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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 PROGRAMJohn AlfordSam Houston State University
Differential Equation Models • A first-order ordinary differential equation (ODE) has the general form
Differential Equation Models • A first-order ODE together with an initial condition is called an initial value problem (IVP). ODEINITIAL CONDITION
Differential Equation Models • When there is no explicit dependence on t, the equation is autonomous • Unless otherwise stated, we now assume autonomous ODE
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
Differential Equation Models • integrate
Differential Equation Models • A linear autonomous IVP has the form (*) where a and b are constants
Differential Equation Models • The solution of (*) is (You should check this) Is this the only solution?
Differential Equation Models Existence and Uniqueness Theorem for an IVP
Differential Equation Models • Example of non-uniqueness of solutions It is easy to check that this IVP has a constant solution
Differential Equation Models • Others? (separate variables) After integrating both sides
Differential Equation Models • Must satisfy initial condition • Solve for x to get another solution to the initial value problem
Differential Equation Models Which path do we choose if we start from t=0?
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)
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.
Differential Equation Models • In order to determine if an equilibrium x* is locally asymptotically stable, let to get the perturbation equation
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?
Differential Equation Models • Use the following observations and to get why?
Differential Equation Models • Thus, assuming small yields that an approximation to the perturbation equation is the equation why?
Differential Equation Models • The approximation is called the linearization of the original ODE about the equilibrium why?
Differential Equation Models • Let and assume • Two types of solutions to linearization • decaying exponential • growing exponential why?
Differential Equation Models Fixed Point Stability Theorem
Differential Equation Models • Application of stability theorem: • Fixed points:
Differential Equation Models • Differentiate f with respect to x • Substitute fixed points
Differential Equation Models • Fixed Point Stability Theorem shows • x=0 is unstable and x=K is stable • NOTICE: stability depends on the parameter r!
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
Differential Equation Models • Example: • Fixed points:
Differential Equation Models Graph f(x) vs. x
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
Differential Equation Models • Last graph
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:
Differential Equation Models • Last graph • are unstable because • are stable because
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.
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
Bifurcation Theory How Parameters Influence Fixed Points
Bifurcation Theory • Example equation • Here a is a real valued parameter • Fixed points obey
Bifurcation Theory • Fixed points depend on parameter a i) two stable ii) one unstable iii) no fixed points exist
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
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.
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