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MMChapter4_4p8

# The origin is asymptotically stable if the

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The origin is asymptotically stable if the eigenvalues of A are negative or have negative real parts. The origin is stable if the eigenvalues of A are nonpositive or have nonpositive real parts. The origin is unstable if at least one eigenvalue of A is positive or has positive real part. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 4 / 41

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Solutions approach the origin if the origin is asymptotically stable , lim t →∞ ( x ( t ) , y ( t ) ) = (0 , 0) . Based on these definition, it is easy to determine the stability of the origin once the eigenvalues are known. In addition, even without calculating the eigenvalues, the stability can be determined by applying the Routh-Hurwitz criteria to the characteristic polynomial of A : λ 2 - Tr ( A ) λ + det ( A ) = 0 . Asymptotically stability is determined only by the trace and determinant because these two quantities are the coefficients of the characteristic polynomial . According to the Routh-Hurwitz criteria, the eigenvalues lie in the left hand of the complex plane if and only if the coefficients are positive . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 5 / 41
Corollary (4.2) Suppose dX/dt = AX , where A is a constant 2 × 2 matrix with det ( A ) 6 = 0 . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 6 / 41

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Corollary (4.2) Suppose dX/dt = AX , where A is a constant 2 × 2 matrix with det ( A ) 6 = 0 . The origin is asymptotically stable iff Tr ( A ) < 0 and det ( A ) > 0 . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 6 / 41
Corollary (4.2) Suppose dX/dt = AX , where A is a constant 2 × 2 matrix with det ( A ) 6 = 0 . The origin is asymptotically stable iff Tr ( A ) < 0 and det ( A ) > 0 . The origin is stable iff Tr ( A ) 0 and det ( A ) > 0 . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 6 / 41

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Corollary (4.2) Suppose dX/dt = AX , where A is a constant 2 × 2 matrix with det ( A ) 6 = 0 . The origin is asymptotically stable iff Tr ( A ) < 0 and det ( A ) > 0 . The origin is stable iff Tr ( A ) 0 and det ( A ) > 0 . The origin is unstable iff Tr ( A ) > 0 or det ( A ) < 0 . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 6 / 41
Now, we give specific criteria for the origin of a general linear differential system to be classified into one of four types: node, saddle, spiral, or center . Then we apply the previous results to classify the origin as stable or unstable. This classification scheme is based on the fact that the origin is the only fixed point or equilibrium solution of the linear system, det ( A ) 6 = 0. Matrix A has no zero eigenvalues. The classification scheme depends on whether the eigenvalues are real or complex, whether the real eigenvalues are positive or negative, and whether the complex eigenvalues have negative real parts. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 7 / 41

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Outline 1. Phase Plane Analysis 2. Eigenvalues. Real Eigenvalues Complex Eigenvalues Examples 3. Gershgorin’s Theorem 4. An Example: Pharmacokinetics Model 5. References J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 8 / 41
Real Eigenvalues: In the case of real eigenvalues, λ 1 and λ 2 , the corresponding eigenvector V 1 and V 2 are directions along which solutions travel toward or away from the origin. For example, if λ 1 is positive

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