The models that describe our nonlinear phenomeana are defined through the language of mathematics. The equations representing the models may be ordinary differential equations (ODEs), or partial differential equations (PDEs). Moreover, some problems are described by ``algebraic'' equations, or by integral equations, or by some mixture. In addition to the differential equations there may be boundary conditions or initial conditions. Faced with the great variety of possible combinations and formats we basically restrict ourselves to the ODE situation. Many of the ideas and methods can be applied in a similar way to other equations.

**1. A computer experiment**

and satisfy the initial values

for the background see WOBexd2. The symbol stands for a parameter that may take values in a range, say, between 0.125 and 0.26. Take any code for integrating ODE initial-value problems, and integrate the equations for the time interval .

Fig. 1. Computer experiment. vertical axis: *y*_{1},

horizontal axis: time *t*. curves: behavior of *y*_{2}(*t*)

for three values of the parameter :
0.125 (solid line),

0.2 (dashed), 0.26 (dotted).

For the first series of experiments we take constant values of
for each integration, but vary
from one integration to the next.
This treatment of the parameter
as being constant but variable is called *quasistationary* variation.
In Figure 1 we observe the results for three selected values of ,
namely
,
,
and
.
For example, take
:
We observe that after a **transient phase** of, say,
,
the trajectory *y*_{1}(*t*) becomes stationary -- that is, it is attracted by a state with a constant value.
For
there is again a transient phase, after which the trajectory becomes periodic with large oscillations.
The stationary state, and the periodic state respectively attract the neighboring trajectory.
Both attracting states are regular in shape.
Finally, for
the attractor is again stationary, but now on a higher level.
The experiment has shown that the level and the quality of solutions *vary with the parameter*.

Fig. 2. *y*_{1}(*t*) for
and
varying from 0.1 to 0.3

In a second type of experiment we let the parameter drift slowly in a nonstationary way,
.
In order to move with the parameter through a range that matches the first set of experiments, we chose the additional differential equation

We may denote and integrate a coupled system of three ODEs for . The result is seen in Figure 2. Because of the linear variation of , the horizontal axis can be exchanged by an equivalent -axis extending from to , which would indicate the current value of the parameter. Observing the result of Figure 2, we come closer to an explanation of our quasistationary treatment reported above. There appears to be a sudden transition from a more or less stationary state to a strongly oscillating state at ( ). For further increasing

**2. The model problem**

where is a vector function with

Fig. 3. Stable solutions of Eq.(3),
for

**3. An example from chemistry**

which is taken from [Krug & Kuhnert, 1985]. Equation (3) is of the format of Eq.(1), with

Looking back at Figure 2 we may notice that the phenomenon taking place for is of that kind. Here, in the quasistationary setting of Eq.(3), the attractors come out most regularly, as in Figure 3.

Fig. 4. Eq.(3), ,
projection to the (*y*_{1},*y*_{2}) plane.

Fig. 5. Eq.(3),
,
projection to the (*y*_{1},*y*_{2}) plane.

Fig. 6. Trigger Eq.(4), output voltage *y*_{6} versus input voltage .

The implicit function theorem (see any textbook of analysis) specifies sufficient criteria guaranteeing that a branch can be parameterized by . For a specific stationary solution of Eq.(2) the criterion basically requires nonsingularity of the Jacobian matrix . Then there is an interval around such that for all in that interval Eq.(2) has a solution close to .

**5. An example from electrical engineering**

This problem from [Pönisch & Schwetlick, 1982] is of the form , with

Fold bifurcations are frequently occurring in applications. For a boundary value problem (not of the type of Eq.(1)) see the Duffing oscillator WOBexb3, or the catalytic reaction WOBexb1, or the electric power system WOBexd15.

**6. Bifurcation**

Fig. 7. Hopf scenarios.

At a Hopf bifurcation, the branch of stationary solutions does not bifurcate; the Jacobian matrix
is nonsingular. In Figure 7a, the
branch of stationary solutions extends beyond ,
but it is unstable for
.
The Jacobian at a Hopf bifurcation has a pair of purely imaginary eigenvalues
.
Here a branch of periodic orbits is born. In Figure 7a, for
,
the periodic solutions merge into the stationary branch, the amplitude vanishes. The bifurcation is vertical, and the amplitude locally behaves like
.
This is seen in Figure 3, just concentrate on the *y*_{1}-values belonging to the minimum *t*-value. The vertical axes in Figure 7 depict a scalar measure of the solutions, such as

That is, depicts a relative maximum, or minimum of

The situation of Figures 7a, 7b depicts a transition without jump: Passing
when increasing
one experiences a *soft loss* of stability of the stationary state; the bifurcation is *supercritical*. In Figure 7c we illustrate a *subcritical* situation where locally no stable state exists on one side of
(here for
). Globally, this local scenario of Figure 7c often extends to a different situation, see Figure 7d: The branch of unstable periodic orbits bends back, gaining stability at a turning point like situation. Consequently, when we increase
beyond the critical Hopf parameter value
a jump occurs. For
in Figure 7d, there are no neighboring small-amplitude periodic solutions, and the dynamics is immediately attracted by a large-amplitude oscillation. This large jump is the *hard loss* of stability. Note that Figure 7d shows a bistable situation for
.
Note further that the described scenarios may also happen for *de*creasing .

