Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues

Real and different eigenvalues complex eigenvalues

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Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Bifurcation Example 1 Bifurcation Example: Consider the example: ˙ x 1 ˙ x 2 = α 2 - 2 0 x 1 x 2 , which contains a parameter α that affects the behavior of this system We want to determine the different qualitative behaviors for different values of α The eigenvalues satisfy det α - λ 2 - 2 - λ = λ 2 - αλ + 4 = 0 Thus, the eigenvalues satisfy λ = α ± α 2 - 16 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (49/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Bifurcation Example 2 Bifurcation Example: For ˙ x 1 ˙ x 2 = α 2 - 2 0 x 1 x 2 (3) The eigenvalues are λ = α ± α 2 - 16 2 Classifications as α varies are: For α < - 4, System (3) is a Stable Node For α = - 4, System (3) is a Stable Improper Node For - 4 < α < 0, System (3) is a Stable Spiral For α = 0, System (3) is a Center For 0 < α < 4, System (3) is a Unstable Spiral For α = 4, System (3) is a Unstable Improper Node For α > 4, System (3) is a Unstable Node Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (50/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Bifurcation Example 3 Bifurcation Example: Phase Portraits ( α < 0) Observe a smooth transition as eigenvalues change from negative to complex with negative real part α = - 5 α = - 4 α = - 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (51/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Bifurcation Example 4 Bifurcation Example: Phase Portraits ( - 4 < α < 4) Observe the transitions as complex eigenvalues change from negative real part to positive real part - This is a significant part of a Hopf bifurcation α = - 2 α = 0 α = 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (52/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Bifurcation Example 5 Bifurcation Example: Phase Portraits ( α > 0) Observe a smooth transition as eigenvalues change from complex with positive real part to positive real values α = 2 α = 4 α = 5 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (53/54)
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Introduction Solutions of Two 1 st Order Linear DEs
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