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
42
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
1.4 Phase Line and Bifurcation Diag
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
Technical publications may use special diagrams to display qu
research. A complete examinationinformation
of of the field would
impossible.
aboutbethe
equilibrium points of the differential e
A fixed point (or equilibrium point) of a differential equation y 0 = f (y) is a root of the
y ! = f (y).
(1)
equation f (y) = 0. As we have already seenfor autonomous problems fixed points can be
This
independent of x, hence there are no externa
very useful in determining the long
timeequation
behavior is
of solutions.
terms that depend on x. Due to the lack of external controls,
tion is said to be self-governing or autonomous.
Qualitative information about the equilibrium points of the differential equation y 0 = f (y)
can be obtained from special diagrams called phase diagrams.
A phase line diagram for the autonomous equation y ! = f (y
A phase line diagram for the autonomous
equation
y 0 sink,
= f (y)source
is a line segment
with
labels
segment with
labels
or node,
one
for each root of
for so-called sinks, sources or nodes,
forequilibrium;
each root of f see
(y) =Figure
0, i.e. each
i.e. one
each
11. equilibrium.
source
sink
y0
y1
Figure 11. A phase line diagram
autonomous equation y ! = f (y).
labelsofare
borrowed
the theory
of fluids, and they
The names are borrowed fromThe
the theory
fluids
and they from
are defined
as follows:
following special definitions:6
1. Sink An equilibrium y0 which attracts nearby solutions at t = ∞, i.e., there exists
Sink y = y0
The
t→∞ equilibrium y = y0 attracts nearby solu
x = ∞: for some H > 0, |y(0) − y0 | < H
|y(x) − y | decreases to 0 as x → ∞.
2. Source An equilibrium y1 which repels nearby solutions at 0t = ∞, i.e., here exists
Source y = y1
The equilibrium y = y1 repels nearby solut
M > 0 so that if |y(0) − y1 | < M , then |y(x) − y1 | increases as t → ∞.
x = ∞: for some H > 0, |y(0) − y1 | < H
that |y(x) − y1 | increases as x → ∞.
3. Node An equilibrium y2 which is neither a sink or a source. In fluids,
sink means fluid
Node y = y2
The equilibrium y = y2 is neither a sink nor a
M > 0 so that if |y(0) − y0 | < M , then |y(x) − y0 | −−−→ 0
is lost and source means fluid is created.
In fluids, sink
means fluid is lost and source means fluid is cr
1
memory device for these concepts is the kitchen sink, wherein t
is the source and the drain is the sink. The stability test
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 and
a source is unstable. Precisely, an equilibrium y0 is stable provided for given > 0 there
exists some δ > 0 such that |y(0) − y0 | < δ implies y(t) exists for t ≥ 0 and |y(t)?y0 | < .
Theorem 2.1 (Stability Conditions). Let f and f 0 be continuous. The equation y 0 = f (y)
has a sink at y = y0 provided f (y0 ) = 0 and f 0 (y0) < 0. An equilibrium y = y1 is a source
provided f (y1 ) = 0 and f 0 (y1 ) > 0. There is no test when f 0 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) corresponding to the creation and destruction of fixed points. The normal form for this type of
bifurcation is given by the example
x0 = 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 this
√
equilibrium converge to it as time increases. Further, initial conditions near −r diverge
from it.
2
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 if r > 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.
x
r
Note that if instead we consider x0 = 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. x0 = 1 + rx + x2
2. x0 = r − cosh(x)
3. x0 = r + x − ln(1 + x)
3
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
x0 = rx − x2
(2)
In this case there is either one (r = 0) or two (r 6= 0) fixed points. When r = 0 the only
fixed point is x = 0 which is semi-stable (i.e., stable from the right and unstable from the
left). For r 6= 0 there are two fixed points given by x = 0 and x = r. So we note in this case
x = 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.
x
r
Exercise: Analyze the bifurcation properties of the following following problems.
1. x0 = rx + x2
2. x0 = rx − ln(1 + x)
4
3. x0 = 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
x0 = 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
x
r
Super Critical Pitchfork Bifurcation Diagram
Now consider the example
x0 = rx + x3 .
(4)
For this example we obtain the so-called sub-critical pitchfork bifurcation. Notice that solutions blow-up in finite time, i.e., satisfy x(t) → ±∞ as t → a < ∞.
5
x
r
Sub Critical Pitchfork Bifurcation Diagram
Exercise: Analyze the bifurcation properties of the following following problems.
1. x0 = −x + β tanh(x)
2. x0 = rx − 4x3
3. x0 = rx − sin(x)
4. x0 = rx + 4x3
5. x0 = rx − sinh(x)
6. x0 = rx − 4x3
7. x0 = x +
2.4
rx
1 + x2
Hysteresis: a more complicated bifurcation
In this subsection we consider an even more complicated example which contains pitchforkand
saddle node bifurcations. Consider the example
x0 = 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
r
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
Sub Critical Pitchfork Bifurcation Diagram
r > r∗
Exercise: Analyze the bifurcation properties of the following following problems.
2. Note that for r∗ < r < 0 there
are three stable solutions. The initial condition
!
1. x = −x + β tanh(x)
determines which of these three fixed points the solution converges to as time increases.
2. x! = rx − 4x3
3. This example demonstrates an! important physically observed phenomenon known as
3. x = rx − sin(x)
Hysteresis. If we start the system with an initial condition close to x = 0 r∗
3.9.4
Hysteresis: a more complicated bifurcation
In this subsection we consider an even more complicated example which contains pitchforkand
x
saddle node bifurcations. Consider the example
x! = rx + x3 − x5 .
r
r∗
Bifurcation Diagram showing Hysteresis33
7
(14)