version of 15 May 1992

GUCKENHEIMER and HOLMES [1983] describe a simple apparatus which exhibits highly irregular behavior both experimentally and when modelled numerically on a computer. It consists of a flexible steel beam attracted by two magnets and also subjected to a periodic forcing function:

There are three equilibrium positions, one with the beam centered, and one with
it near each magnet. Let us call the angular displacement of the beam x, and
imagine that -[[infinity]] < x < [[infinity]]. The acceleration
d^{2}x/dt^{2 }will depend on elastic forces in the beam, the
magnetic force, friction, and the forcing function, which we will assume is of
the form f(t) = [[gamma]] cos([[omega]]t). The friction can be roughly
modelled as a contribution -d dx/dt, and the other forces as a function of
position only, equalling 0 at x=0, +/-1, like x - x^{3}.^{
}The potential energy of this force would be^{ }-x^{2}/2 +
x^{4}/4, having two "wells" near x=+/-1. ^{ }Thus we are led to
consider **DUFFING's Equation**,

d^{2}x/dt^{2 }+ d dx/dt - x + x^{3 }= [[gamma]]
cos([[omega]]t). (2.1)

This will clearly be a bad model for large x, but we suppose that the beam is stiff, and consider the model valid only for small oscillations.

This chapter of the notes will give a brief description of differential
equations can be understood in terms of the geometry of the trajectories in
state space, or **phase portraiture**. More detailed descriptions can be
found in any good introductory book on ordinary differential equations, but the
best way to develop your intuition is probably to use some of the good computer
software which has been developed for this purpose. I recommend in particular
H. Gollwitzer's *Differential Systems* for the Macintosh and H.
KOÇAK's *Phaser* for MS-DOS machines. To perform a phase-plane
analysis, we put p = dx/dt, and consider the system

f(dx,dt) = p

(2.2)

f(dp,dt) = x - x^{3 }- d p + [[gamma]] cos([[omega]]t)

Let us initially suppose that [[gamma]] = 0, so that we have an autonomous system. We can then readily sketch the phase portrait for the system, and find:

**Exercise **II.1: Sketch these phase portraits for yourself.

In the case d = 0 there is an [[infinity]]-shaped figure, consisting of an
equilibrium at the origin and two special trajectories known as the
**separatrices**. A trajectory enclosed by one separatrix will orbit an
equilibrium point near one magnet (unless the trajectory is the equilibrium
itself), whereas a trajectory outside the separatrices exhibits periodic motion
circling all the equilibria. The motion on the separatrices is nonperiodic.
It takes an infinite time for the system to travel on the separatrix for one
circuit.

**Definition**. If for some zo [[propersubset]] S, [[phi]]t(zo) = zo for
all t, then zo is said to be an **equilibrium** (also **fixed point**,
**critical point**, or **rest point**).

An equilibrium zo is **stable** if for any R > 0, there exists a value r,
0 < r <= R, such that if dist(zo, z) < r, then dist([[phi]]t(z), zo)
< R. In words: All points near zo follow trajectories that remain near zo.
The slightly complicated wording of the precise definition allows the possibly
that nearby points may move a bit farther away from zo, although not very
far.

**Example**. For (2.2) with [[gamma]]=0, the equilibria at (+/-1,0) are
stable, and the equilibrium at (0,0) is unstable.

**Definition**. An equilibrium zo is **asymptotically stable**, or
**attractive**, if for some R > 0, whenever dist(zo, z) < R, then
[[phi]]t(z) -> zo as t -> [[infinity]]. In words: All points near zo
are pulled into zo. An **attracting set** S is a set (generally more
complicated than a single equilibrium) with the analogous property: For some
neighborhood U of S, if z [[propersubset]] U, then dist(S, [[phi]]t(z)) ->
0.

**Example**. For (2.2) with [[gamma]]=0, and with friction, d > 0, the
equilibria (+/-1, 0) are asymptotically stable.

The trajectories which tend asymptotically to an equilibrium zo = (0,0) as t
-> [[infinity]] constitute its **stable manifold**
W^{s}(zo). Similarly, its **unstable manifold** W^{u}(zo)
comprises the trajectories that tend to zo as t -> -[[infinity]]. Notice
that when d = 0, the stable and unstable manifolds are actually the same set,
and that for a stable equilibrium zo the unstable manifold consists only of zo
itself.

Turning on an external force f(t) can be thought of as causing the system not to stay on one of the trajectories drawn above, but instead to move constantly from one trajectory to another. Let us suppose that there is some friction in the system, d > 0. It is physically plausible that if the strength [[gamma]] is small and a trajectory is initially near one of the asymptotically stable equilibria, then the perturbed trajectory also remains near the equilibrium, and the motion is periodic with some period 2[[pi]]/[[omega]]. A greater [[gamma]] could easily cause transitions from motion in one well to motion in the other well. One might expect the transitions to occur periodically, with some period related to the driving period 2[[pi]]/[[omega]], but that is not what is observed. Instead, for [[gamma]] greater than some critical value, the system makes irregular transitions from one well to the other. Between the transitions, it executes a certain number of oscillations within a given well, but the number of oscillations after a transition bears no apparent relationship to the number before the transition.

