant Manifolds Liapunov Functions Textbooks Perko: Differential Equations and Dynamical Systems

Possible topics

Topic	ACM 101b	AM 125b	CDS 140a	Jordan/Smith	Perko
Existence and uniqueness of solutions	x		x	x	x
Linear ODEs, with inputs (convolution equation)	x		x
Jordan form			x		x
Transform methods (eg, Laplace, Fourier)	x
Spectral theory for linear ODEs	x
Phase space diagrams, sinks, sources, saddles, limit cycles, etc			x	x	x
Poincare-Bendixon theory			x	x	x
Analytical methods for solving nonlinear ODEs	x			x
Stability of equilibrium points, Lyapunov stability			x	x	x
Invariant manifolds (stable, unstable, center)			x	x	x
Stability of limit cycles, orbital stability, Poincare maps			x	x	x
Attractors and structural stability			?		x
Bifurcations, including saddle node, pitchfork, Hopf, etc	x		x	x	x
Floquet theory, parametric resonance				x	x
Hamiltonian dynamics, variational principles			x		x
Lagrange's equations, energy-based stability methods			x
Hamiltonian dynamics, variational principles			x
Ergodicity
Boundary value problems	x
Perturbation methods	x			x
Asymptotic and approximation methods	x
Singular perturbation theory				?
Method of averaging				x
Integral equations		x
Partial differential equations	?

Course Listings

ACM 95/100 abc. Introductory Methods of Applied Mathematics. 12 units (4-0-8); first, second, third terms. Prerequisites: Ma 1 abc, Ma 2 ab (may be taken concurrently), or equivalents. First term: complex analysis: analyticity, Laurent series, singularities, branch cuts, contour integration, residue calculus. Second term: ordinary differential equations. Linear initial value problems: Laplace transforms, series solutions. Linear boundary value problems: eigenvalue problems, Fourier series, Sturm-Liouville theory, eigenfunction expansions, the Fredholm alternative, Green’s functions, nonlinear equations, stability theory, Lyapunov functions, numerical methods. Third term: linear partial differential equations: heat equation separation of variables, Fourier transforms, special functions, Green’s functions, wave equation, Laplace equation, method of characteristics, numerical methods. Instructors: Pierce, Bruno.

ACM 101 abc. Methods of Applied Mathematics I. 9 units (3-0-6); first, second, third terms. Prerequisite: ACM 95/100 abc. Analytical methods for the formulation and solution of initial and boundary value problems for ordinary and partial differential equations. Techniques include the use of complex variables, generalized eigenfunction expansions, transform methods and applied spectral theory, linear operators, nonlinear methods, asymptotic and approximate methods, Weiner-Hopf, and integral equations. Instructors: Guo, Hou.

ACM 106 abc. Introductory Methods of Computational Mathematics. 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 1 abc, Ma 2 ab, ACM 11, ACM 95/100 abc or equivalent. The sequence covers the introductory methods in both theory and implementation of numerical linear algebra, approximation theory, ordinary differential equations, and partial differential equations. The course covers methods such as direct and iterative solution of large linear systems; eigenvalue and vector computations; function minimization; nonlinear algebraic solvers; preconditioning; time-frequency transforms (Fourier, wavelet, etc.); root finding; data fitting; interpolation and approximation of functions; numerical quadrature; numerical integration of systems of ODEs (initial and boundary value problems); finite difference, element, and volume methods for PDEs; level set methods. Programming is a significant part of the course. Instructor: Yan.

ACM/CS 114. Parallel Algorithms for Scientific Applications. 9 units (3-0-6); second term. Prerequisites: ACM 11, 106 or equivalent. Introduction to parallel program design for numerically intensive scientific applications. Parallel programming methods; distributed-memory model with message passing using the message passing interface; shared-memory model with threads using open MP, CUDA; object-based models using a problem-solving environment with parallel objects. Parallel numerical algorithms: numerical methods for linear algebraic systems, such as LU decomposition, QR method, CG solvers; parallel implementations of numerical methods for PDEs, including finite-difference, finite-element; particle-based simulations. Performance measurement, scaling and parallel efficiency, load balancing strategies. Instructor: Aivazis.

ACM 210 ab. Numerical Methods for PDEs. 9 units (3-0-6); second, third terms. Prerequisite: ACM 11, 106 or instructor’s permission. Finite difference and finite volume methods for hyperbolic problems. Stability and error analysis of nonoscillatory numerical schemes: i) linear convection: Lax equivalence theorem, consistency, stability, convergence, truncation error, CFL condition, Fourier stability analysis, von Neumann condition, maximum principle, amplitude and phase errors, group velocity, modified equation analysis, Fourier and eigenvalue stability of systems, spectra and pseudospectra of nonnormal matrices, Kreiss matrix theorem, boundary condition analysis, group velocity and GKS normal mode analysis; ii) conservation laws: weak solutions, entropy conditions, Riemann problems, shocks, contacts, rarefactions, discrete conservation, Lax-Wendroff theorem, Godunov’s method, Roe’s linearization, TVD schemes, high-resolution schemes, flux and slope limiters, systems and multiple dimensions, characteristic boundary conditions; iii) adjoint equations: sensitivity analysis, boundary conditions, optimal shape design, error analysis. Interface problems, level set methods for multiphase flows, boundary integral methods, fast summation algorithms, stability issues. Spectral methods: Fourier spectral methods on infinite and periodic domains. Chebyshev spectral methods on finite domains. Spectral element methods and h-p refinement. Multiscale finite element methods for elliptic problems with multiscale coefficients. Instructor: Guo.

