and if e1,…,eM
are the roots of the equations
it follows from (4) that
The right side of (7)
vanishes only on the surface of an ellipsoid E, and
is positive only in the interior of E. The surfaces
of constant Q are concentric spheres. If S denotes
a particular one of these spheres whose interior R contains
the ellipsoid E, it is evident that each trajectory
eventually becomes trapped within R.
3. The instability of nonperiodic flow
In this section we shall establish
one of the most important properties of deterministic nonperiodic
flow, namely, its instability with respect to modifications
of small amplitude. We shall find it convenient to do this
by identifying the solutions of the governing equations
with trajectories in phase space. We shall use such symbols
as P(t) (variable argument) to denote trajectories,
and such symbols as P or P(t) (no argument
or constant argument) to denote points, the latter symbol
denoting the specific point through which P(t) passes
at time t0.
We shall deal with a phase space Γ
in which a unique trajectory passes through each point,
and where the passage of time defines a continuous deformation
of any region of Γ into another region, so that
if the points P1(t0), P2(t0),
… approach P0(t0) as a
limit, the points P1(t0+τ),
P2(t0+τ), must approach P0(t0+τ)
as a limit. We shall furthermore require that the trajectories
be uniformly bounded as t→ ∞; that is,
there must be a bounded region R, such that every
trajectory ultimately remains with R. Our procedure
is influenced by the work of Birkhoff (1927) on dynamical
systems, but differs in that Birkhoff was concerned mainly
with conservative systems. A rather detailed treatment
of dynamical systems has been given by Nemytskii and Stepanov
(1960), and rigorous proofs of some of the theorems which
we shall present are to be found in that source.
We shall first classify the trajectories
in three different manners, namely, according to the absence
or presence of transient properties, according to the stability
or instability of the trajectories with respect to small
modifications, and according to the presence or absence
of periodic behavior. Since any trajectory P(t) is
bounded, it must possess at least one limit point P0,
a point which it approaches arbitrarily closely arbitrarily
often. More precisely, P0 is a limit point
of P(t) if for any ε > 0 and any time t1
there exists a time t2(ε,t1)>t1
such that |P(t2)-P0| <
ε. Here
|
absolute-value signs denote distance in phase
space. Because Γ is continuously deformed as t
varies, every point on the trajectory through P0
is also a limit point of P(t), and the set of limit
points of P(t) forms a trajectory, or a set of trajectories,
called the limiting trajectories of P(t). A
limiting trajectory is obviously contained within R in
its entirety.
If a trajectory is contained among its own
limiting trajectories, it will be called central; otherwise
it will be called noncentral. A central trajectory
passes arbitrarily closely arbitrarily often to any point
through which it has previously passed, and, in this sense
at least, separate sufficiently long segments of a central
trajectory are statistically similar. A noncentral trajectory
remains a certain distance away from any point through which
it has previously passed. It must approach its entire set
of limit points asymptotically, although it need Dot approach
any particular limiting trajectory asymptotically. Its instantaneous
distance from its closest limit point is therefore a transient
quantity, which becomes arbitrarily small as t→
∞.
A trajectory P(t) will be called
stable at a point P(t1) if any other trajectory
passing sufficiently dose to P(t1) at time
t1 remains dose to P(t) as t→
∞; i.e., P(t) is stable at P(t1)
if for any ε > 0 there exists a δ(ε,t1)
such that if |P1(t1) – P(t1)|<δ
and t2>t1, |P1(t2)
- P(t2)| < ε. Otherwise P(t) will
be called unstable at P(t1). Because
Γ is continuously deformed as t varies,
a trajectory which is stable at one point is stable at every
point, and will be called a stable trajectory. A trajectory
unstable at one point is unstable at every point, and will
be called an unstable trajectory. In the special case
that P(t) is confined to one point, this definition
of stability coincides with the familiar concept of stability
of steady flow.
A stable trajectory P(t) will be
called uniformly stable if the distance within which a neighboring
trajectory must approach a point P(t1), in
order to be certain of remaining dose to P(t) as t→
∞, itself possesses a positive lower bound as t1→
∞; i.e., P(t) is uniformly stable if for any
ε > 0 there exists a δ(ε) > 0
and a time t0(ε) such that if t1>t0
and |P1(t1) – P(t1)|<δ
and t2>t1, |P1(t2)
- P(t2)| < ε. A limiting trajectory
P0(t) of a uniformly stable trajectory P(t)
must be uniformly stable itself, since all trajectories
passing sufficiently dose to Po(t) must pass arbitrarily
dose to some point of P(t) and so must remain dose
to P(t), and hence to P0(t), as t→
∞.
Since each point lies on a unique trajectory,
any trajectory passing through a point through which it has
previously passed must continue to repeat its past behavior,
and so must be periodic. A trajectory P(t) will
be called quasi-periodic if for some arbitrarily large
time interval τ, P(t+τ) ultimately remains
arbitrarily close to P(t), i.e., P(t) is quasi-periodic
if for any ε > 0 and for any time interval τ0,
there exists a τ(ε,τ0)
> τ0 and a time t1(ε,τ0)
such that if t2>t1, |P1(t2+τ)
- P(t2)| |