Difference between revisions of "Differential equations and dynamical systems courses"

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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.
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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 101b, AM 125b and CDS 140a.
  
  
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'''Catalog listing'''
 
'''Catalog listing'''
nalytical 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.
+
<font color=blue>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. </font> Taught concurrently with CDS 140 and AM 125. Instructors: Marsden, Murray.
  
 
'''Dependent courses''':
 
'''Dependent courses''':
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=== CDS 140a:   ===
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=== CDS 140a: Introduction to Dynamics ===
 
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''' Partially overlapping courses'''
 
''' Partially overlapping courses'''
* ACM 106b, AM 125b
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* ACM 101b, AM 125b
 
|
 
|
 
'''Topics ([http://www.cds.caltech.edu/~marsden/cds140a-09/home Fall 2009]) '''
 
'''Topics ([http://www.cds.caltech.edu/~marsden/cds140a-09/home Fall 2009]) '''
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== Textbooks ==
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== Possible topics ==
  
 
{| border=1
 
{| border=1
 
|-
 
|-
 
| Topic
 
| Topic
| Bender
+
| ACM 101b
| Jordan and Smith
+
| AM 125b
 +
| CDS 140a
 +
| Jordan/Smith
 
| Perko
 
| Perko
 
| Verhulst
 
| Verhulst
{{cds140 chapter|title=Second-order differential equations in the phase plane}}
+
{{cds140 section|title=Existence and uniqueness of solutions|acm=x|cds=x|perko=x|jordan=x|bender=x|verhulst=x|am=x}}
{{cds140 section|title=Phase diagram for the pendulum equation|jordan=x}}
+
{{cds140 section|title=Linear ODEs, with inputs (convolution equation)|acm=x|cds=x|am=x|perko=x}}
{{cds140 section|title=Autonomous equations in the phase plane|jordan=x}}
+
{{cds140 section|title=Matrix exponential, Jordan form|acm=x|am=x|cds=x|perko=x|verhulst=-}}
{{cds140 section|title=Mechanical analogy for the conservative system|jordan=x}}
+
{{cds140 section|title=Transform methods (eg, Laplace, Fourier)|acm=x}}
{{cds140 section|title=The damped linear oscillator|jordan=x}}
+
{{cds140 section|title=Phase space diagrams, sinks, sources, saddles, limit cycles, etc|cds=x|perko=x|jordan=x|bender=x|verhulst=x|am=x}}
{{cds140 section|title=Nonlinear damping: limit cycles|jordan=x}}
+
{{cds140 section|title=Poincare-Bendixon theory|cds=x|perko=x|jordan=x|verhulst=x|am=x}}
{{cds140 section|title=Some applications|jordan=x}}
+
{{cds140 section|title=Analytical methods for solving nonlinear ODEs|acm=x|jordan=x|bender=x|am=?}}
{{cds140 section|title=Parameter-dependent conservative systems|jordan=x}}
+
{{cds140 section|title=Stability of equilibrium points, Lyapunov stability|cds=x|perko=x|jordan=x|bender=s|verhulst=x|am=x}}
{{cds140 section|title=Graphical representation of solutions}}
+
{{cds140 section|title=Invariant manifolds (stable, unstable, center)|cds=x|perko=x|jordan=x|verhulst=-}}
 
+
{{cds140 section|title=Stability of limit cycles, orbital stability, Poincare maps|cds=x|perko=x|jordan=x|verhulst=x|am=x}}
{{cds140 chapter|title=Plane autonomous systems and linearization}}
+
{{cds140 section|title=Attractors and structural stability|cds=?|perko=x|am=?}}
{{cds140 section|title=The general phase plane|jordan=x}}
+
{{cds140 section|title=Bifurcations, including saddle node, pitchfork, Hopf, etc|cds=?|acm=x|perko=x|jordan=x|verhulst=x|am=x}}
{{cds140 section|title=Some population models|jordan=x}}
+
{{cds140 section|title=Floquet theory, parametric resonance|perko=x|jordan=x|verhulst=x|am=x}}
{{cds140 section|title=Linear approximation at equilibrium points    |jordan=x}}
+
{{cds140 section|title=Lagrange's equations (variational principles), energy-based stability methods|cds=x}}
{{cds140 section|title=The general solution of linear autonomous plane system|jordan=x}}
+
{{cds140 section|title=Hamiltonian dynamics|cds=x}}
{{cds140 section|title=The phase paths of linear autonomous plane systems    |jordan=x}}
+
{{cds140 section|title=Ergodicity}}
{{cds140 section|title=Scaling in the phase diagram for a linear autonomous system  |jordan=x}}
+
{{cds140 section|title=Boundary value problems, Green functions|acm=x|am=x}}
{{cds140 section|title=Constructing a phase diagram  |jordan=x}}
+
{{cds140 section|title=Perturbation methods|acm=x|jordan=x|verhulst=x}}
{{cds140 section|title=Hamiltonian systems  |jordan=x}}
+
{{cds140 section|title=Asymptotic and approximation methods|acm=x|verhulst=x}}
 
+
{{cds140 section|title=Singular perturbation theory|jordan=?}}
{{cds140 chapter|title=Geometrical aspects of plane autonomous systems}}
+
{{cds140 section|title=Method of averaging|jordan=x|verhulst=x}}
{{cds140 section|title=The index of a point|jordan=x}}
+
{{cds140 section|title=Integral equations|acm=x}}
{{cds140 section|title=The index at infinity|jordan=x}}
+
{{cds140 section|title=Partial differential equations|acm=?}}
{{cds140 section|title=The phase diagram at infinity|jordan=x}}
+
{{cds140 section|title=Numerical methods|acm=?}}
{{cds140 section|title=Limit cycles and other closed paths|jordan=x}}
+
{{cds140 section|title=Computation of the phase diagram|jordan=x}}
+
{{cds140 section|title=Homoclinic and heteroclinic paths|jordan=x}}
+
 
+
{{cds140 chapter|title=Periodic solutions; averaging methods}}
+
{{cds140 section|title=An energy-balance method for limit cycles|jordan=x}}
+
{{cds140 section|title=Amplitude and frequency estimates: polar coordinates|jordan=x}}
+
{{cds140 section|title=An averaging method for spiral phase paths|jordan=x}}
+
{{cds140 section|title=Periodic solutions: harmonic balance|jordan=x}}
+
{{cds140 section|title=The equivalent linear equation by harmonic balance|jordan=x}}
+
 
+
{{cds140 chapter|title=Perturbation methods}}
+
{{cds140 section|title=Nonautonomous systems: forced oscillations|jordan=x}}
+
{{cds140 section|title=The direct perturbation method for the undamped Duffing’s equation|jordan=x}}
+
{{cds140 section|title=Forced oscillations far from resonance|jordan=x}}
+
{{cds140 section|title=Forced oscillations near resonance with weak excitation|jordan=x}}
+
{{cds140 section|title=The amplitude equation for the undamped pendulum|jordan=x}}
+
{{cds140 section|title=The amplitude equation for a damped pendulum|jordan=x}}
+
{{cds140 section|title=Soft and hard springs|jordan=x}}
+
{{cds140 section|title=Amplitude–phase perturbation for the pendulum equation|jordan=x}}
+
{{cds140 section|title=Periodic solutions of autonomous equations (Lindstedt’s method)|jordan=x}}
+
{{cds140 section|title= Forced oscillation of a self-excited equation|jordan=x}}
+
{{cds140 section|title= The perturbation method and Fourier series|jordan=x}}
+
{{cds140 section|title= Homoclinic bifurcation: an example|jordan=x}}
+
 
+
{{cds140 chapter|title=Singular perturbation methods}}
+
{{cds140 section|title=Non-uniform approximations to functions on an interval|jordan=x}}
+
{{cds140 section|title=Coordinate perturbation|jordan=x}}
+
{{cds140 section|title=Lighthill’s method|jordan=x}}
+
{{cds140 section|title=Time-scaling for series solutions of autonomous equations|jordan=x}}
+
{{cds140 section|title=The multiple-scale technique applied to saddle points and nodes|jordan=x}}
+
{{cds140 section|title=Matching approximations on an interval|jordan=x}}
+
{{cds140 section|title=A matching technique for differential equations|jordan=x}}
+
 
+
{{cds140 chapter|title=Forced oscillations: harmonic and subharmonic response, stability, and entrainment}}
+
{{cds140 section|title=General forced periodic solutions|jordan=x}}
+
{{cds140 section|title=Harmonic solutions, transients, and stability for Duffing’s equation|jordan=x}}
+
{{cds140 section|title=The jump phenomenon|jordan=x}}
+
{{cds140 section|title=Harmonic oscillations, stability, and transients for the forced van der Pol equation|jordan=x}}
+
{{cds140 section|title=Frequency entrainment for the van der Pol equation|jordan=x}}
+
{{cds140 section|title= Subharmonics of Duffing’s equation by perturbation|jordan=x}}
+
{{cds140 section|title=Stability and transients for subharmonics of Duffing’s equation|jordan=x}}
+
 
+
{{cds140 chapter|title=Stability}}
+
{{cds140 section|title=Poincaré stability (stability of paths)|jordan=x}}
+
{{cds140 section|title=Paths and solution curves for general systems|jordan=x}}
+
{{cds140 section|title=Stability of time solutions: Liapunov stability|jordan=x}}
+
{{cds140 section|title=Liapunov stability of plane autonomous linear systems|jordan=x}}
+
{{cds140 section|title=Structure of the solutions of n-dimensional linear systems|jordan=x}}
+
{{cds140 section|title=Structure of n-dimensional inhomogeneous linear systems|jordan=x}}
+
{{cds140 section|title=Stability and boundedness for linear systems|jordan=x}}
+
{{cds140 section|title=Stability of linear systems with constant coefficients|jordan=x}}
+
{{cds140 section|title=Linear approximation at equilibrium points for first-order systems in n variables|jordan=x}}
+
{{cds140 section|title= Stability of a class of non-autonomous linear systems in n dimensions|jordan=x}}
+
{{cds140 section|title= Stability of the zero solutions of nearly linear systems|jordan=x}}
+
 
