Duffing equations are used to model certain mechanical or electrical oscillators. One frequently mentioned derivation as a one-mode approximation of the deflections
of a symmetric beam modelled by

with

is summarized in [R. Seydel: Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos. Second Edition. Springer Interdisciplinary Applied Mathematics 1994.]

It is an instructive exercise to consider the case
(no driving force) with the compressive force
as bifurcation parameter. For
there is a pitchfork bifurcation. At this value of *Euler's first buckling load* the underlying beam starts to bend. Up to symmetry, one stable equilibrium exists for each
.
This is satisfied by our parameters.

*First-order System:*

The harmonic forcing term
on the right-hand side provokes a
response of the system (beam, electric current).
The harmonic oscillations obey the boundary conditions

Because the period

The transformation

leads to the first-order system

with boundary conditions

Redefining the normalized time by

*Bifurcations:*

A main part of the bifurcation diagram is shown in Figures 1, 2.
The branching behavior is rich for
small values of .
As
Figure 3 shows, there are many
branches, turning points, and bifurcation points for
values of the parameter
The solid curve in this branching
diagram represents oscillations, with phase diagrams being
symmetric with respect to the origin as in Figure 4. For decreasing
,
each loop of this ``main" branch attaches a further wiggle to
the oscillation. The wiggles indicate that the small-amplitude
oscillation around any of the stable equilibria
takes much time to collect enough energy before a transition to
the other attracting basin is possible; compare the phase diagram
in Figure 4 and the time dependence in Figure 5
.
At first glance, the many wiggles might give the impression
that such solutions are not harmonic; but they do have period *T*.
Each of the closed branches in Figure 3 (dashed and
dotted curves) is attached to the main branch via two pitchfork
bifurcations. Accordingly, the phase diagrams of these ``secondary"
branches are asymmetric with respect to the
origin; Figure 6 shows one such phase plot.
The closed branch in Figure 7 (drawn in Figure 3 in a dashed line)
appears to have sharp corners, and one might expect difficulties
in tracing this particular branch. The plotted behavior refers
only to the dependence *y*_{1} (0) versus ;
the corresponding graph
of *y*_{2} (0) behaves more smoothly (Figure 8).

The table includes 20 bifurcation points of the
above Duffing equation with accurate values of the amplitude. The above Duffing equation also exhibits *subharmonic*
oscillations; then the period of the system is an integral multiple of the
period of the external frequency.
Branches of subharmonic solutions bifurcate from the harmonic solutions (not listed in the table). Stability, and period doubling scenarios have been calculated by K. Riedel (Ulm 1996). For example, a period doubling cascade starts at
.

*Chaotic solutions:*

For a range of -values ther exist chaotic attractors.
One such attractor is illustrated by the Poincaré map / stroboscopic map
,
,
*k*=1,2,...,8000,
,
in Figure 9.

Figure 1

Bifurcation diagram

Figure 2

Bifurcation diagram
versus .

Figure 3

Detail of bifurcation diagram, *u*(0) versus .

Figure 4

phase-plane for
,
symmetric oscillation.

Figure 5

*u*(*t*) for
,
*t* is a normalized time, *t*=1 corresponds to
.

Figure 6

plane for
,
asymmetric oscillation.

Figure 7

Bifurcation diagram of a closed branch of asymmetric solutions, *u*(0) versus .

Figure 8

Same as in Figure 7, but
versus .

Figure 9

Duffing equation,
plane, Poincaré set of the stroboscopic map,
,
fractal dimension *D*=2.381.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9