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
dx
 f (t , x)
dt
Differential Equation Models

A first-order ODE together with an
initial condition is called an initial
value problem (IVP).
ODE
dx / dt  f (t , x)
INITIAL CONDITION
x(t0 )  x0
Differential Equation Models

When there is no explicit
dependence on t, the equation is
autonomous
dx
 f (x )
dt

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
dx
1
 f ( x) 
dx  1dt
dt
f ( x)
Differential Equation Models
– integrate

1
dx   1dt
f ( x)
Differential Equation Models

A linear autonomous IVP has the
form
dx
 ax  b,
dt
x(0)  x0
(*)
where a and b are constants
Differential Equation Models

The solution of (*) is
(ax0  b)e  b
x(t ) 
a
at
(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
dx
1/ 3
 x , x(0)  0
dt
It is easy to check that this IVP has a
constant solution
x(t )  0 for all t
Differential Equation Models

Others? (separate variables)
dx
1/ 3
x
dt

x
1 / 3
dx  1 dt
After integrating both sides
3 2/3
x t C
2
Differential Equation Models

Must satisfy initial condition
3 2/3
x t
2
x(0)  0  C  0 

Solve for x to get another solution
to the initial value problem
2 
x(t )   t 
3 
3/2
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
Differential Equation Models
Differential Equation Models
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
 (t )  x(t )  x
*
to get
dx
d
 f ( x) 
 f ( x)
dt
dt
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*)



f ( x)  f ( x )  f ' ( x ) x  x  f ' ' ( ) x  x
*
*
*
 /2
* 2
where  is a number between x and x*
and prime on f indicates derivative with
respect to x
Differential Equation Models

Use the following observations
f (x )  0
*
and
x  x small

*
to get

x  x   0
* 2

f ( x)  f ' ( x ) x  x  f ' ( x ) 
*
*
*
Differential Equation Models

Thus, assuming small
xx
*
yields that an approximation to the
perturbation equation
d
 f ( x) 
dt
is the equation
d
*
 f '(x )
dt
Differential Equation Models

The approximation
d
*
 f '(x )
dt
is called the linearization of the original
ODE about the equilibrium
Differential Equation Models

Let
d
 
dt

0
f ' ( x )   and assume
*

  0e
t
Two types of solutions to linearization
–
0
decaying exponential
–
 0
growing exponential
Differential Equation Models
Fixed Point Stability Theorem
Differential Equation Models

Application of stability theorem:
dx
x

 rx1  ,
dt
 K

r  0, K  0
Fixed points:
x

f ( x)  rx1    0  x  0, x  K
 K
Differential Equation Models

Differentiate f with respect to x
 2x 
f ' ( x)  r 1  
K


Substitute fixed points
f ' (0)  r  0 and f ' ( K )  r  0
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
dx
 f (x )
dt
–
The zeros of the graph for
are the fixed points
f ( x) vs. x
Differential Equation Models

Example:
dx
 x(1  x)
dt

Fixed points:
x 0
*
1
and
x 1
*
2
Differential Equation Models
Graph f(x) vs. x
Differential Equation Models
dx
 f (x )
dt
Differential Equation Models

Imagine a particle which moves along
the x-axis (one-dimension) according to
dx / dt  f ( x)
f ( x)  0
f ( x)  0
f ( x)  0
 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
x xx
*
1
*
2
and x  x  x
*
3
*
4
 f ( x)  0 (arrowsright)
x  x , x  x  x , and x  x
*
1
*
2
*
3
*
4
 f ( x)  0 (arrowsleft)
Differential Equation Models
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:
f ' ( x )  0  localasymptoticstabilityat x
*
*
f ' ( x )  0  instability at x
*
*
Differential Equation Models

Last graph
–
*
1
*
3
x , x
are unstable because
f ' ( x )  0 and f ' ( x )  0
*
1
–
*
2
*
4
x , x
*
3
are stable because
f ' ( x2* )  0 and f ' ( x4* )  0
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
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
dx
2
ax
dt


Here a is a real valued parameter
Fixed points obey
ax 0  x a
2
2
Bifurcation Theory
a0
Bifurcation Theory
a0
Bifurcation Theory
a0
Bifurcation Theory

Fixed points depend on parameter a
i) two stable
a  0  x  a and x   a
*
1
ii) one unstable
*
2
a0  x 0
iii) no fixed points exist
*
a0
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
Bifurcation Theory

Example equation
dx
 a  x  ln(1  x)
dt


Here a is a real valued parameter
Fixed points obey
a  x  ln(1  x)  0  x  ?
Bifurcation Theory

Define
f 2 ( x)  a and f1 ( x)  ln(1  x)  x

Then
dx
 f 2 ( x)  f1 ( x)
dt
Bifurcation Theory

Fixed points obey
f 2 ( x)  f1 ( x)  0


f 2 ( x)  f1 ( x)
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
dx
 f 2 ( x)  f1 ( x)
dt
Bifurcation Theory

Arrows point right when graph 2 is
above graph 1
f 2 ( x)  f1 ( x)

Arrows point left when graph 2 is
below graph 1
f 2 ( x)  f1 ( x)
Bifurcation Theory

Stability can also be determined
using (local asymptotic) stability
theorem (do the calculus!)
Bifurcation Theory

First, differentiate
d
1
f ' ( x)  a  x  ln(1  x)  1 
dx
1 x

After a little algebra
x
f ' ( x) 
1 x
Bifurcation Theory

If a<0, there are two fixed points
– The one on the left is stable since
1  x  0
*
1
 f ' (x )  0
*
1
– The one on the right is unstable since
x 0
*
2
 f ' (x )  0
*
2
Bifurcation Theory

OK- LURE students, what does the
bifurcation diagram look like??
(see next slide)
Bifurcation Theory
Bifurcation Theory

Use Strogatz’s Nonlinear Dynamics and Chaos
to learn about the following bifurcations
–
–
–
A) transcritical bifurcation (pg. 50-52)
(do problem 3.2.4 on page 80)
B) pitchfork bifurcation (pg. 55-60)
(do problem 3.4.3 on page 82)
C) imperfection bifurcations,
catastrophes (pg. 69-73)
(do problem 3.6.2 (a) and (b) only on
page 86)