A good suspect for the cause for this chaotic motion is the saddle-type equilibrium at (0,0), since in any small neighborhood, there are trajectories of the autonomous system executing eight radically different types of motion when d = 0:

stationary motion at the equilibrium

periodic motion about either well (2)

periodic motion about both wells

nonperiodic motion away from the equilibrium (2)

asymptotic approach towards equilibrium (2)

**Exercise **II.2. Classify the different types of motion when d > 0.

**Exercise **II.3. Set [[omega]] = 2[[pi]], and study (2.2)
numerically, plotting the state of the system in the x-p plane at times t = 1,
2, 3, .... Choose various values of [[gamma]]. Compare with
[GUCKENHEIMER-HOLMES, 1983, [[section]]2.2].

**Some further remarks about phase portraits.**

Consider a differentiable dynamical system specified by a system of two
equations of motion for two variables, x(t) and y(t). The most efficient way
to get a qualitative idea of the motion is first to locate the equilibria, and
analyze the motion near the equilibria by linearizing the system - assuming
that the coordinates are near those of the equilibrium and making a TAYLOR
expansion of the vector field, keeping only the leading term. We assume that
the equilibrium is **nondegenerate**, i.e., that the leading term in the
expansion is first-order in the displacement from equilibrium (in every
direction). As we shall see, only a few types of motion are possible near the
equilibrium (this is special to two dimensions), and usually the motion can be
understood by analyzing a two-
by-
two
matrix. Then plot a few representative vectors of the vector field away from
the equilibria and connect them with flow lines.

Geometrically, only the kinds of behavior shown on the next page are possible near a nondegenerate equilibrium:

Usually it is possible to classify equilibria by linearization, i.e., for an n-dimensional system, writing

v(z) = A (z-zo) + o(dist(z,zo)),

where A is an nxn matrix with real coefficients, viz.,

Ajk = [[partialdiff]]vj(zo)/[[partialdiff]]xk. (2.3)

There is a compact notation for this: A = Dv(zo). A will be called the
**linearization**, or **Jacobian**, of v(z) at zo. The equilibrium is
**nondegenerate** when Dv(zo) is a nondegenerate matrix. Neglecting the
higher-order terms would leave us with a very simple system to analyze,
depending on the eigenvalue analysis of the matrix A.

**Example**. When we linearize DUFFING's equation around the three
equilibria, the matrices A are:

zo = (0,0): A = b(a(0 1,1 -d))

(2.4)

zo = (+/-1,0): A = b(a(0 1,-2 -d ))

If [[lambda]] is a positive eigenvalue of A, i.e., Au = [[lambda]] u, for a
nonzero vector u [[propersubset]] R^{n }and [[lambda]] > 0, then the
trajectory of the linearized system, dz/dt = A z, passing through z = u, is a
ray, [[phi]]t(u) = e^{[[lambda]]t }u, fleeing the origin. Similarly,
if [[lambda]] < 0, there is a trajectory that is a ray oriented towards the
origin. Even if [[lambda]] is complex, if Re [[lambda]] > 0, then there is
an unstable subspace leaving the origin, and if Re [[lambda]] < 0, then
there is a stable subspace entering the origin. The **center subspace**
consists of the eigenvectors with purely imaginary eigenvalues, and
linearization in this case does not tell the whole story about the nature of an
equilibrium of a nonlinear system. Even the qualitative features of the motion
will depend on the higher-order terms in the expansion of the vector field in
powers of z-zo.

**Example**. The eigenvalues of the linearizations of DUFFING's equation
are:

zo = (0,0): b(-d +/- r(d^{2} + 4) )/2 (one positive, one
negative)

zo = (+/-1,0): b(-d +/- r(d^{2} - 8 ) )/2 (Both complex with
negative

real part when d < 2r(2), and

both negative when d > 2r(2)).

This shows that there is one curve passing through the origin consisting of points which are attracted to it, while there is another consisting of points that get expelled. On the other hand, there is a two-dimensional region about each of the other equilibria that gets attracted to it. The origin is always a saddle. If d < 2r(2), then (+/-1,0) are attracting spirals, while if d >= 2r(2), then (+/-1,0) are attracting nodes.

**Exercise **II.4. Find the eigenvectors of the matrices A. When the
eigenvectors are real, they specify the tangent lines to trajectories that are
attracted to or repelled from the equilibria. Use this information to improve
your sketch of the phase portrait. Can the eigenvectors be nonreal, and, if
so, what would that mean?

