Phase Diagrams See http://mathworld.wolfram.com/PhasePortrait.html for more details

Stability A Dynamic System is said to be stable if the system approaches zero as t→∞. What conditions are necessary for a system to be stable? Hint: what kinds of values do we need for λ? Recall:

Stability A Dynamic System is said to be stable if the system approaches zero as t→∞. What conditions are necessary for a system to be stable? All eigenvalues must be negative. For differential equations If values are complex then the real portion must be negative

Steady State A Dynamic system is said to be Unstable if the system “blows up” as t→∞. What conditions are needed for unstability? A Dynamic System is said to represent a steady state if the system neither approach zero nor infinity as t →∞. What conditions are necessary for a steady state?

Stability What conditions are needed for unstability? One or more positive eigenvalues A Dynamic System is said to represent a steady state if the system approaches a constant value (other than zero) at t →∞. eigenvalues all negative (real portion is negative in the case of complex numbers)

Difference equations vs. Differential equations Stable if all │λ│<1 Steady state if one or more λ=1 and all other │λ│<1 Unstable or “blows up” if one or more │λ│>1 Differential equations Stable if all λ <0 Steady state if one or more λ = 0 and all other λ <0 Unstable or “blows up” if one or more λ > 0

Example 1

Example 1 Plug in some points

Example 1 Plot the results as vectors

Example 1 Follow some of the paths from different starting points (different initial conditions) Notice that the path varies greatly depending upon the initial conditions Usually just the paths are drawn without the vector field One negative and one positive eigenvalue results in a saddle point

What do the eigenvalues tell you about a system? What if the eigenvalues are both negative? What if the eigenvalues are both positive?

Stability from a phase portrait Positive distinct eigenvalues with eigenvector as the asymptote Negative distinct eigenvalues with eigenvector as the asymptote

From your knowledge of complex numbers From your knowledge of complex numbers. What do you think the phase portrait will look like if the eigenvalues are complex? Hint consider Complex Eigenvalues with Re > 0 Complex Eigenvalues with Re < 0

Example 2 Complex Eigenvalues with Re < 0 Complex Eigenvalues

Complex Eigvenvalues with Re > 0 Complex Eigvenvalues with Re < 0

What do you think the phase portrait will look like when the eigenvalues are purely imaginary

Predict the eigenvalues for this system Eigenvalues are purely complex This is because you end up with eix in your answer e ix = cosx + isinx acosx + bsinx forms and ellipse Center is stable

Describe the phase portrait [ ] -1 0 x’t = 0 -3 x(t)

Describe the phase portrait [ ] -1 0 x’t = 0 -3 x(t) By inspection the eigen values are -1 and -3 Therefore the system is asymptotically stable the phase portrait will consist of curves that run towards the origin of the form at the right The eigenvectors are <1,0> and <0,1> therefore the aysmptotes will be on the x and y axis

Homework: worksheet 8.5 1-9 all

What do you think the phase portrait will look like when there is one positive (a repeated eigenvalue) eigenvalue? What do you think the phase portrait will look like when there is one negative eigenvalue?

Notice that the direction of the arrows affects stability Single positive eigenvalue The eigenvector is the asymptote Single negative eigenvalue Eigenvector is the asymptote

Learn more at: http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx