Difference between revisions of "Differential equations and dynamical systems courses"
(→Textbooks) 

Line 80:  Line 80:  
== Textbooks ==  == Textbooks ==  
+  
+  { border=1  
+    
+   Topic  
+   Bender  
+   Jordan and Smith  
+   Perko  
+   Verhulst  
+  {{cds140 chaptertitle=Secondorder differential equations in the phase plane}}  
+  {{cds140 sectiontitle=Phase diagram for the pendulum equationjordan=x}}  
+  {{cds140 sectiontitle=Autonomous equations in the phase planejordan=x}}  
+  {{cds140 sectiontitle=Mechanical analogy for the conservative systemjordan=x}}  
+  {{cds140 sectiontitle=The damped linear oscillatorjordan=x}}  
+  {{cds140 sectiontitle=Nonlinear damping: limit cyclesjordan=x}}  
+  {{cds140 sectiontitle=Some applicationsjordan=x}}  
+  {{cds140 sectiontitle=Parameterdependent conservative systemsjordan=x}}  
+  {{cds140 sectiontitle=Graphical representation of solutions}}  
+  
+  {{cds140 chaptertitle=Plane autonomous systems and linearization}}  
+  {{cds140 sectiontitle=The general phase planejordan=x}}  
+  {{cds140 sectiontitle=Some population modelsjordan=x}}  
+  {{cds140 sectiontitle=Linear approximation at equilibrium points jordan=x}}  
+  {{cds140 sectiontitle=The general solution of linear autonomous plane systemjordan=x}}  
+  {{cds140 sectiontitle=The phase paths of linear autonomous plane systems jordan=x}}  
+  {{cds140 sectiontitle=Scaling in the phase diagram for a linear autonomous system jordan=x}}  
+  {{cds140 sectiontitle=Constructing a phase diagram jordan=x}}  
+  {{cds140 sectiontitle=Hamiltonian systems jordan=x}}  
+  
+  {{cds140 chaptertitle=Geometrical aspects of plane autonomous systems}}  
+  {{cds140 sectiontitle=The index of a pointjordan=x}}  
+  {{cds140 sectiontitle=The index at infinityjordan=x}}  
+  {{cds140 sectiontitle=The phase diagram at infinityjordan=x}}  
+  {{cds140 sectiontitle=Limit cycles and other closed pathsjordan=x}}  
+  {{cds140 sectiontitle=Computation of the phase diagramjordan=x}}  
+  {{cds140 sectiontitle=Homoclinic and heteroclinic pathsjordan=x}}  
+  
+  {{cds140 chaptertitle=Periodic solutions; averaging methods}}  
+  {{cds140 sectiontitle=An energybalance method for limit cyclesjordan=x}}  
+  {{cds140 sectiontitle=Amplitude and frequency estimates: polar coordinatesjordan=x}}  
+  {{cds140 sectiontitle=An averaging method for spiral phase pathsjordan=x}}  
+  {{cds140 sectiontitle=Periodic solutions: harmonic balancejordan=x}}  
+  {{cds140 sectiontitle=The equivalent linear equation by harmonic balancejordan=x}}  
+  
+  {{cds140 chaptertitle=Perturbation methods}}  
+  {{cds140 sectiontitle=Nonautonomous systems: forced oscillationsjordan=x}}  
+  {{cds140 sectiontitle=The direct perturbation method for the undamped Duffing’s equationjordan=x}}  
+  {{cds140 sectiontitle=Forced oscillations far from resonancejordan=x}}  
+  {{cds140 sectiontitle=Forced oscillations near resonance with weak excitationjordan=x}}  
+  {{cds140 sectiontitle=The amplitude equation for the undamped pendulumjordan=x}}  
+  {{cds140 sectiontitle=The amplitude equation for a damped pendulumjordan=x}}  
+  {{cds140 sectiontitle=Soft and hard springsjordan=x}}  
+  {{cds140 sectiontitle=Amplitude–phase perturbation for the pendulum equationjordan=x}}  
+  {{cds140 sectiontitle=Periodic solutions of autonomous equations (Lindstedt’s method)jordan=x}}  
+  {{cds140 sectiontitle= Forced oscillation of a selfexcited equationjordan=x}}  
+  {{cds140 sectiontitle= The perturbation method and Fourier seriesjordan=x}}  
+  {{cds140 sectiontitle= Homoclinic bifurcation: an examplejordan=x}}  
+  
+  {{cds140 chaptertitle=Singular perturbation methods}}  
+  {{cds140 sectiontitle=Nonuniform approximations to functions on an intervaljordan=x}}  
+  {{cds140 sectiontitle=Coordinate perturbationjordan=x}}  
+  {{cds140 sectiontitle=Lighthill’s methodjordan=x}}  
+  {{cds140 sectiontitle=Timescaling for series solutions of autonomous equationsjordan=x}}  
+  {{cds140 sectiontitle=The multiplescale technique applied to saddle points and nodesjordan=x}}  
+  {{cds140 sectiontitle=Matching approximations on an intervaljordan=x}}  
+  {{cds140 sectiontitle=A matching technique for differential equationsjordan=x}}  
+  
+  {{cds140 chaptertitle=Forced oscillations: harmonic and subharmonic response, stability, and entrainment}}  
+  {{cds140 sectiontitle=General forced periodic solutionsjordan=x}}  
+  {{cds140 sectiontitle=Harmonic solutions, transients, and stability for Duffing’s equationjordan=x}}  
+  {{cds140 sectiontitle=The jump phenomenonjordan=x}}  
+  {{cds140 sectiontitle=Harmonic oscillations, stability, and transients for the forced van der Pol equationjordan=x}}  
+  {{cds140 sectiontitle=Frequency entrainment for the van der Pol equationjordan=x}}  
+  {{cds140 sectiontitle= Subharmonics of Duffing’s equation by perturbationjordan=x}}  