**Definition**. An equilibrium zo of a differentiable dynamical system is
**hyperbolic** iff the linearization at zo has no center subspace.

If the equilibrium is hyperbolic, then an important theorem due to HARTMAN and GROBMAN states that the linearization gives a somewhat accurate picture of the flow:

**Theorem II.1**. *Let* [[phi]]t *be the flow of a differentiable
dynamical system on a manifold. Write the equations of motion as*

dz/dt = v(z).

*If* zo *is a hyperbolic equilibrium, then there are a time* T>0,
*a neighborhood* U *of* zo, *and a homeomorphism* h: U -> V,
*some subset of* R^{n},^{ }*such that if* 0
<=t<=T *and if* z *and* [[phi]]t(z) [[propersubset]] U,
*then*

h([[phi]]t(z)) = exp(t Dv(zo)) h(z). (2.5)

In this theorem, the exponential matrix exp(tA) stands for the solution operator of a constant-coefficient system

f(dy,dt) = A y.

Recall that a **homeomorphism** is a continuous, continuously invertible
bijection. This theorem almost states that a coordinate transformation can be
made so that the flow on U becomes the flow of a constant-coefficient system.
What one would really need to be able to do this is for h to be a
**diffeomorphism**, i.e., a homeomorphism with some degree of
differentiability. Some additional assumptions are necessary to guarantee
this, and there are examples showing that the problem may sometimes be serious.
Roughly, according to STERNBERG's theorem, h is a diffeomorphism provided that
there are no **resonances**, i.e., certain rational relationships among the
eigenvalues of the linearization (see [RUELLE, 1989]. A relationship such as
(2.5) is known as a **topological conjugacy**, and it means that there is a
dictionary for translating from [[phi]]t to the possibly simpler flow exp(t
Dv(zo)) and back by:

More abstractly, if the linear flow is [[psi]]t, then we can write this relationship as

[[phi]]t = h^{-1}[[ring]][[psi]]t[[ring]]h,

or, equivalently,

h[[ring]][[phi]]t = [[psi]]t[[ring]]h,

or

[[psi]]t = h[[ring]][[phi]]t[[ring]]h^{-1}.

**Exercise** II.5. Find a finite-dimensional example of a bijection
(one-to-one and onto) which is continuous but which has a discontinuous
inverse.

**Exercise** II.6. By transforming to polar coördinates, show that
linearization fails to give the correct qualitative flow for the system

f(dx,dt) = [[alpha]](x^{2}+y^{2})x - y

f(dy,dt) = [[alpha]](x^{2}+y^{2})y + x

near the equilibrium (0,0) when [[alpha]]!=0.

Another important theorem is known as the **stable manifold theorem**.

**Definition **Let zo be an equilibrium of a differentiable dynamical
system, and let U be a small neighborhood of zo. The **local stable **and
**unstable manifolds **are

W^{s}loc(zo) = {z[[propersubset]]U such that [[phi]]t(z)
[[propersubset]] U for all t >= 0, and

limt->[[infinity]] [[phi]]t(z) = zo}

W^{u}loc(zo) = {z[[propersubset]]U such that [[phi]]t(z)
[[propersubset]] U for all t <= 0, and

limt->-[[infinity]] [[phi]]t(z) = zo}

These, obviously, depend on U. The** stable **and **unstable manifolds
**are

and^{}

^{}
^{}

^{}The stable manifold consists of points that eventually find
themselves in the local stable manifold, and are then sucked into the
equilibrium; it no longer depends on U. It is sometimes called the **basin of
attraction** of zo. The unstable manifold is similar, but with time
reversed. It is also possible to define stable and unstable manifolds for more
general sets than individual equilibria.

**Theorem**. *Let* zo *be a hyperbolic equilibrium of a
differential dynamical system. Then there are local stable and unstable
manifolds for* zo *having the same dimensions as those of the linearized
system at* zo, *and being tangent to them at* zo.

Since we still do not know that the flow is diffeomorphic, this statement does not quite state that the structure of the stable manifold is the same as for the linearized problem. One might look like a spiral and the other a node, for example, even when both are two-dimensional surfaces that are tangent.

**Exercise **II.7. Verify that the homeomorphism given in
polar-coördinates by (r,[[theta]])->(r,[[theta]]+1/r) except at the
origin, which is fixed, can change a node into a spiral.

**Exercises **II.8.

a) What are the stable and unstable manifolds for DUFFING's equation (time-independent)?

b) Perform a complete phase-plane analysis of

f(d,dt) b(a(x,y)) = b(a(x - x^{3} + (1+x)y,x - x^{3} +
(2-x)y)).

Include in your analysis a) locating all the equilibria; b) linearizing the system and classifying the equilibria; and c) a sketch of the flow.