The origin is classified as a spiral or focus if a 6

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solutions are closed curves, encircling the origin. The origin is classified as a spiral (or focus) if a 6 = 0 and a center if a = 0. 3. Spiral or Focus: Eigenvalues have nonzero real part ( a 6 = 0). 4. Center: Eigenvalues are purely imaginary ( a = 0), λ 1 , 2 = ± ib . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 12 / 41
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Examples: Example 4.11 Let A = 0 - 1 1 0 . The eigenvalues of A are ± i , so that the origin is a center. The solution to dX/dt = AX can be found by noting dx/dt = - y and dy/dt = x so that dy dx = dy/dt dx/dt = - x y . Separating variables and integrating y 2 2 + x 2 2 = c. This latter equation is a circle centered at the origin. Solutions travel in a counterclockwise direction on circles surrounding the origin. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 13 / 41
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Example 4.12 Let A = 0 1 1 0 . The eigenvalues of A are ± 1, so that the origin is a saddle. The solution to dX/dt = AX can be found by noting dy/dx = x/y . Separating variables and integrating y 2 2 - x 2 2 = c. This latter equation is a hyperbola centered at the origin. Figure 2: The origin is a saddle. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 14 / 41
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Example 4.13 Let A = 1 0 0 3 . The eigenvalues of A are 1 and 3. The origin is a unstable node. The solution to dX/dt = AX can be found by integrating dx dt = x and dy dy = 3 y so that x ( t ) = x 0 e t and y ( t ) = y 0 e 3 t . Solutions in the phase plane have the form X ( t ) = x 0 e t 1 0 + y 0 e 3 t 0 1 . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 15 / 41
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The classification schemes can be related to the signs of the trace and determinant of A and the discriminant of the characteristic polynomial. Let τ = Tr ( A ) and δ = det ( A ) . Recall that the eigenvalues, the roots of the characteristic polynomial λ 2 - Tr ( A ) λ + det ( A ) = λ 2 - τλ + δ satisfy λ 1 , 2 = τ ± τ 2 - 4 δ 2 . The discriminant is denoted by γ and defined as follows: γ = τ 2 - 4 δ. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 16 / 41
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The following classification scheme summarizes the dynamics according to the sign of the discriminant, γ , that is, according to whether the eigenvalues are real or complex conjugates. Improper and proper nodes are not distinguished. Eigenvalues are real ( γ > 0): Unstable node if τ > 0 and δ > 0 ( λ 1 , 2 > 0) Saddle point if δ < 0 ( λ 1 < 0 < λ 2 ) Stable node if τ < 0 and δ > 0 ( λ 1 , 2 < 0) J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 17 / 41
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Eigenvalues are complex conjugates a ± bi ( γ < 0): Unstable spiral if τ > 0 ( a > 0) Neutral center if τ = 0 ( a = 0) Stable spiral or stable focus if τ < 0 ( a < 0) The classification scheme is illustrated in the τ - δ plane in Figure 3. Note that asymptotic stability requires τ < 0 and δ > 0 (the trace is negative and the determinant is positive, Corollary 4.2 (1)). J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 18 / 41
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τ saddle S node S spiral U spiral U node center τ 2 = 4 δ δ Figure 3: Stability diagram in the τ - δ plane. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 19 / 41
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Example 4.14 Determine the conditions on a so that the zero equilibrium of the following system is a stable spiral: dx dt = y and dy dy = - x + ay.
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