Review of last 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.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.

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 S2010 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.

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.

Final exam review continuedSome chapter 8 and chapter 9 problems. Subspace problems from chapter 4.Nonlinear mechanical systems.Hard and soft springs. Nonlinear pendulum. Undamped pendulum. Damped pendulum. Phase diagrams. Energy conservation laws and separatrices.