dx/dt = F(x,
(t)),In this work, we consider mainly adiabatic differential equations of the form
dx/dt = F(x,(t)),
where x is the vector of
dynamic variables, t = t is the slow time,
the small adiabatic
parameter, and (t) the slowly varying parameter. The main idea is to
establish connections between solutions of this equation and
those of the associated family of frozen dynamical systems dx/dt
= F(x,).
Detailed results are contained in my Ph.D. thesis, and a summary of mathematical results can be found this proceedings. A non-technical description is given here, several physical applications are found in this paper, while chaotic hysteresis is briefly described in this letter.
)
is a hyperbolic equilibrium
branch of F(x,
),
we show the existence of a particular solution of the
adiabatic system, which tracks the branch at a distance
of order . We call
it an adiabatic solution. It can be expanded in
a power series of the adiabatic parameter .
Even when the function F
is analytic, this series does not converge in general,
but can be truncated at exponentially small order. (t)).
If x*(t) is an
equilibrium branch arriving at the origin, adiabatic
solutions no longer track this branch at a distance of
order , but at a distance
scaling with some other power of .
We developed a method to compute this exponent in a
simple way, using Newton's polygon. It allows to
determine scaling laws of hysteresis cycles in an easy,
geometric way. In the vectorial case, the same method can
be applied after using a centre manifold reduction.
dx/dt = A(t) x.We provide a method to
diagonalize this equation dynamically, when its
eigenvalues have different real parts. Eigenvalue
crossings can be studied by a local analysis. In some
cases, these crossings lead to a similar behaviour than
bifurcations. . In the
analytic case, they can be truncated to exponentially
small order. dx/dt = A(t) x + b(x,t), is to try to eliminate the
nonlinear term b by a change of variables. This
turns out to be possible in the nonresonant case.
Resonant terms can be dealed with by a local analysis,
which is again similar to the analysis of bifurcations. Chaotic hysteresis: Consider a damped pendulum on a table, rotating with a slowly modulated angular frequency. When the frequency is small, the downward position of the pendulum is stable. When the frequency is sufficiently high, this position becomes unstable, and the pendulum makes an angle with the vertical. When the frequency is modulated periodically, the system displays hysteresis: the pendulum leaves the vertical position at a frequency which is higher than when he drops back. Moreover, the pendulum does not always choose the same asymmetric equilibrium: for some values of the parameter, it may visit these positions in a chaotic sequence. We explain this behaviour by computing an asymptotic expression for the Poincaré map, using the methods described above (see this letter).
The rotating pendulum is mounted on a table rotating with angular frequency W. It experiences friction, gravity and an intertial torque. When the frequency W is varied periodically, this system sometimes displays chaotic hysteresis: each time the frequency becomes large, the pendulum leaves the origin for one of two possible equilibria at an angle. The sequence of visited equilibria is chaotic for some values of the parameters.