AM 125 abc. Engineering Mathematical Principles. 9 units (3-0-6); first, second, third terms. Prerequisite: ACM 95/100 abc. Topics include linear spaces, operators and matrices, integral equations, variational principles, ordinary and partial differential equations, stability, perturbation theory. Applications to problems in engineering and science are stressed. Instructor: Beck.

AM/CE 151 ab. Dynamics and Vibration. 9 units (3-0-6); second, third terms. Equilibrium concepts, conservative and dissipative systems, Lagrange’s equations, differential equations of motion for discrete single and multi degree-of-freedom systems, natural frequencies and mode shapes of these systems (Eigen value problem associated with the governing equations), phase plane analysis of vibrating systems, forms of damping and energy dissipated in damped systems, response to simple force pulses, harmonic and earthquake excitation, response spectrum concepts, vibration isolation, seismic instruments, dynamics of continuous systems, Hamilton’s principle, axial vibration of rods and membranes, transverse vibration of strings, beams (Bernoulli-Euler and Timoshenko beam theory), and plates, traveling and standing wave solutions to motion of continuous systems, Rayleigh quotient and the Rayleigh-Ritz method to approximate natural frequencies and mode shapes of discrete and continuous systems, frequency domain solutions to dynamical systems, stability criteria for dynamical systems, and introduction to nonlinear systems and random vibration theory. Instructor: Krishnan.

CDS 140 ab. Introduction to Dynamics. 9 units (3-0-6); first, second terms. Prerequisite: ACM 95 or equivalent. Basics in topics in dynamics in Euclidean space, including equilibria, stability, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Attractors and structural stability. The Euler-Lagrange equations, mechanical systems, small oscillations, dissipation, energy as a Lyapunov function, conservation laws. Introduction to simple bifurcations and eigenvalue crossing conditions. Discussion of bifurcations in applications, invariant manifolds, the method of averaging, Melnikov’s method, and the Smale horseshoe. Instructors: Marsden, staff.

CDS 205. Geometric Mechanics. 9 units (3-0-6); third term. Prerequisites: CDS 202, CDS 140. The geometry and dynamics of Lagrangian and Hamiltonian systems, including symplectic and Poisson manifolds, variational principles, Lie groups, momentum maps, rigid-body dynamics, Euler-Poincaré equations, stability, and an introduction to reduction theory. More advanced topics (taught in a course the following year) will include reduction theory, fluid dynamics, the energy momentum method, geometric phases, bifurcation theory for mechanical systems, and nonholonomic systems. Given in alternate years; not offered 2009–10.

Ma 2 ab. Differential Equations, Probability and Statistics. 9 units (4-0-5); first, second terms. Prerequisite: Ma 1 abc. Ordinary differential equations, probability, statistics. Instructors: Flach, Makarov, Borodin.

Ma/ACM 142 abc. Ordinary and Partial Differential Equations. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 108. Ma 109 is desirable. The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics. Instructors: Zinchenko, Kang. Not offered 2009–10.

Ma 147 abc. Dynamical Systems. 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 108, Ma 109, or equivalent. First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Instructor: Makarov. Not offered 2009–10.

Ph 20, 21, 22. Computational Physics Laboratory. A series of courses on the application of computational techniques to simulate or solve simple physical systems, with the intent of aiding both physics understanding and programming ability. Instructors: Mach, Prince.

• 20. 6 units (0-6-0); first, second, third terms. Introduction to scientific computing with applications to physics. Use of numerical algorithms and symbolic manipulation packages for solution of physical problems. Numerical integration and numerical solution of differential equations of motion. Simulation of orbital mechanics.

• 21. 6 units (0-6-0); second, third terms. Prerequisite: Ph 20 or equivalent experience with programming and numerical techniques. Introduction to numerical algorithms for scientific computing. Root-finding, Runge-Kutta methods, Monte Carlo techniques, numerical solution of partial differential equations, minimization techniques such as neural networks. Applications to problems in classical mechanics and discrete- element electromagnetism.

• 22. 6 units (0-6-0); third term. Prerequisite: Ph 20 or equivalent experience with programming and numerical techniques. Introduction to scientific computing on parallel computers. Introduction to parallel computing and multiprocessing. Message passing on networked workstations. Algorithm decomposition and parallelization. Numerical solution of N-body systems on multiprocessor computers.

Ph 129 abc. Mathematical Methods of Physics. 9 units (3-0-6); first, second, third terms. Prerequisites: Ph 106 abc and ACM 95/100 abc or Ma 108 abc, or equivalents. Mathematical methods and their application in physics. First term includes analytic and numerical methods for solving differential equations, integral equations, and transforms, and other applications of real analysis. Second term covers group theoretic methods in physics. Third term focuses on probability and statistics in physics. The three terms can be taken independently. Instructors: Porter, Ooguri.