Edwards-Penney, sections 6.1, 6.2, 6.3, 6.4 The textbook topics, definitions and theorems

Edwards-Penney 6.1, 6.2, 6.3, 6.4 (11.8 K, txt, 05 Apr 2015)

Slides on Dynamical Systems: Systems theory and examples (730.9 K, pdf, 09 Apr 2014)Manuscript: Laplace second order systems, spring-mass, boxcars, earthquakes (273.7 K, pdf, 14 Mar 2016)Slides: Dynamical systems (1037.6 K, pdf, 05 Apr 2015)Manuscript: Introduction to dynamical systems (144.9 K, pdf, 14 Mar 2016)Slides: Phase Portraits for dynamical systems (221.2 K, pdf, 14 Mar 2016)Slides: Stability for dynamical systems (158.2 K, pdf, 14 Mar 2016)Slides: Nonlinear classification spiral, node, center, saddle (97.9 K, pdf, 14 Mar 2016)Slides: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)Slides

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: Each vector solution u(t) is a vector linear combination of Euler atoms. An atom has limit zero at t=infinity if and only if real part of the root is negative. A root with zero real part has Euler atoms x^n or pure sine and cosine times x^n. Theorem. 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 calculates 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 5.3.: Extra Credit Sources (4.0 K, html, 20 Apr 2017)HTML

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 ==> asymptotic 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.

Nonlinear stability theoryWhen 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.

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 6.1, 6.2.

Nonlinear mechanical systems.Hard and soft springs. Nonlinear pendulum. Undamped pendulum. Damped pendulum. Phase diagrams. Energy conservation laws and separatrices.

Review of the week's 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.Spiral, 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.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.

Slides on Dynamical Systems: Systems theory and examples (730.9 K, pdf, 09 Apr 2014)Manuscript: Laplace second order systems, spring-mass,boxcars, earthquakes (273.7 K, pdf, 14 Mar 2016)Slides: Introduction to dynamical systems (144.9 K, pdf, 14 Mar 2016)Slides: Phase Portraits for dynamical systems (221.2 K, pdf, 14 Mar 2016)Slides: Stability for dynamical systems (158.2 K, pdf, 14 Mar 2016)Slides: Nonlinear classification spiral, node, center, saddle (97.9 K, pdf, 14 Mar 2016)Slides: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)SlidesReferences for Eigenanalysis and Systems of Differential Equations.: Algebraic eigenanalysis (173.4 K, pdf, 14 Mar 2016)Slides: What's eigenanalysis 2008 (161.5 K, pdf, 14 Mar 2016)Slides: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)Slides: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007)Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.4 K, pdf, 14 Mar 2016)Slides: Cayley-Hamilton-Ziebur for second order systems (130.4 K, pdf, 22 Mar 2017)Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)Slides: Systems of DE examples and theory (730.9 K, pdf, 09 Apr 2014)Manuscript: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)Slides: Lawrence Page's pagerank algorithm (0.0 K, txt, 31 Dec 1969)Text: History of telecom companies (0.0 K, txt, 31 Dec 1969)TextSystems of Differential Equations references: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 20 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. (188.3 K, pdf, 14 Mar 2016)Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016)Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016)Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016)Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016)Slides: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)MAPLE: DE systems, examples, theory (730.9 K, pdf, 09 Apr 2014)Manuscript: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 20 Aug 2008)Slides: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)Manuscript: Laplace theory 2008 (544.6 K, pdf, 07 Mar 2017)Manuscript: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 22 Feb 2015)Transparencies: Laplace theory problem notes (0.0 K, txt, 31 Dec 1969)Text: Final exam study guide (0.0 K, txt, 31 Dec 1969)Text