Survey of Methods for solving a 2x2 dynamical system1. Cayley-Hamilton-Ziebur method for u'=Au Solution: u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n) Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0 Vectors: Symbols vec(c_1), ..., vec(c_n) are not arbitrary. They are determined from A and u(0). Algorithm outlined above for 2x2. 2. Laplace resolvent L(u)=(s I - A)^(-1) u(0) 3. Eigenanalysis u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2 4. 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 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.

Exam 3 ReviewEigenvalues A 4x4 matrix. Block determinant theorem. Eigenvectors for a 4x4. lambda=5,5,3i,-3i One panel for lambda=5 First frame is A-5I with 0 appended Find rref Apply last frame algorithm Scalar general solution Take partials on t1, t2to find v1,v2 Eigenpairs are (5,v1), (5,v2) One panel for lambda=3i Same outline as lambda=5 Get one eigenpair (3i,v3) Other eigenpair=(-3i,v4) where v4 is the conjugate of v3. Shortest trial solution in undetermined coefficients. Second shifting theorem in Laplace theory.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. <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 F2009 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 and second order systems. For ch5, deeper problems on variation of parameters and undetermined coefficients, resonance, and beats.

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.

Gustafson: 1-3pm