Examples for a hard loss of stability are the computer experiment (WOBexd2), the bogie model (WOBexd10), the nerve model WOBexd14, and the power system WOBexd15. Examples for a soft loss of stability are again the computer experiment (the ``right'' bifurcation), the Brusselator (WOBexd1), and the reaction of Eq.(3).

In any arbitrary example of the type of Eq.(1) one must expect the occurrence of a fold bifurcation, or of a Hopf bifurcation.
There are other bifurcations which are less likely to be found in a general equation.
But many equations involve some symmetry.
Often the symmetry in the equations reflects the common situation that a model consists of two or more identical parts that are coupled.
When two identical subsystems are suitably coupled, one can exchange their states by a simple reflection.
The related symmetry is the *Z*_{2}-symmetry.
For equations with a *Z*_{2}-symmetry the **pitchfork bifurcation** is common too.

For a *Z*_{2}-pitchfork bifurcation, the emanating branch consists of solutions that lose their symmetry. This phenomenon is called **symmetry breaking**.
Schematically, the related bifurcation diagrams resemble those of Figure 7 of the Hopf scenario; compare Figure 8.
The two asymmetric half-branches can be identified because they are transformed into each other by the underlying reflection that describes the exchange of the states of the subsystem.

Fig. 8. *Z*_{2}-Pitchfork Scenario.

Examples of such pitchfork bifurcations are the Brusselator reaction WOBexa1, the flipflop WOBexa11, WOBexd1, the superconductivity WOBexb2, and the Duffing oscillator WOBexb3.

**7. Period doubling and chaos**

The local stability of a periodic orbit
is determined by the eigenvalues of the monodromy matrix
where
is the matrix function that solves the linear matrix initial-value problem

These eigenvalues of are called

Dynamically, the period doubling bifurcation (or **flip bifurcation**) is the following scenario: Assume a branch of periodic solutions parameterized by
with stable orbits on one side of
(say,
)
and a multiplier crossing the unit circle with
.
Then, locally, there are periodic orbits with the double period near .
These double-periodic orbits form a new branch that emerges at .
Note that the periods vary with ,
and the factor 2 of period doubling holds only asymptotically for
.
The situation typically is as in the left part of Figure 9, for
.

Fig. 9. Cascade of period doublings, schematically.

The ``new'' branch with the ``double'' period can experience a period doubling, too (in Figure 9, for
). There are many important applications were an infinite chain of such period doublings occurs for
,
,
,
In the supercritical case, the stability is exchanged to the branches of the double period. After some period doublings the period has become so large that the orbit looks irregular. As has been shown, the bifurcation values
satisfy a universal scaling law,

This scaling law, named after Feigenbaum, has a remarkable consequence: There is an accumulation point of the sequence of period doubling bifurcations. Passing means that the ``period'' has reached infinity. The resulting solution is fully aperiodic, and is ``chaotic.'' In this scenario the irregularity of a chaotic solution can be explained by the infinite number of unstable periodic orbits (UPOs) that are ``left behind'' at the infinite sequence of period doublings. Imagine the state space is packed with UPOs, all repelling any trajectory that searches a path through that area. This state of continuously being pushed by UPOs may explain the sensitive dependence on the initial conditions that is a characterisitic criterion of chaos.

Fig. 10. Eq.(7), bifurcation diagram *y*_{1}(0) versus .

**8. An example: isothermal reaction**

There is a Hopf bifurcation at , and a sequence of period doubling bifurcations, compare the branching diagram Figure 10. The critical parameter values of period doubling are

These four values allow to calculate two of the Feigenbaum ratios in Eq.(6). The ratios are 0.13, and 0.19. The asymptotic law allows to estimate the accumulation point where chaos sets in. From Eq.(6) we derive the estimate

Applying this to our sequence of period doublings we estimate that is as close as . Figures 11a, 11b show phase portraits of a periodic orbit of the ``double'' period for , and of the fourfold period ( ). Figure 12 depicts an apparently chaotic solution for . The sequence of bifurcations from stationary state ( ) to ``period four'' ( ) is illustrated by Figure 13. Another interesting example with period doublings is the voltage collapse problem WOBexd15.

