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LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University. Some Theoretical Considerations. Differential Equation Models. A first-order ordinary differential equation (ODE) has the general form. Differential Equation Models.

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LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

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  1. LURE 2009 SUMMER PROGRAMJohn AlfordSam Houston State University

  2. Some Theoretical Considerations

  3. Differential Equation Models • A first-order ordinary differential equation (ODE) has the general form

  4. Differential Equation Models • A first-order ODE together with an initial condition is called an initial value problem (IVP). ODEINITIAL CONDITION

  5. Differential Equation Models • When there is no explicit dependence on t, the equation is autonomous • Unless otherwise stated, we now assume autonomous ODE

  6. Differential Equation Models • We may be able to solve an autonomous ode by separating variables (see chapter 9.1 and 9.2 in Thomas’ calculus textbook!) • separate

  7. Differential Equation Models • integrate

  8. Differential Equation Models • A linear autonomous IVP has the form (*) where a and b are constants

  9. Differential Equation Models • The solution of (*) is (You should check this) Is this the only solution?

  10. Differential Equation Models Existence and Uniqueness Theorem for an IVP

  11. Differential Equation Models • Example of non-uniqueness of solutions It is easy to check that this IVP has a constant solution

  12. Differential Equation Models • Others? (separate variables) After integrating both sides

  13. Differential Equation Models • Must satisfy initial condition • Solve for x to get another solution to the initial value problem

  14. Differential Equation Models Which path do we choose if we start from t=0?

  15. Differential Equation Models • Existence and uniqueness theorem does not tell us how to find a solution (just that there is one and only one solution) • We could spend all summer talking about how to solve ODE IVPs (but we won’t)

  16. Differential Equation Models

  17. Differential Equation Models

  18. Differential Equation Models

  19. Differential Equation Models • We might say • A fixed point is locally stable if starting close (enough) guarantees that you stay close. • A fixed point is locally asymptotically stable if all sufficiently small perturbations produce small excursions that eventually return to the equilibrium.

  20. Differential Equation Models • In order to determine if an equilibrium x* is locally asymptotically stable, let to get the perturbation equation

  21. Differential Equation Models • Use Taylor’s formula (Cal II) to expand f(x) about the equilibrium (assume f has at least two continuous derivatives with respect to x in an interval containing x*) where is a number between x and x* and prime on f indicates derivative with respect to x why?

  22. Differential Equation Models • Use the following observations and to get why?

  23. Differential Equation Models • Thus, assuming small yields that an approximation to the perturbation equation is the equation why?

  24. Differential Equation Models • The approximation is called the linearization of the original ODE about the equilibrium why?

  25. Differential Equation Models • Let and assume • Two types of solutions to linearization • decaying exponential • growing exponential why?

  26. Differential Equation Models Fixed Point Stability Theorem

  27. Differential Equation Models • Application of stability theorem: • Fixed points:

  28. Differential Equation Models • Differentiate f with respect to x • Substitute fixed points

  29. Differential Equation Models • Fixed Point Stability Theorem shows • x=0 is unstable and x=K is stable • NOTICE: stability depends on the parameter r!

  30. Differential Equation Models • A Geometrical (Graphical) Approach to Stability of Fixed Points • Consider an autonomous first order ODE • The zeros of the graph for • are the fixed points

  31. Differential Equation Models • Example: • Fixed points:

  32. Differential Equation Models Graph f(x) vs. x

  33. Differential Equation Models

  34. Differential Equation Models • Imagine a particle which moves along the x-axis (one-dimension) according to particle moves right particle moves left particle is fixed This movement can be shown using arrows on the x-axis

  35. Differential Equation Models • Last graph

  36. Differential Equation Models

  37. Differential Equation Models • Theorem for local asymptotic stability of a fixed point used the sign of the derivative of f(x) evaluated at a fixed point:

  38. Differential Equation Models • Last graph • are unstable because • are stable because

  39. Differential Equation Models • Fixed points that are locally asymptotically stable are denoted with a solid dot on the x-axis • Fixed points that are unstable are denoted with an open dot on the x-axis.

  40. Differential Equation Models

  41. Differential Equation Models • Putting the arrows on the x-axis along with the open circles or closed dots at the fixed points is called plotting thephase lineon the x-axis

  42. Bifurcation Theory How Parameters Influence Fixed Points

  43. Bifurcation Theory • Example equation • Here a is a real valued parameter • Fixed points obey

  44. Bifurcation Theory

  45. Bifurcation Theory

  46. Bifurcation Theory

  47. Bifurcation Theory • Fixed points depend on parameter a i) two stable ii) one unstable iii) no fixed points exist

  48. Bifurcation Theory • The parameter values at which qualitative changes in the dynamics occur are called bifurcation points. • Some possible qualitative changes in dynamics • The number of fixed points change • The stability of fixed points change

  49. Bifurcation Theory • In the previous example, there was a bifurcation point at a=0. • For a>0 there were two fixed points • For a<0 there were no fixed points • When the number of fixed points changes at a parameter value, we say that a saddle-node bifurcation has occurred.

  50. Bifurcation Theory • Bifurcation Diagram • fixed points on the vertical axis and parameter on the horizontal axis • sections of the graph that depict unstable fixed points are plotted dashed; sections of the graph that depict stable fixed points are solid • the following slide shows a bifurcation diagram for the previous example

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