View
40
Download
3
Category
Preview:
DESCRIPTION
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
Citation preview
LURE 2009 SUMMER PROGRAMLURE 2009 SUMMER PROGRAMJohn AlfordJohn Alford
Sam Houston State UniversitySam Houston State University
Some Theoretical ConsiderationsSome Theoretical Considerations
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
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
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
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)(
Differential Equation ModelsDifferential Equation Models
– integrateintegrate
dtdxxf
1)(
1
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
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
Differential Equation ModelsDifferential Equation ModelsExistence and Uniqueness Theorem for an IVPExistence and Uniqueness Theorem for an IVP
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)(
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
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
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??
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)
Differential Equation ModelsDifferential Equation Models
Differential Equation ModelsDifferential Equation Models
Differential Equation ModelsDifferential Equation Models
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.
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
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
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
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
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
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
Differential Equation ModelsDifferential Equation Models
Fixed Point Stability TheoremFixed Point Stability Theorem
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)(
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
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!!
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.)(
Differential Equation ModelsDifferential Equation Models
Example:Example:
Fixed points:Fixed points:
)1( xxdtdx
1 and 0 *2
*1 xx
Differential Equation ModelsDifferential Equation Models
Graph f(x) vs. x
Differential Equation ModelsDifferential Equation Models
)(xfdtdx
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
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
Differential Equation ModelsDifferential Equation Models
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
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
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.
Differential Equation ModelsDifferential Equation Models
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
Bifurcation TheoryBifurcation Theory
How Parameters Influence Fixed PointsHow Parameters Influence Fixed Points
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
Bifurcation TheoryBifurcation Theory
0a
Bifurcation TheoryBifurcation Theory
0a
Bifurcation TheoryBifurcation Theory
0a
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
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
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.
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
Bifurcation TheoryBifurcation Theory
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
Bifurcation TheoryBifurcation Theory
DefineDefine
ThenThen
xxxfaxf )1ln()( and )( 12
)()( 12 xfxfdtdx
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
Bifurcation TheoryBifurcation Theory
aa= 1= 1
Bifurcation TheoryBifurcation Theory
aa= 0= 0
Bifurcation TheoryBifurcation Theory
aa= -1= -1
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
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
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
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!)
Bifurcation TheoryBifurcation Theory
First, differentiateFirst, differentiate
After a little algebraAfter a little algebra
x
xxadxdxf
111)1ln()('
xxxf
1
)('
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
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)
Bifurcation TheoryBifurcation Theory
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)
Recommended