4330_ds_2

8/2/2019 4330_ds_2

1/7

Dynamical Systems: Part 2

2 Bifurcation Theory

In practical applications that involve differential equations it very often happens that the

differential equation contains parameters and the value of these parameters are often only

known approximately. In particular they are generally determined by measurements which

are not exact. For that reason it is important to study the behavior of solutions and examine

their dependence on the parameters. This study leads to the area referred to as bifurcation

theory. It can happen that a slight variation in a parameter can have significant impact on

the solution. Bifurcation theory is a very deep and complicated area involving lots of current

research. A complete examination of of the field would be impossible.

A fixed point (or equilibrium point) of a differential equation y = f(y) is a root of the

equation f(y) = 0. As we have already seenfor autonomous problems fixed points can be

very useful in determining the long time behavior of solutions.

Qualitative information about the equilibrium points of the differential equation y = f(y)

can be obtained from special diagrams called phase diagrams.

A phase line diagram for the autonomous equation y = f(y) is a line segment with labels

for so-called sinks, sources or nodes, one for each root of f(y) = 0, i.e. each equilibrium.

sinksource

y0 y1

The names are borrowed from the theory of fluids and they are defined as follows:

1. Sink An equilibrium y0 which attracts nearby solutions at t = , i.e., there exists

M > 0 so that if |y(0) y0| < M, then |y(x) y0|t

02. Source An equilibrium y1 which repels nearby solutions at t = , i.e., here exists

M > 0 so that if |y(0) y1| < M, then |y(x) y1| increases as t .

3. Node An equilibrium y2 which is neither a sink or a source. In fluids, sink means fluid

is lost and source means fluid is created.

1

8/2/2019 4330_ds_2

2/7

Stability Test: The term stable means that solutions that start near the equilibrium will

stay nearby as t . The term unstable means not stable. Therefore, a sink is stable anda source is unstable. Precisely, an equilibrium y0 is stable provided for given > 0 there

exists some > 0 such that|y(0)

y0|

< implies y(t) exists for t

0 and|y(t)?y0

|< .

Theorem 2.1 (Stability Conditions). Let f and f be continuous. The equation y = f(y)

has a sink at y = y0 provided f(y0) = 0 and f(y0) < 0. An equilibrium y = y1 is a source

provided f(y1) = 0 and f(y1) > 0. There is no test when f

is zero at an equilibrium.

Our objective in this section (for first order equations) is to briefly examine the three

simplest types of bifurcations: 1) Saddle Node; 2) Transcritical; 3) Pitchfork .

2.1 Saddle Bode Bifurcation

We begin with the Saddle Node bifurcation (also called the blue sky bifurcation) corre-

sponding to the creation and destruction of fixed points. The normal form for this type of

bifurcation is given by the example

x = r + x2 (1)

The three cases of r < 0, r = 0 and r > 0 give very different structure for the solutions.

r < 0 r = 0 r > 0

We observe that there is a bifurcation at r = 0. For r < 0 there are two fixed points

given by x = r. The equilibrium x = r is stable, i.e., solutions beginning near thisequilibrium converge to it as time increases. Further, initial conditions near

r divergefrom it.

2

8/2/2019 4330_ds_2

3/7

At r = 0 there is a single fixed point at x = 0 and initial conditions less than zero give

solutions that converge to zero while positive initial conditions give solutions that increase

without bound.

Finally ifr > 0 there are no fixed points at all. For any initial condition solutions increase

without bound.

There are several ways we depict this type of bifurcation one of which is the so called

bifurcation diagram.

r

x

Note that if instead we consider x = r

x2 the the so-called phase line can be drawn as

r < 0 r = 0 r > 0

Exercise: Analyze the bifurcation properties of the following following problems.

1. x = 1 + rx + x2

2.x

=r cosh(

x)

3. x = r + x ln(1 + x)

3

8/2/2019 4330_ds_2

4/7

2.2 Transcritical Bifurcation

Next we consider the transcritical bifurcation corresponding to the exchange of stability of

fixed points. The normal form for this type of bifurcation is given by the example

x = rx x2 (2)

In this case there is either one (r = 0) or two (r = 0) fixed points. When r = 0 the onlyfixed point is x = 0 which is semi-stable (i.e., stable from the right and unstable from the

left). For r = 0 there are two fixed points given by x = 0 and x = r. So we note in this casex = 0 is a fixed point for all r. For r < 0 the nonzero fixed point is unstable but for r > 0

the nonzero fixed point becomes stable. Thus we say that the stability of this fixed point

has switched from unstable to stable.

r < 0 r = 0 r > 0

Bifurcation diagram for a transcritical bifurcation.

r

x


1. x = rx + x2

2. x = rx ln(1 + x)

4

8/2/2019 4330_ds_2

5/7

3. x = x rx(1 x)

2.3 Pitchfork Bifurcation

Finally we consider the pitchfork bifurcation. The normal form for this type of bifurcation

is given by the example

x = rx x3 (3)

The cases of r 0 and r > 0, once again, give very different structure for the solutions.

r < 0 r = 0 r > 0

r

x

Super Critical Pitchfork Bifurcation Diagram

Now consider the example

x = rx + x3. (4)

For this example we obtain the so-called sub-critical pitchfork bifurcation. Notice that solu-

tions blow-up in finite time, i.e., satisfy x(t) as t a < .

5

8/2/2019 4330_ds_2

6/7

r

x

Sub Critical Pitchfork Bifurcation Diagram


1. x = x + tanh(x)

2. x = rx 4x3

3. x = rx sin(x)

4. x = rx + 4x3

5. x = rx sinh(x)

6. x = rx 4x3

7. x = x +rx

1 + x2

2.4 Hysteresis: a more complicated bifurcation

In this subsection we consider an even more complicated example which contains pitchforkand

saddle node bifurcations. Consider the example

x = rx + x3 x5. (5)

1. For small initial conditions the bifurcation diagram looks just like the sub-critical

bifurcation diagram. The origin is locally stable for r < 0 and the two branches are

unstable. The two backward unstable branches bifurcated from r = 0. The term x5

6

8/2/2019 4330_ds_2

7/7

has now created a new phenomenon: at a value of r < 0, denoted by r, the unstable

branches turn around around and become stable. These new branches exist for all

r > r

2. Note that for r < r < 0 there are three stable solutions. The initial condition

determines which of these three fixed points the solution converges to as time increases.

3. This example demonstrates an important physically observed phenomenon known as

Hysteresis. If we start the system with an initial condition close to x = 0 r

r

x

r

Bifurcation Diagram showing Hysteresis

7

Documents

4330_ds_2