lecture21 - Differential Equations & Linear...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 21 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Three main patterns Contents Three main patterns Repeated eigenvalues Summary of three main cases For second order systems with real coefficients, we have completed three main cases that can occur. 1. Eigenvalues are real and have opposite signs; x = is a saddle point. 2. Eigenvalues are real and have same sign but are unequal; x = is a node. 3. Eigenvalues are complex with nonzero real part; x = is a spiral point. Saddle points When eigenvalues are real and have different signs, the origin is a saddle point. A saddle point is always unstable because almost all trajectories depart from it. Nodes When eigenvalues are real, different and have same sign, the origin is a node. If the eigenvalues are negative, the node is stable because all trajectories approach it. If the eigenvalues are positive, the node is unstable because all trajectories depart from it. Spiral points When eigenvalues are complex with nonzero real part, the origin is a spiral point. If the real part is negative, the spiral point is asymptotically stable because all trajectories approach it. If the real part is positive, the spiral point is unstable because all trajectories depart from it. Example Example The system x = α 2- 2 0 x contains a parameter α . Describe how the solutions depend qualitatively on α ; in particular, find the critical values of α at which the qualitative behavior of the...
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lecture21 - Differential Equations & Linear...

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