We continue with 2D flows
with an example of a subtle
bifurcation:
- Van der Pol oscillator with Hopf bifurcation
The equations are:
- x' = y
- y' = -x -m(x2 - 1)y
Here x is voltage, and m (nonnegative) the control parameter.
See Strogatz, p. 198 and DGB Pt. 4.
Also, we introduce chaotic attractors
in two contexts, 3D flows and 2D cascades,
in the Maple environment, with a single example:
- the Ueda attractor in the forced Duffing system
In the 3D ring model, the equations are:
- x' = y
- y' = - ky - x3 + Bcos(z)
- z' = 1
where x is the position of an oscillating
mass on a spring, y its velocity,
z is the phase angle of the forcing term,
and k (friction, positive) and B
(forcing amplitude, nonnegative)
are control parameters.
See Ueda, p. 158, and DGB Pt. 4.
The Maple models
- Duffing scheme
- The unforced system here is the Duffing oscillator
- Download: [Maple worksheet]
- Download: [Maple script (text)]
- Forced Duffing
- The forced system exhibits Ueda's chaotic
attractor for k = 0.1, B = 12.
- This model plots the 3D trajectories
projected into the 2D state space of the Duffing system
- Download: [Maple worksheet]
- Download: [Maple script (text)]
- Poincare section
- Same model, but plots just a few trajectory points
of the Poincare cascade in 2D
- Download: [Maple worksheet]
- Download: [Maple script (text)]
- Poincare cascade
- Same, but allows longer runs
- Download: [Maple worksheet]
- Download: [Maple script (text)]
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