In this work, we consider mainly adiabatic
differential equations of the form
where x is the vector of
dynamic variables, τ = εt is the slow time, ε
the small adiabatic parameter, and λ(τ) 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
Detailed results are contained in
my Ph.D. thesis, and a summary of mathematical
results can be found in 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.
solutions: If x*(λ)
is a hyperbolic equilibrium
branch of F(x, λ),
we show the existence of a particular solution of the
adiabatic system, which tracks the branch at a distance
of order ε. We call
it an adiabatic solution. It can be expanded in
a power series of the adiabatic parameter ε.
Even when the function F
is analytic, this series does not converge in general,
but can be truncated at exponentially small order.
Assume x is a scalar variable, and the origin is
a bifurcation point of f(x,τ)
= F(x, λ(τ)).
If x*(τ) is an
equilibrium branch arriving at the origin, adiabatic
solutions no longer track this branch at a distance of
order ε, but at a distance
scaling with some other power of ε.
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.
- Linear systems:
When linearized around an adiabatic solution, the
adiabatic system takes the form ε
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
manifolds: Adiabatic solutions tracking a
hyperbolic equilibrium admit stable and unstable
invariant manifolds. We showed that in a neighbourhood of
the branch, they can be represented by graphs, admitting
power series in ε. In the
analytic case, they can be truncated to exponentially
- Dynamic normal
forms: Another way to simplify a
nonlinear equation of the form εdx/dτ
= A(τ)x + b(x,τ), 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.
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 Ω. It
experiences friction, gravity and an intertial torque.
When the frequency Ω 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.
hysteresis: Consider a lattice system
with infinite range interaction (Curie-Weiss model), and
Glauber spin flip dynamics. Its magnetization obeys an
effective mean field equation. We studied this equation
for one- and two-dimensional spins and a periodically
varying magnetic field, in order to investigate the
mechanism of magnetization reversal, dynamic phase
transitions, and the shape and area of hysteresis cycles.