+
{{cds140 chapter|title=Stability by solution perturbation: Mathieu's equation}}
+
{{cds140 section|title=The stability of forced oscillations by solution perturbation|jordan=x}}
+
{{cds140 section|title=Equations with periodic coefficients (Floquet theory)|jordan=x}}
+
{{cds140 section|title=Mathieu’s equation arising from a Duffing equation|jordan=x}}
+
{{cds140 section|title=Transition curves for Mathieu’s equation by perturbation|jordan=x}}
+
{{cds140 section|title=Mathieu’s damped equation arising from a Duffing equation|jordan=x}}
+
 
+
{{cds140 chapter|title=Liapunov methods for determining stability of the zero solution}}
+
{{cds140 section|title=Introducing the Liapunov method|jordan=x}}
+
{{cds140 section|title=Topographic systems and the Poincaré–Bendixson theorem|jordan=x}}
+
{{cds140 section|title=Liapunov stability of the zero solution|jordan=x}}
+
{{cds140 section|title=Asymptotic stability of the zero solution|jordan=x}}
+
{{cds140 section|title=Extending weak Liapunov functions to asymptotic stability|jordan=x}}
+
{{cds140 section|title=A more general theory for autonomous systems|jordan=x}}
+
{{cds140 section|title=A test for instability of the zero solution: n dimensions|jordan=x}}
+
{{cds140 section|title=Stability and the linear approximation in two dimensions|jordan=x}}
+
{{cds140 section|title=Exponential function of a matrix|jordan=x}}
+
{{cds140 section|title= Stability and the linear approximation for nth order autonomous systems|jordan=x}}
+
{{cds140 section|title= Special systems|jordan=x}}
+
 
+
{{cds140 chapter|title=The existence of periodic solutions}}
+
{{cds140 section|title=The Poincaré–Bendixson theorem and periodic solutions|jordan=x}}
+
{{cds140 section|title=A theorem on the existence of a centre|jordan=x}}
+
{{cds140 section|title=A theorem on the existence of a limit cycle|jordan=x}}
+
{{cds140 section|title=Van der Pol’s equation with large parameter|jordan=x}}
+
 
+
{{cds140 chapter|title=Bifurcations and manifolds}}
+
{{cds140 section|title=Examples of simple bifurcations|jordan=x}}
+
{{cds140 section|title=The fold and the cusp|jordan=x}}
+
{{cds140 section|title=Further types of bifurcation|jordan=x}}
+
{{cds140 section|title=Hopf bifurcations|jordan=x}}
+
{{cds140 section|title=Higher-order systems: manifolds|jordan=x}}
+
{{cds140 section|title=Linear approximation: centre manifolds|jordan=x}}
+
 
+
{{cds140 chapter|title=Poincaré sequences, homoclinic bifurcation, and chaos}}
+
{{cds140 section|title=Poincaré sequences|jordan=x}}
+
{{cds140 section|title=Poincaré sections for nonautonomous systems|jordan=x}}
+
{{cds140 section|title=Subharmonics and period doubling|jordan=x}}
+
{{cds140 section|title=Homoclinic paths, strange attractors and chaos|jordan=x}}
+
{{cds140 section|title=The Duffing oscillator|jordan=x}}
+
{{cds140 section|title=A discrete system: the logistic difference equation|jordan=x}}
+
{{cds140 section|title=Liapunov exponents and difference equations|jordan=x}}
+
{{cds140 section|title=Homoclinic bifurcation for forced systems|jordan=x}}
+
{{cds140 section|title=The horseshoe map|jordan=x}}
+
{{cds140 section|title= Melnikov’s method for detecting homoclinic bifurcation|jordan=x}}
+
{{cds140 section|title= Liapunov exponents and differential equations|jordan=x}}
+
{{cds140 section|title= Power spectra|jordan=x}}
+
{{cds140 section|title= Some further features of chaotic oscillations|jordan=x}}
+
 
+
 
|}
 
|}
|-
 
  
 
== Course Listings ==
 
== Course Listings ==

Latest revision as of 18:03, 14 November 2010

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 101b, 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: Introduction to Dynamics

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 101b, AM 125b

Topics (Fall 2009)

  1. Introductory Material
  2. Linear Systems
  3. Vector Fields and Flows
  4. Stability and Linearization
  5. Mechanical Systems
  6. Invariant Manifolds
  7. Liapunov Functions

Textbooks

  • Perko: Differential Equations and Dynamical Systems

Possible topics

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

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.