Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues

# Real and different eigenvalues complex eigenvalues

• 54

This preview shows page 49 - 54 out of 54 pages.

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)

Subscribe to view the full document.

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)
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)

Subscribe to view the full document.

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)
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)

Subscribe to view the full document.

Introduction Solutions of Two 1 st Order Linear DEs
• Fall '08
• staff

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes