Homoclinic Intersections By homoclinic intersections we refer to transverse
crossings of the stable and unstable manifolds of a Poincaré map saddle-point.
Fig. 6.13 shows phase plane orbits for the Duffing equation (6.1), when there is
no damping and no forcing. There is a saddle at (0, 0) and two centers at (±1, 0).
A homoclinic orbit surrounds each center, connecting the stable and unstable
branches of the saddle (recall from Sect. 3.4.3 that a homoclinic orbit connects a
singular point with itself). When damping is added the centers become stable foci,
and the homoclinic orbits break up. The saddle remains a saddle. Add then forcing
to the system and the orbits will move around in a more complicated manner,
though the presence of the saddle and the foci for the unforced problem will remain
to affect the flow locally. The addition of forcing brings an extra state variable to the
system, since the force depends explicitly on time. The system becomes
three-dimensional, so that orbits may now intersect in the phase plane. This has
nothing to do with homoclinic intersections.
We then consider instead the Poincaré map (PM) representation of the dynamics
of a system. For the Duffing equation we sample the phase plane orbits at discrete
times in phase with the driving force. In the Poincaré map, the continuous flow of
phase plane orbits convert into sequences of points. These points may lie along
certain curves in the PM. Such curves are called orbits (of the PM), just as for the
phase plane. As we have seen, periodic motion reveals itself in the PM as one or
more points. Even if there is only a single point, this point is called an orbit for the
PM. Similarly, for quasiperiodic motion we obtain a closed orbit in the PM. If the
system is in a transient or chaotic state the PM-points will move around as governed
by the dynamics of the particular map.
The mapping of points in the PM is affected by the presence and the nature of
fixed points – just as flows of phase plane orbits are locally affected by singular
points. Near fixed points, new PM points will map according to the nature of the
fixed point. For example, if a PM point is near a stable focus, then the next
PM-point will map closer to this fixed point.
Though it is not obvious, the PM for the forced Duffing equation has a
saddle-point near (0,0). Near this PM saddle, points are repelled from certain
directions while attracted from other. Fig. 6.14 shows a sketch of a PM for the
Duffing system (6.1) when the forcing is weak. The saddle near (0,0) has stable and
unstable manifolds W
s and W
u , similar to the stable and unstable branches of a
phase plane saddle. These manifolds are invariant for the map, meaning that any
PM point on W
s will map on W
s , and similarly for W
u
. Points on W
s will map closer
and closer to the saddle, whereas point on W
u will map closer and closer to the two
stable foci. The foci represent regular periodic motion in the phase plane (for the
magnetically buckled beam each PM focus corresponds to periodic oscillations
about one of the magnets). We note that W
s and W
u do not intersect.
However, in some cases the stable and unstable manifolds of a PM saddle may
intersect. Fig. 6.15 shows a PM for the Duffing equation when the force has been
raised to a level causing W
s (in solid line) and W
u (dashed) to intersect. This is called
homoclinic intersection, and the points of intersection are homoclinic points. Let us
consider a PM point falling at the intersection marked I in Fig. 6.15. Since this point
6.5 Tools for Predicting the Onset of Chaos
351
crossings of the stable and unstable manifolds of a Poincaré map saddle-point.
Fig. 6.13 shows phase plane orbits for the Duffing equation (6.1), when there is
no damping and no forcing. There is a saddle at (0, 0) and two centers at (±1, 0).
A homoclinic orbit surrounds each center, connecting the stable and unstable
branches of the saddle (recall from Sect. 3.4.3 that a homoclinic orbit connects a
singular point with itself). When damping is added the centers become stable foci,
and the homoclinic orbits break up. The saddle remains a saddle. Add then forcing
to the system and the orbits will move around in a more complicated manner,
though the presence of the saddle and the foci for the unforced problem will remain
to affect the flow locally. The addition of forcing brings an extra state variable to the
system, since the force depends explicitly on time. The system becomes
three-dimensional, so that orbits may now intersect in the phase plane. This has
nothing to do with homoclinic intersections.
We then consider instead the Poincaré map (PM) representation of the dynamics
of a system. For the Duffing equation we sample the phase plane orbits at discrete
times in phase with the driving force. In the Poincaré map, the continuous flow of
phase plane orbits convert into sequences of points. These points may lie along
certain curves in the PM. Such curves are called orbits (of the PM), just as for the
phase plane. As we have seen, periodic motion reveals itself in the PM as one or
more points. Even if there is only a single point, this point is called an orbit for the
PM. Similarly, for quasiperiodic motion we obtain a closed orbit in the PM. If the
system is in a transient or chaotic state the PM-points will move around as governed
by the dynamics of the particular map.
The mapping of points in the PM is affected by the presence and the nature of
fixed points – just as flows of phase plane orbits are locally affected by singular
points. Near fixed points, new PM points will map according to the nature of the
fixed point. For example, if a PM point is near a stable focus, then the next
PM-point will map closer to this fixed point.
Though it is not obvious, the PM for the forced Duffing equation has a
saddle-point near (0,0). Near this PM saddle, points are repelled from certain
directions while attracted from other. Fig. 6.14 shows a sketch of a PM for the
Duffing system (6.1) when the forcing is weak. The saddle near (0,0) has stable and
unstable manifolds W
s and W
u , similar to the stable and unstable branches of a
phase plane saddle. These manifolds are invariant for the map, meaning that any
PM point on W
s will map on W
s , and similarly for W
u
. Points on W
s will map closer
and closer to the saddle, whereas point on W
u will map closer and closer to the two
stable foci. The foci represent regular periodic motion in the phase plane (for the
magnetically buckled beam each PM focus corresponds to periodic oscillations
about one of the magnets). We note that W
s and W
u do not intersect.
However, in some cases the stable and unstable manifolds of a PM saddle may
intersect. Fig. 6.15 shows a PM for the Duffing equation when the force has been
raised to a level causing W
s (in solid line) and W
u (dashed) to intersect. This is called
homoclinic intersection, and the points of intersection are homoclinic points. Let us
consider a PM point falling at the intersection marked I in Fig. 6.15. Since this point
6.5 Tools for Predicting the Onset of Chaos
351