+  {{cds140 sectiontitle=Stability and transients for subharmonics of Duffing’s equationjordan=x}}  
+  
+  {{cds140 chaptertitle=Stability}}  
+  {{cds140 sectiontitle=Poincaré stability (stability of paths)jordan=x}}  
+  {{cds140 sectiontitle=Paths and solution curves for general systemsjordan=x}}  
+  {{cds140 sectiontitle=Stability of time solutions: Liapunov stabilityjordan=x}}  
+  {{cds140 sectiontitle=Liapunov stability of plane autonomous linear systemsjordan=x}}  
+  {{cds140 sectiontitle=Structure of the solutions of ndimensional linear systemsjordan=x}}  
+  {{cds140 sectiontitle=Structure of ndimensional inhomogeneous linear systemsjordan=x}}  
+  {{cds140 sectiontitle=Stability and boundedness for linear systemsjordan=x}}  
+  {{cds140 sectiontitle=Stability of linear systems with constant coefficientsjordan=x}}  
+  {{cds140 sectiontitle=Linear approximation at equilibrium points for firstorder systems in n variablesjordan=x}}  
+  {{cds140 sectiontitle= Stability of a class of nonautonomous linear systems in n dimensionsjordan=x}}  
+  {{cds140 sectiontitle= Stability of the zero solutions of nearly linear systemsjordan=x}}  
+  
+  {{cds140 chaptertitle=Stability by solution perturbation: Mathieu's equation}}  
+  {{cds140 sectiontitle=The stability of forced oscillations by solution perturbationjordan=x}}  
+  {{cds140 sectiontitle=Equations with periodic coefficients (Floquet theory)jordan=x}}  
+  {{cds140 sectiontitle=Mathieu’s equation arising from a Duffing equationjordan=x}}  
+  {{cds140 sectiontitle=Transition curves for Mathieu’s equation by perturbationjordan=x}}  
+  {{cds140 sectiontitle=Mathieu’s damped equation arising from a Duffing equationjordan=x}}  
+  
+  {{cds140 chaptertitle=Liapunov methods for determining stability of the zero solution}}  
+  {{cds140 sectiontitle=Introducing the Liapunov methodjordan=x}}  
+  {{cds140 sectiontitle=Topographic systems and the Poincaré–Bendixson theoremjordan=x}}  
+  {{cds140 sectiontitle=Liapunov stability of the zero solutionjordan=x}}  
+  {{cds140 sectiontitle=Asymptotic stability of the zero solutionjordan=x}}  
+  {{cds140 sectiontitle=Extending weak Liapunov functions to asymptotic stabilityjordan=x}}  
+  {{cds140 sectiontitle=A more general theory for autonomous systemsjordan=x}}  
+  {{cds140 sectiontitle=A test for instability of the zero solution: n dimensionsjordan=x}}  
+  {{cds140 sectiontitle=Stability and the linear approximation in two dimensionsjordan=x}}  
+  {{cds140 sectiontitle=Exponential function of a matrixjordan=x}}  
+  {{cds140 sectiontitle= Stability and the linear approximation for nth order autonomous systemsjordan=x}}  
+  {{cds140 sectiontitle= Special systemsjordan=x}}  
+  
+  {{cds140 chaptertitle=The existence of periodic solutions}}  
+  {{cds140 sectiontitle=The Poincaré–Bendixson theorem and periodic solutionsjordan=x}}  
+  {{cds140 sectiontitle=A theorem on the existence of a centrejordan=x}}  
+  {{cds140 sectiontitle=A theorem on the existence of a limit cyclejordan=x}}  
+  {{cds140 sectiontitle=Van der Pol’s equation with large parameterjordan=x}}  
+  
+  {{cds140 chaptertitle=Bifurcations and manifolds}}  
+  {{cds140 sectiontitle=Examples of simple bifurcationsjordan=x}}  
+  {{cds140 sectiontitle=The fold and the cuspjordan=x}}  
+  {{cds140 sectiontitle=Further types of bifurcationjordan=x}}  
+  {{cds140 sectiontitle=Hopf bifurcationsjordan=x}}  
+  {{cds140 sectiontitle=Higherorder systems: manifoldsjordan=x}}  
+  {{cds140 sectiontitle=Linear approximation: centre manifoldsjordan=x}}  
+  
+  {{cds140 chaptertitle=Poincaré sequences, homoclinic bifurcation, and chaos}}  
+  {{cds140 sectiontitle=Poincaré sequencesjordan=x}}  
+  {{cds140 sectiontitle=Poincaré sections for nonautonomous systemsjordan=x}}  
+  {{cds140 sectiontitle=Subharmonics and period doublingjordan=x}}  
+  {{cds140 sectiontitle=Homoclinic paths, strange attractors and chaosjordan=x}}  
+  {{cds140 sectiontitle=The Duffing oscillatorjordan=x}}  
+  {{cds140 sectiontitle=A discrete system: the logistic difference equationjordan=x}}  
+  {{cds140 sectiontitle=Liapunov exponents and difference equationsjordan=x}}  
+  {{cds140 sectiontitle=Homoclinic bifurcation for forced systemsjordan=x}}  
+  {{cds140 sectiontitle=The horseshoe mapjordan=x}}  
+  {{cds140 sectiontitle= Melnikov’s method for detecting homoclinic bifurcationjordan=x}}  
+  {{cds140 sectiontitle= Liapunov exponents and differential equationsjordan=x}}  
+  {{cds140 sectiontitle= Power spectrajordan=x}}  
+  {{cds140 sectiontitle= Some further features of chaotic oscillationsjordan=x}}  
+  
+  }  
+    
== Course Listings ==  == Course Listings == 
Revision as of 02:57, 22 October 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 106b, AM 125b and CDS 140a.
Contents
Overview of current course sequence
ACM 101b: Methods of Applied Mathematics
 
