II.  DUFFING's Equation and Equilibria of Differentiable Dynamical Systems.

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  -∞ < x < ∞.  The acceleration d2x/dt2 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) = g cos(wt).  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 - x3.  The potential energy of this force would be -x2/2 + x4/4, having two “wells” near x=±1.  Thus we are led to consider Duffing’s Equation,

	d2x/dt2 + d dx/dt - x + x3 =  g cos(wt).				(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 - x3 - d p + g cos(wt)

Let us initially suppose that g = 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 ∞-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 Œ S,  ft(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(ft(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 g=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 ft(z) Æ zo as t Æ ∞.  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 Œ U, then dist(S, ft(z)) Æ 0.

Example.  For (2.2) with g=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 Æ ∞ constitute its stable manifold Ws(zo).  Similarly, its unstable manifold Wu(zo) comprises the trajectories that tend to zo as t Æ -∞.  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 g 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 2π/w.  A greater g 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 2π/w, but that is not what is observed.  Instead, for g 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 w = 2π, 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 g.  Compare with [Guckenheimer-Holmes, 1983, §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 n¥n matrix with real coefficients, viz.,

		Ajk = ∂vj(zo)/∂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 l is a positive eigenvalue of A, i.e., Au = l u, for a nonzero vector u Œ Rn and l > 0, then the trajectory of the linearized system,   dz/dt = A z, passing through z = u, is a ray, ft(u) = elt u, fleeing the origin.  Similarly, if l < 0, there is a trajectory that is a ray oriented towards the origin.  Even if l is complex, if Re l > 0, then there is an unstable subspace leaving the origin, and if Re l < 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(d2 + 4) )/2      (one positive, one negative)

	zo = (±1,0):	b(-d  ±  r(d2 - 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  ft 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 Æ V, some subset of  Rn, such that if  0 ≤t≤T and if  z and  ft(z) Œ U, then

	h(ft(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 ft to the possibly simpler flow exp(t Dv(zo)) and back by:
	
More abstractly, if the linear flow is yt, then we can write this relationship as
	ft = h-1˚yt˚h,
or, equivalently, 
	h˚ft = yt˚h, 
or 
	yt = h˚ft˚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) =  a(x2+y2)x - y
	f(dy,dt) =  a(x2+y2)y + x

near the equilibrium (0,0) when a≠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 

	Wsloc(zo) = {zŒU such that ft(z) Œ  U for all t ≥  0, and 
			limtÆ∞ ft(z) = zo}

	Wuloc(zo) = {zŒU such that ft(z) Œ  U for all t ≤  0, and 
			limtÆ-∞ ft(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,q)Æ(r,q+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 - x3 + (1+x)y,x - x3 + (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.