Edwards-Penney, sections 9.1 to 9.4 The textbook topics, definitions and theorems

Edwards-Penney 9.1 to 9.4 (12.1 K, txt, 17 Apr 2013)

Putzer's method for the 2x2 matrix exponential. Solution of u'=Au is: u(t) = exp(A t)u(0) THEOREM: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I), Lambda Symbols: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0. The DE System: r1'(t) = lambda_1 r1(t), r1(0)=0, r2'(t) = lambda_2 r2(t) + r1(t), r2(0)=0 See the slides and manuscript on systems for proofs and details. THEOREM. The formula can be used as e^{r1 t} - e^{r2 t} e^{At} = e^{r1 t} I + ------------------- (A-r1 I) r1 - r2 where r1=lambda_1, r2=lambda_2 are the eigenavalues of A. EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]), A=matrix([[2,3],[0,4]]) using the matrix exponential, Zeibur's method, Laplace resolvent and eigenanalysis. EXAMPLE. Solve a non-homogeneous system u'=Au+F(t), u(0)=vector([0,0]), A=matrix([[2,3],[0,4]]), F(t)=vector([3,1]) using variation of parameters.

Dynamical Systems TopicsEquilibria. Stability. Instability. Asymptotic stability. Classification of equilibria for u'=Au when det(A) is not zero, for the 2x2 case.

Spiral, saddle, center, node.Linearization theory. Jacobian.Detecting stability: Re(lambda)<0 ==> asym. stability. Stability at t=-infinity classifiesUnstablesolutions. Maple phase diagram tools. Demonstration for the example x' = x + y, y' = 1 - x^2 How to detect saddle, spiral, node, center in the linear case using Zeibur's method and examples. Limitations: In the case of a node, we cannot sub-classify as improper or proper using the Zeibur method and examples. The finer sub-classifications require the exponential matrix e^{At} or else a synthetic eigenvalue theorem which calculated the sub-classification.

Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.Extra Credit Maple Project: Earthquakes. Explore a 5-story or 7-story building and the resonant frequencies of oscillation of the building which might make it destruct during an earthquake. See Edwards-Penney, application section in 7.4.

Review of topicsPhase diagram. Stability and the three pictures: Node, Center, SpiralDetecting stability and instability for u'=Au at x=y=0: Main Theorem: Re(lambda)<0 ==> asym. stability. Stable center picture. Definition of stability. Stability at t=-infinity classifiesUnstablesolutions.Maple DemonstrationMaple phase diagram tools. Example x' = x + y, y' = 1 - x^2Spiral, saddle, center, node.Classification pictures Set 1: Stable node, stable center, stable spiral Set 2: Unstable node, unstable saddle, unstable spiral How to detect saddle, spiral, node, center in the linear case using Zeibur's method and examples.Limitations: In the case of a node, we cannot sub-classify as improper or proper using the Zeibur method and examples. The finer sub-classifications require the exponential matrix e^{At} or else a synthetic eigenvalue theorem which calculates the sub-classification.

B>Nonlinear stability theory When the linearized classification and stability transfers to the nonlinear system. stability of almost linear [nonlinear] systems, phase diagrams, classification of nonlinear systems.Nonlinear stabilityphase diagrams, classification. Using DEtools and DEplot in maple to make phase diagrams. Jacobian.

Spiral, saddle, center, node.Linearization theory. Jacobian.Algebraic Detection of Linear stability for u'=Au: Rule: det(A) not zero of all classifications! Re(lambda)<0 ==>asymptotic stabilityRe(lambda)=0 and lambda not zero ==>Center pictureStability at t=-infinity classifiesUnstablesolutions. When testing stability, we check t=infinity and t=-infinity.Nonlinear stability theory u'=f(u)When the linearized classification and stability transfers to the nonlinear system. stability of almost linear [nonlinear] systems, phase diagrams, classification of nonlinear systems.Final exam review started.Cover today ch8 and some of ch10. Review packet distributed on the web.Final exam detailsLess contact with ch3, ch4, ch6 due their appearance on exams 1,2,3. Since F2008, there are extra chapters 8,9 on the final. A good sample is the S2012 final exam. Chapters 5,6,7,10 will undergo changes and spins. For ch10, more contact with the second shifting theorem and the Dirac Delta. For ch7-ch8, there are additional methods for solving DE, especially Cayley-Hamilton-Ziebur, matrix exp(At) and the Laplace resolvent for first order systems. For ch5, deeper problems on the topics of variation of parameters and undetermined coefficients, resonance, and beats.

Final exam reviewSome chapter 8 and chapter 9 problems.Nonlinear stabilityphase diagrams, classification. Predator-Prey systems. How to tell which is the predator and which is the prey. Calculations for equilibrium points, linearization, classification of equilibria, impact on the phase diagram. Using DEtools and DEplot in maple to make phase diagrams. Exercises 9.1, 9.2.

Slides on Dynamical Systems: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)Manuscript: Laplace second order systems, spring-mass,boxcars, earthquakes (288.1 K, pdf, 04 Mar 2012)Slides: Introduction to dynamical systems (158.0 K, pdf, 04 Mar 2012)Slides: Phase Portraits for dynamical systems (239.3 K, pdf, 04 Mar 2012)Slides: Stability for dynamical systems (170.8 K, pdf, 04 Mar 2012)Slides: Nonlinear classification spiral, node, center, saddle (75.3 K, pdf, 12 Dec 2009)Slides: Matrix Exponential, Putzer Formula, Variation Parameters (130.1 K, pdf, 04 Mar 2012)SlidesReferences for Eigenanalysis and Systems of Differential Equations.: Algebraic eigenanalysis (187.6 K, pdf, 04 Mar 2012)Manuscript: What's eigenanalysis 2008 (174.2 K, pdf, 04 Mar 2012)Manuscript: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)Manuscript: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007)Manuscript: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (152.9 K, pdf, 04 Mar 2012)Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012)Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)Slides: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)Manuscript: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012)Slides: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)TextSystems of Differential Equations references: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)SlidesExtra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.Laplace theory references: Laplace and Newton calculus. Photos. (200.2 K, pdf, 04 Mar 2012)Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012)Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012)Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012)Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012)Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013)Slides: Maple Lab 7. Laplace applications (156.0 K, pdf, 04 Dec 2012)MAPLE: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)Manuscript: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012)Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012)Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)Slides: Heaviside's method 2008 (186.8 K, pdf, 20 Oct 2009)Manuscript: Laplace theory 2008 (351.3 K, pdf, 09 Apr 2013)Manuscript: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010)Transparencies: Laplace theory problem notes (17.2 K, txt, 03 Dec 2012)Text: Final exam study guide (8.2 K, txt, 05 Dec 2012)Text