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LURE 2009 SUMMER PROGRAM LURE 2009 SUMMER PROGRAM John Alford John Alford Sam Houston State University Sam Houston State University

LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

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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. - PowerPoint PPT Presentation

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Page 1: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

LURE 2009 SUMMER PROGRAMLURE 2009 SUMMER PROGRAMJohn AlfordJohn Alford

Sam Houston State UniversitySam Houston State University

Page 2: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Some Theoretical ConsiderationsSome Theoretical Considerations

Page 3: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

A first-order A first-order ordinary differential ordinary differential equationequation (ODE) has the general (ODE) has the general formform

),( xtfdtdx

Page 4: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

A first-order ODE together with an A first-order ODE together with an initial conditioninitial condition is is called an initial called an initial value problem (IVP)value problem (IVP)..

ODEODE INITIAL CONDITIONINITIAL CONDITION

00 )( ),(/ xtxxtfdtdx

Page 5: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

When there is no explicit When there is no explicit dependence on dependence on tt, the equation is , the equation is autonomousautonomous

Unless otherwise stated, we now Unless otherwise stated, we now assume autonomous ODEassume autonomous ODE

)(xfdtdx

Page 6: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

We may be able to We may be able to solvesolve an an autonomous ode by autonomous ode by separating separating variables (see chapter 9.1 and 9.2 variables (see chapter 9.1 and 9.2 in Thomas’ calculus textbook!)in Thomas’ calculus textbook!)

– separateseparate

dtdxxf

xfdtdx 1

)(1)(

Page 7: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

– integrateintegrate

dtdxxf

1)(

1

Page 8: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

A A linearlinear autonomous IVP has the autonomous IVP has the formform

(*)(*)

where where aa and and bb are constants are constants

0)0( , xxbaxdtdx

Page 9: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

The solution of (*) isThe solution of (*) is

(You should check this)(You should check this)

Is this the only solution?Is this the only solution?

abebaxtx

at

)()( 0

Page 10: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation ModelsExistence and Uniqueness Theorem for an IVPExistence and Uniqueness Theorem for an IVP

Page 11: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models Example of Example of non-uniquenessnon-uniqueness of solutions of solutions

It is easy to check that this IVP has a It is easy to check that this IVP has a constantconstant solution solution

0)0( ,3/1 xxdtdx

ttx allfor 0)(

Page 12: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models Others? (separate variables)Others? (separate variables)

After integrating both sidesAfter integrating both sides

dtdxxxdtdx 1 3/13/1

23 3/2 Ctx

Page 13: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models Must satisfy initial conditionMust satisfy initial condition

Solve for Solve for xx to get to get anotheranother solution solution to the initial value problemto the initial value problem

23 0 0)0( 3/2 txCx

32)(

3/2

ttx

Page 14: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Which path do we choose if we start from Which path do we choose if we start from t=0t=0??

Page 15: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Existence and uniqueness theorem Existence and uniqueness theorem does not tell us does not tell us howhow to find a to find a solution (just that there is one and solution (just that there is one and only one solution)only one solution)

We could spend all summer talking We could spend all summer talking about how to solve ODE IVPs (but about how to solve ODE IVPs (but we won’t)we won’t)

Page 16: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Page 17: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Page 18: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Page 19: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

We might sayWe might say– A fixed point is A fixed point is locally stablelocally stable if starting close if starting close

(enough) guarantees that you stay close.(enough) guarantees that you stay close.

– A fixed point is A fixed point is locally asymptotically stablelocally asymptotically stable if if all sufficiently small perturbations produce all sufficiently small perturbations produce small excursions that eventually return to the small excursions that eventually return to the equilibrium.equilibrium.

Page 20: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

In order to determine if an equilibrium In order to determine if an equilibrium x*x* is locally asymptotically stable, letis locally asymptotically stable, let

to getto get

the the perturbation equationperturbation equation

*)()( xtxt

)( )( xfdtdxf

dtdx

Page 21: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Use Use Taylor’s formulaTaylor’s formula (Cal II) to expand (Cal II) to expand f(x)f(x) about the equilibrium (assume about the equilibrium (assume f f has has at least two continuous derivatives with at least two continuous derivatives with respect to respect to xx in an interval containing in an interval containing xx*)*)

where is a number between where is a number between xx and and xx* * and prime on and prime on ff indicates derivative with indicates derivative with respect to respect to xx

why?why?

2/ )('' )(' )()(2 **** xxfxxxfxfxf

Page 22: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Use the following observationsUse the following observations

andand

to getto get

why?why?