AM 125b: Engineering Mathematical Principles
 
CDS 140a:

Textbooks
Topic  Bender  Jordan and Smith  Perko  Verhulst  
Secondorder differential equations in the phase plane  

x  

x  

x  

x  

x  

x  

x  


Plane autonomous systems and linearization  

x  

x  

x  

x  

x  

x  

x  

x  
Geometrical aspects of plane autonomous systems  

x  

x  

x  

x  

x  

x  
Periodic solutions; averaging methods  

x  

x  

x  

x  

x  
Perturbation methods  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  
Singular perturbation methods  

x  

x  

x  

x  

x  

x  

x  
Forced oscillations: harmonic and subharmonic response, stability, and entrainment  

x  

x  

x  

x  

x  

x  

x  
Stability  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  
Stability by solution perturbation: Mathieu's equation  

x  

x  

x  

x  

x  
Liapunov methods for determining stability of the zero solution  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  
The existence of periodic solutions  

x  

x  

x  

x  
Bifurcations and manifolds  

x  

x  

x  

x  

x  

x  
Poincaré sequences, homoclinic bifurcation, and chaos  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x  

x 

Course Listings
ACM 95/100 abc. Introductory Methods of Applied Mathematics. 12 units (408); 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, SturmLiouville 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 (306); 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, WeinerHopf, and integral equations. Instructors: Guo, Hou.
ACM 106 abc. Introductory Methods of Computational Mathematics. 9 units (306); 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; timefrequency 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 (306); second term. Prerequisites: ACM 11, 106 or equivalent. Introduction to parallel program design for numerically intensive scientific applications. Parallel programming methods; distributedmemory model with message passing using the message passing interface; sharedmemory model with threads using open MP, CUDA; objectbased models using a problemsolving 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 finitedifference, finiteelement; particlebased simulations. Performance measurement, scaling and parallel efficiency, load balancing strategies. Instructor: Aivazis.
ACM 210 ab. Numerical Methods for PDEs. 9 units (306); 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, LaxWendroff theorem, Godunov’s method, Roe’s linearization, TVD schemes, highresolution 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 hp refinement. Multiscale finite element methods for elliptic problems with multiscale coefficients. Instructor: Guo.
AM 125 abc. Engineering Mathematical Principles. 9 units (306); 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 (306); second, third terms. Equilibrium concepts, conservative and dissipative systems, Lagrange’s equations, differential equations of motion for discrete single and multi degreeoffreedom 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 (BernoulliEuler and Timoshenko beam theory), and plates, traveling and standing wave solutions to motion of continuous systems, Rayleigh quotient and the RayleighRitz 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 (306); first, second terms. Prerequisite: ACM 95 or equivalent. Basics in topics in dynamics in Euclidean space, including equilibria, stability, Lyapunov functions, periodic solutions, PoincaréBendixon theory, Poincaré maps. Attractors and structural stability. The EulerLagrange 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 (306); 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, rigidbody dynamics, EulerPoincaré 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 (405); 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 (306); 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 (306); 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 (060); 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 (060); second, third terms. Prerequisite: Ph 20 or equivalent experience with programming and numerical techniques. Introduction to numerical algorithms for scientific computing. Rootfinding, RungeKutta 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 (060); 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 Nbody systems on multiprocessor computers.
Ph 129 abc. Mathematical Methods of Physics. 9 units (306); 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.