LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Some Theoretical Considerations Some Theoretical Considerations

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). ODE INITIAL CONDITION ODE INITIAL 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 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 It is easy to check that this IVP has a constant solution

Differential Equation Models  Others? (separate variables) After integrating both sides 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 to get the perturbation equation 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 where is a number between x and x* and prime on f indicates derivative with respect to x why? why?

Differential Equation Models  Use the following observations and to get why? why?

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

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

Differential Equation Models  Let and assume  Two types of solutions to linearization – decaying exponential – growing exponential why? 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 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 right particle moves left particle moves left particle is fixed particle is fixed This movement can be shown using arrows on the x-axis 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 the phase line on 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 i) two stable ii) one unstable 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

Bifurcation Theory

 Example equation  Here a is a real valued parameter  Fixed points obey

Bifurcation Theory  Define  Then

Bifurcation Theory  Fixed points obey  For different values of a, graph each function on the same grid and determine if graphs intersect. The x-values at intersection (if any) are fixed points.

Bifurcation Theory  a= 1

Bifurcation Theory  a= 0

Bifurcation Theory  a= -1

Bifurcation Theory  From graphical analysis, there appear to be three qualitatively different cases –a>0 no fixed points –a=0 one fixed point –a<0 two fixed points  A saddle-node bifurcation occurs at the bifurcation value a=0

Bifurcation Theory  Stability can be determined graphically also by plotting the phase line (direction arrows along the x-axis) using the sign of the right side of the ode

Bifurcation Theory  Arrows point right when graph 2 is above graph 1  Arrows point left when graph 2 is below graph 1

Bifurcation Theory  Stability can also be determined using (local asymptotic) stability theorem (do the calculus!)

Bifurcation Theory  First, differentiate  After a little algebra

Bifurcation Theory  If a<0, there are two fixed points –The one on the left is stable since –The one on the right is unstable since

Bifurcation Theory  OK- LURE students, what does the bifurcation diagram look like?? (see next slide) (see next slide)

Bifurcation Theory

 Use Strogatz’s Nonlinear Dynamics and Chaos to learn about the following bifurcations –A) transcritical bifurcation (pg ) (do problem on page 80) (do problem on page 80) –B) pitchfork bifurcation (pg ) (do problem on page 82) (do problem on page 82) –C) imperfection bifurcations, catastrophes (pg ) (do problem (a) and (b) only on page 86)