Differential Equations MATH 33B
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This 1 page Class Notes was uploaded by Kaylin Wehner on Friday September 4, 2015. The Class Notes belongs to MATH 33B at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 102 views. For similar materials see /class/177836/math-33b-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.
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Date Created: 09/04/15
Math 33B Final Exam Review Sheet 1 Classi cation of the Equilibria of y Ay where A is a 2 gtlt 2 matrix with trace T and determinant D Generic Cases 0 Real eigenvalues one positive one negative D lt 0 Saddle o Distinct real e values both negative D gt 0 T2 7 4D gt 0 T lt 0 Nodal Sink o Distinct real e values both positive D gt 0 T274D gt 0 T gt 0 Nodal Source 0 Complex e values with negative real part T2 7 4D lt 0 T lt 0 Spiral Sink 0 Complex e values with positive real part T2 7 4D lt 0 T gt 0 Spiral Source Degenerate Cases 0 Purely imaginary eigenvalues D gt 0 T 0 Center 0 Negative e value of multiplicity two T2 7 4D 0 T lt 0 Degenerate Sink Like a nodal sink but a solution may curve signi cantly as it approaches the origin like its almost77 a spiral 0 Positive e value of multiplicity two T2 7 4D 0 T gt 0 Degenerate Source Like a nodal source but a solution may curve signi cantly as it leaves the origin like its almost77 a spiral 0 One of the eigenvalues is 0 D 0 No Catchy Name In this case there are in nitely many equilibria arranged along a line in the plane If T lt 0 they are all stable If T gt 0 they are all unstable If T 0 they are neither All other solutions are straight lines 96 may 2 Qualitative Analysis of Autonomous Nonlinear Systems y 9mg 0 The znullcline is the set of points Ly where fy 0 ie where the arrows all point straight up or down The ynullcline is the set of points Ly where gxy 0 ie where the arrows all point straight left or right 0 A point xmyo is an equilibrium if f0y0 0 and gxoy0 0 ie if it is on both nullclines Q Q o The Jacobian of the system is the matrix J lt3 3 E E o The linearization of the system at an equilibrium zoy0 is the homogeneous linear system y J07yoY o If the linearization at zoy0 is one of the ve generic types saddle nodal sink nodal source spiral sink spiral source then the equilibrium at zoy0 is of the same type
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