Fig. 11a. Eq.(7), periodic orbit,
,
projection to (*y*_{1},*y*_{2}) plane.

Fig.11b. Eq.(7), periodic orbit,
,
projection to (*y*_{1},*y*_{2}) plane.

Fig. 12. Eq.(7), chaotic orbit,
,
projection to (*y*_{1},*y*_{2}) plane.

Fig. 13. Eq.(7),
for
,
periods scaled to unity.

**9. Other bifurcations**

First we return to the pitchfork. The simplest equation with pitchfork is
.
The simplest equations exhibiting a certain phenomenon are called *normal forms*. As mentioned above, in case the underlying equation supports a *Z*_{2}-symmetry, the pitchfork scenario is also likely to occur. The geometrical analogy of a pitchfork illustration with that of a Hopf bifurcation is no coincidence since the normal form of Hopf bifurcation is (in polar coordinates ,
)

Solutions of this normal form include the stationary state , and the periodic state . In both cases holds, and we have a pitchfork characterization of the amplitude. If the equation supports no symmetry, then to have a pitchfork more parameters than just are required. Classifications of related bifurcations of higher

Branches of periodic orbits can show more bifurcation phenomena than just period doubling. The multipliers can cross the unit circle of the complex plane at +1. Then we encounter, for example, a turning point, or a pitchfork bifurcation.
The turning point of periodic orbits is sometimes called **cyclic fold bifurcation**. When the crossing is with nonzero imaginary part, there is a bifurcation into a torus-like object.

Periodic orbits are born in a Hopf bifurcation, and may end in a **homoclinic orbit**. To explain the simplest such scenario imagine in a plane a stationary state of saddle type. This saddle has a pair of entering trajectories and a pair of leaving trajectories. In case the leaving trajectory bends back such that it is identical with the entering trajectory we have a loop with infinite period. This is a homoclinic orbit. A periodic stable orbit close to a saddle is depicted in the phase portrait of Figure 14. Assume this is the situation for
,
and the periodic orbit and the saddle approach each other for
.
Then we encounter a homoclinic orbit for
and no periodic orbit for
.

Fig. 14. Situation close to a homoclinic orbit.

This simplest scenario of a homoclinic orbit illustrates how an unstable stationary state annihilates a periodic orbit. Analog phenomena are also possible with other unstable states. For example, a collision of an unstable periodic orbit might terminate a chaotic attractor. Generally, unstable states play a fundamental role in organizing dynamical behavior. This situation stresses the importance of calculating also unstable states. An example of the decisive role an unstable state plays in ``killing'' the operating regime is provided by models of voltage collapse, see WOBexd15.

**10. Multi-parameter problems**

Fig. 15. Cusp scenario, with fold curves.

As a simple example, imagine a hysteresis situation as depicted in Figure 6. The range of bistability between the values
of the two turning points may shrink when a second parameter
is varied, see Figure 15. Denote
the specific value where both -values coincide. Then, on one side of
(say for
)
a bistable situation with jump phenomena exists whereas on the other side no bistability, and no bifurcation of that kind exists. Exactly
for
the bifurcation is a *hysteresis point*, the two adjacent turning points have collapsed into a point of inflection. The value
is seen as an *organizing center* separating two completely different dynamical situations. This is illustrated by the parameter chart of Figure 16.
The figure depicts also a jump-free transition from one operation point in the parameter plane to another.
A related example is provided by a catalytic reaction (WOBexb1).
The knowledge of the bifurcation curves allows to find easily a curve of parameter combinations
that detours and avoids the jumps triggered by turning points.

Fig. 16. Jump-free path (dotted) in the parameter plane.

If in addition to
and ,
a third parameter is considered to be freely variable, the bifurcation curves in the parameter plane extend to *bifurcation surfaces* in the parameter space. Possible bifurcation scenarios become even more involved. A theory of singularites has been established to analyze related higher-order bifurcations [Golubitsky & Schaeffer, 1985].

**11. Other problems**

The transformation

Problems of this type describe, for example,

comprise all ODE settings. For vanishing diffusion ( ) the model problem of Eq.(1) results. On the other hand, for a stationary situation ( , one space variable) the PDE (12) reduces to an ODE boundary-value problem of type (11). The former case models purely temporal dynamics of the reaction-diffusion problem, whereas the latter case concentrates on purely spatial patterns. In the general case, temporal and spatial phenomena may be interrelated in complicated ways.

**12. Historical and bibliographical remarks**

**References**

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