448
elle 1981) relied essentially on the method of delays, based in turn on the
Whitney (1936) embedding lemma and the Mane (1981)-Takens (1981)
theorems. Heuristically, here is the sequence of steps involved. It is easy
to transform a single nonlinear ordinary differential equation (ODE) of
order n,
x(n) = F(x(n-l), ... , X(j), ... , X),
into a system of first-order ODEs,
Xi = fi(Xl , ... , Xj, ... ,Xn), 1:S: i, j :s: n;
here x(n) == d n X/dtn and X == dx/dt. It suffices to write
X == Xl, Xl = X 2 , ... , Xn = F(Xl, ... ,Xn),
so that !1 = X 2 , h = X 3 ,· .. , fn = F.
(2)
(3)
(4)
Equation (3) is considered as a fairly general description of a differentiable dynamical system in continuous time (Arnold 1973, 1983) and we are
interested in the case in which, say, only a time series Xj(t) were known.
For the solutions of such a system to be irregular, i.e., other than (asymptotically) steady or periodic, three or more d-o-f are necessary. Can one
then go from (3) to (2) just as easily as in the opposite direction? The
answer, in general is "no"; hence, a slightly more sophisticated procedure
needs to be applied. This procedure, in some sense, tries to imitate the
Yule-Walker inference of (1) from X(t), t = 0, ... , N. First of all, one acknowledges that the data X(tk) = X k are typically given at discrete times
tk = kilt only. Next, one admits that, at first, it is hard to get actually
the right-hand sides fi; instead one attempts to reconstruct the invariant
set on which the solutions of (3) that satisfy certain constraints lie.
In the case of conservative, Hamiltonian systems (Lichtenberg and Lieberman 1991), deterministically irregular motion is often called "stochastic."
In this case, there are typically (unique) solutions through every point in
phase space and the irregularity is associated with the intricate structure
of cantori (Wiggins 1988), complicated sets of folded tori characterized by
a given energy of the solutions lying on them. These cantori have, in particular, finite and fractional dimension, being fractals (Mandelbrot 1982).
Hamiltonian systems, however, are - mathematically speaking - structurally unstable (Smale 1967) in the (function) space of all differentiable
dynamical systems, while - physically speaking - "open" systems in
which energy is gained externally and dissipated internally - abound in
elle 1981) relied essentially on the method of delays, based in turn on the
Whitney (1936) embedding lemma and the Mane (1981)-Takens (1981)
theorems. Heuristically, here is the sequence of steps involved. It is easy
to transform a single nonlinear ordinary differential equation (ODE) of
order n,
x(n) = F(x(n-l), ... , X(j), ... , X),
into a system of first-order ODEs,
Xi = fi(Xl , ... , Xj, ... ,Xn), 1:S: i, j :s: n;
here x(n) == d n X/dtn and X == dx/dt. It suffices to write
X == Xl, Xl = X 2 , ... , Xn = F(Xl, ... ,Xn),
so that !1 = X 2 , h = X 3 ,· .. , fn = F.
(2)
(3)
(4)
Equation (3) is considered as a fairly general description of a differentiable dynamical system in continuous time (Arnold 1973, 1983) and we are
interested in the case in which, say, only a time series Xj(t) were known.
For the solutions of such a system to be irregular, i.e., other than (asymptotically) steady or periodic, three or more d-o-f are necessary. Can one
then go from (3) to (2) just as easily as in the opposite direction? The
answer, in general is "no"; hence, a slightly more sophisticated procedure
needs to be applied. This procedure, in some sense, tries to imitate the
Yule-Walker inference of (1) from X(t), t = 0, ... , N. First of all, one acknowledges that the data X(tk) = X k are typically given at discrete times
tk = kilt only. Next, one admits that, at first, it is hard to get actually
the right-hand sides fi; instead one attempts to reconstruct the invariant
set on which the solutions of (3) that satisfy certain constraints lie.
In the case of conservative, Hamiltonian systems (Lichtenberg and Lieberman 1991), deterministically irregular motion is often called "stochastic."
In this case, there are typically (unique) solutions through every point in
phase space and the irregularity is associated with the intricate structure
of cantori (Wiggins 1988), complicated sets of folded tori characterized by
a given energy of the solutions lying on them. These cantori have, in particular, finite and fractional dimension, being fractals (Mandelbrot 1982).
Hamiltonian systems, however, are - mathematically speaking - structurally unstable (Smale 1967) in the (function) space of all differentiable
dynamical systems, while - physically speaking - "open" systems in
which energy is gained externally and dissipated internally - abound in