MMChapter4_4p8

# Solutions will travel along v 1 away from the origin

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, solutions will travel along V 1 , away from the origin. If λ 2 is negative , solutions will travel along V 2 , toward the origin. In general, solutions travel in a direction which is a linear combination of V 1 and V 2 . The origin is classified as either a node or a saddle . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 9 / 41

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1. Node: Both eigenvalues have the same sign and may be distinct or equal, λ 1 λ 2 < 0 or 0 < λ 1 λ 2 . The origin can be further classified as proper or improper . A node is called proper when the eigenvalues are equal and there are two linearly independent eigenvectors; otherwise it is called improper . A proper node is also referred as a star point or star solution . The term degenerate node is also used to refer to a node when the two eigenvalues of the matrix A are equal. In Figure 1, the improper node in the upper left corner has two distinct eigenvalues; it is not degenerate. But the other two nodes, to the right are degenerate nodes (the eigenvalues are equal). If there is only one independent eigenvector, the dynamics are illustrated in the center figure and if there are two independent eigenvectors, the dynamics are illustrated in the upper right corner (star solution). J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 10 / 41
1. Node: Both eigenvalues have the same sign and may be distinct or equal, λ 1 λ 2 < 0 or 0 < λ 1 λ 2 . The origin can be further classified as proper or improper . A node is called proper when the eigenvalues are equal and there are two linearly independent eigenvectors; otherwise it is called improper . A proper node is also referred as a star point or star solution . The term degenerate node is also used to refer to a node when the two eigenvalues of the matrix A are equal. In Figure 1, the improper node in the upper left corner has two distinct eigenvalues; it is not degenerate. But the other two nodes, to the right are degenerate nodes (the eigenvalues are equal). If there is only one independent eigenvector, the dynamics are illustrated in the center figure and if there are two independent eigenvectors, the dynamics are illustrated in the upper right corner (star solution). 2. Saddle: Eigenvalues λ 1 and λ 2 have opposite signs λ 1 λ 2 < 0 (e.g., λ 1 < 0 < λ 2 ). J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 10 / 41

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Improper and proper node,spiral, saddle and center. Stable improper node Stable improper node Stable proper node Stable spiral Center Saddle Figure 1: Graph of solutions for an improper and proper node, spiral, saddle, and center. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 11 / 41
Complex Eigenvalues: In the case of complex eigenvalues, λ 1 , 2 = a ± ib , b 6 = 0. Because solutions to the linear system dX/dt = AX include factors with cos ( bt ) and sin ( bt ), solutions spiral around the equilibrium. If the real part a < 0, then the solutions with e at cos ( bt ) or e at sin ( bt ) spiral inward, toward the origin. But if the real part a > 0, then the solutions spiral outward, away from the origin. Finally, if the real part a = 0, then solutions are closed curves, encircling the origin. The origin is classified

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