0)( * xf

0 small 2* *xxxx

)(' )(' )( *** xfxxxfxf

Page 23: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Thus, assuming smallThus, assuming small

yields that an approximation to the yields that an approximation to the perturbation equationperturbation equation

is the equationis the equation

why?why?

*xx

)(' * xfdtd

)( xfdtd

Page 24: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

The approximationThe approximation

is called the is called the linearizationlinearization of the original of the original

ODE about the equilibriumODE about the equilibrium

why?why?

)(' * xfdtd

Page 25: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Let and assumeLet and assume

TwoTwo types of solutions to linearization types of solutions to linearization

– decayingdecaying exponential exponential

– growinggrowing exponential exponential

why?why?

tedtd

0

)(' *xf 0

0

0

Page 26: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Fixed Point Stability TheoremFixed Point Stability Theorem

Page 27: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Application of stability theorem:Application of stability theorem:

Fixed points:Fixed points:

0 ,0 ,1

Kr

Kxrx

dtdx

KxxKxrxxf

,0 01)(

Page 28: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Differentiate Differentiate ff with respect to with respect to xx

Substitute fixed pointsSubstitute fixed points

Kxrxf 21)('

0)(' and 0)0(' rKfrf

Page 29: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Fixed Point Stability Theorem showsFixed Point Stability Theorem shows

– x=0x=0 is unstable and is unstable and x=Kx=K is stable is stable

NOTICE: stability depends on the NOTICE: stability depends on the parameter parameter rr!!

Page 30: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

A A GeometricalGeometrical (Graphical) Approach to (Graphical) Approach to Stability of Fixed PointsStability of Fixed Points

– Consider an Consider an autonomousautonomous first order ODE first order ODE

– The The zeroszeros of the graph for of the graph for are the fixed points are the fixed points

)(xfdtdx

xxf vs.)(

Page 31: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Example:Example:

Fixed points:Fixed points:

)1( xxdtdx

1 and 0 *2

*1 xx

Page 32: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Graph f(x) vs. x

Page 33: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

)(xfdtdx

Page 34: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Imagine a particle which moves along Imagine a particle which moves along the the xx-axis (one-dimension) according to-axis (one-dimension) according to

particle moves right particle moves right particle moves leftparticle moves left particle is fixedparticle is fixed This movement can be shown using arrows This movement can be shown using arrows

on the x-axison the x-axis

)(/ xfdtdx

0)( xf 0)( xf 0)( xf

Page 35: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Last graphLast graph

right) (arrows 0)( and *

4*3

*2

*1

xfxxxxxx

left) (arrows 0)( and , , *

4*3

*2

*1

xfxxxxxxx

Page 36: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Page 37: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Theorem for local asymptotic stability of Theorem for local asymptotic stability of a fixed point used the sign of the a fixed point used the sign of the derivative of derivative of f(x)f(x) evaluated at a fixed evaluated at a fixed point:point:

at stability asymptotic local 0)(' ** xxf

at y instabilit 0)(' ** xxf

Page 38: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Last graphLast graph

– are are unstableunstable because because

– are are stablestable because because

0)(' and 0)(' *3

*1 xfxf

0)(' and 0)(' *4

*2 xfxf

, *3

*1 xx

, *4

*2 xx

Page 39: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Fixed points that are locally Fixed points that are locally asymptotically stable are denoted with a asymptotically stable are denoted with a solid dot on the solid dot on the xx-axis-axis

Fixed points that are unstable are Fixed points that are unstable are denoted with an open dot on the denoted with an open dot on the xx-axis.-axis.

Page 40: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Page 41: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Differential Equation ModelsDifferential Equation Models

Putting the arrows on the Putting the arrows on the xx-axis along -axis along with the open circles or closed dots at with the open circles or closed dots at the fixed points is called plotting thethe fixed points is called plotting the phase linephase line on theon the x-axis x-axis

Page 42: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

How Parameters Influence Fixed PointsHow Parameters Influence Fixed Points

Page 43: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Example equationExample equation

Here Here aa is a real valued parameter is a real valued parameter Fixed points obeyFixed points obey

2xadtdx

axxa 22 0

Page 44: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

0a

Page 45: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

0a

Page 46: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

0a

Page 47: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Fixed points depend on parameter Fixed points depend on parameter aa i) two stablei) two stable

ii) one unstableii) one unstable

iii) no fixed points existiii) no fixed points exist 0a

0 0 * xa

and 0 *2

*1 axaxa

Page 48: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

The parameter values at which The parameter values at which qualitative changes in the dynamics qualitative changes in the dynamics occur are called occur are called bifurcation pointsbifurcation points..

Some Some possiblepossible qualitative changes qualitative changes in dynamicsin dynamics

– The number of fixed points changeThe number of fixed points change– The stability of fixed points changeThe stability of fixed points change

Page 49: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

In the previous example, there was In the previous example, there was a bifurcation point at a bifurcation point at a=0a=0..

– For a>0 there were two fixed pointsFor a>0 there were two fixed points– For a<0 there were no fixed pointsFor a<0 there were no fixed points

When the number of fixed points When the number of fixed points changes at a parameter value, we changes at a parameter value, we say that a say that a saddle-node bifurcationsaddle-node bifurcation has occurred.has occurred.

Page 50: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Bifurcation DiagramBifurcation Diagram– fixed points on the vertical axis and fixed points on the vertical axis and

parameter on the horizontal axisparameter on the horizontal axis– sections of the graph that depict sections of the graph that depict

unstable fixed points are plotted unstable fixed points are plotted dasheddashed; sections of the graph that ; sections of the graph that depict stable fixed points are depict stable fixed points are solidsolid

– the following slide shows a bifurcation the following slide shows a bifurcation diagram for the previous examplediagram for the previous example

Page 51: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Page 52: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Example equationExample equation

Here Here aa is a real valued parameter is a real valued parameter Fixed points obeyFixed points obey

)1ln( xxadtdx

? 0)1ln( xxxa

Page 53: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

DefineDefine

ThenThen

xxxfaxf )1ln()( and )( 12

)()( 12 xfxfdtdx

Page 54: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Fixed points obeyFixed points obey

For different values of For different values of aa, graph each , graph each function on the same grid and function on the same grid and determine if graphs intersect. The determine if graphs intersect. The xx-values at intersection (if any) are -values at intersection (if any) are fixed points.fixed points.

)()( 0)()( 1212 xfxfxfxf

Page 55: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

aa= 1= 1

Page 56: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

aa= 0= 0

Page 57: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

aa= -1= -1

Page 58: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

From graphical analysis, there From graphical analysis, there appear to be three appear to be three qualitativelyqualitatively different casesdifferent cases

– a>0 no fixed pointsa>0 no fixed points– a=0 one fixed pointa=0 one fixed point– a<0 two fixed pointsa<0 two fixed points

A A saddle-nodesaddle-node bifurcation occurs at bifurcation occurs at the bifurcation value the bifurcation value aa=0=0

Page 59: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

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

)()( 12 xfxfdtdx

Page 60: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Arrows point Arrows point rightright when graph 2 is when graph 2 is aboveabove graph 1 graph 1

Arrows point Arrows point left left when graph 2 is when graph 2 is belowbelow graph 1 graph 1

)()( 12 xfxf

)()( 12 xfxf

Page 61: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

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

Page 62: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

First, differentiateFirst, differentiate

After a little algebraAfter a little algebra

x

xxadxdxf

111)1ln()('

xxxf

1

)('

Page 63: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

If If aa<0, there are <0, there are twotwo fixed points fixed points– The one on the left is stable sinceThe one on the left is stable since

– The one on the right is unstable sinceThe one on the right is unstable since

0)(' 01 *1

*1 xfx

0)(' 0 *2

*2 xfx

Page 64: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

OK- LURE students, what does the OK- LURE students, what does the bifurcation diagram look like??bifurcation diagram look like??

(see next slide)(see next slide)

Page 65: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Page 66: LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University

Bifurcation TheoryBifurcation Theory

Use Strogatz’s Nonlinear Dynamics and Chaos Use Strogatz’s Nonlinear Dynamics and Chaos to learn about the following bifurcationsto learn about the following bifurcations

– A) transcritical bifurcation (pg. 50-52)A) transcritical bifurcation (pg. 50-52) (do problem 3.2.4 on page 80)(do problem 3.2.4 on page 80)– B) pitchfork bifurcation (pg. 55-60)B) pitchfork bifurcation (pg. 55-60) (do problem 3.4.3 on page 82)(do problem 3.4.3 on page 82)– C) imperfection bifurcations, C) imperfection bifurcations,

catastrophes (pg. 69-73) catastrophes (pg. 69-73) (do problem 3.6.2 (a) and (b) only on (do problem 3.6.2 (a) and (b) only on

page 86)page 86)