SOME ASPECTS OF THE THEORY OF NON-LINEAR
VIBRATIONS
M. L.
CARTWRIGHT
Much of the early work on non-linear oscillations was motivated by
an interest in the periodic solutions of the equations involved; stable periodic
oscillations were desired in the corresponding physical systems and the most
important problems were the determination of the conditions for the existence
of such oscillations, the magnitudes of the amplitudes and the way in which the
amplitude varied through resonance, and in the case of autonomous systems
a very precise estimate of the period was required. These were the early problems of the radio engineers; their systems operated at high frequencies so that
the oscillations built up to their steady state in a very short interval of time,
and the transient features were of minor importance.
I wish now to focus attention on two other classes of problem; they are by
no means new, but I believe them to be of increasing importance on account of
their applications to automatic control theory. The first is the problem of
finding conditions which exclude unwanted oscillations so that the engineer
can operate the corresponding system safely. The other is the problem of
determining how rapidly a given system will settle down to its final steady
state, or rather, what form of non-linearity within the limits available will give
the most satisfactory response to a given input signal. In the latter form the
problem seems to demand a complete mastery of the mathematical form of the
solutions of the equation as the non-linear function varies.
Let us first review the contributions of the existing theory to the problem
of excluding unwanted oscillations. Autonomous second degree equations can
be handled by phase plane and other methods, and present no great difficulty
in this respect, but difficulties arise with forced oscillations. Topological methods are seldom precise enough to exclude all unwanted oscillations and
approximations of nearly linear systems usually ignore the possibility of oscillations starting outside the range of validity of the approximations. However the
conditions for synchronization of the system represented by
(1)
x — k (1 — x2) x + x = p k cos X t, X near 1, k small,
were obtained by considering it as a nearly linear system; van der Pol and
others obtained conditions for the suppression or attraction of the unwanted
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so-called free oscillation by the forced by putting x — bx cos t + b2 sin t and
supposing that bl9 b2 vary slowly. The problem is reduced to the consideration
of first order equations in blt b2 (or corresponding equations in phase and amplitude) which are by no means nearly linear, and it is by no means easy to determine the qualitative behaviour of the solutions as A varies. In fact Mr Gillies
has shown that the results which I stated for a certain intermediate range of A
are not quite correct. The introduction of an unsymmetrical term cue, say, into
the damping coefficient makes no difference unless it is very large, but Gillies
has shown that for fixed a > 0 the solutions of
x — k (1 + a k~% x — x2) x -\- x = p k cos A t, X near 1, k small,
behave very differently from those of (1).
The systems of the automatic control engineer usually have positive
damping and at one time it was thought to be impossible for a stable periodic
solution to exist in such systems with period different from that of the forcing
term, but subharmonics of order 3 were observed in a physical system corresponding to an equation of the form
x -f k x + x — ßx3 = p cos Xt, X near 3, ß k, small.
The existence of such solutions can be verified, but this is a negative and unsatisfactory answer to the problem. However Levinson dealt with the case of a
linear restoring force and positive damping, and Littlewood and I obtained
conditions under which all solutions of the equation
(2)
'x
+ k f (x) x + g (xt k) = k p (t), p (t) periodic,
converge to a single solution. The essential features of these conditions are
first that k ^ 1, / (x) ^ 1, g (0) = 0, g' (x, k) ^ 1 which imply that
\x\ < £Q> \%\ < Vo> ^O'^o independent of k. Further
(3)
2 ^ Max \g"
(x,k)\<k,
l«l<*0
which means that if the restoring form is not too strongly non-linear in the
range considered there is no stable oscillation other than the periodic one which
is expected.
It is noteworthy that the proof depends on the fact that the damping
k f (x) is independent of x or t. Let
F (*) = J""/ (x) dx,z = x2-x1,Af
= f (a2) - / fo),A g = g (x2) - g(x1),
o
where x2 (t), xv (t) are any two solutions of (2). Then substituting xlt x2 in (2)
and subtracting we have
d IAF
\
Ag
dt \ z
I
z
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Multiplying by z and integrating, and then using integration by parts and
substituting for z in the result from (4) we obtain after some manipulation
f* z2dt = 0 (1) as* -» oo
o
when (3) holds, and from this convergence of all solutions is deduced. Further
results of this type seem highly desirable, but although the method works for
more general p (t) it seems to fail completely if / (x) depends on x or t as well as x.
The theory of synchronous motors which was studied some years ago by
Lyon and Edgerton, Tricomi and others, and again more recently, in particular
by Hayes, involves the equation
0 + a 0 + sin 0 = ß,
where a > 0, ß > 0. If ß < 1 under suitable conditions the system will synchronize; that is to say that 0 will tend to a value 0O such that sin 0O = ß. The
problem of the unwanted oscillation arises here in two distinct forms; first is
given a and ß determine ß0 if possible so that a sudden change from ß to ß1< ß0
will not give rise to an unwanted oscillation; the second is given ß < 1 find a
value a 0 such that for a < a 0 no unwanted oscillation is possible. In mathematical terms the first is a problem of determining a safe range of initial values. It
was established early that a0, ß0 exist and that they can be chosen so that for
a > a0, ß1 > ß0 the unwanted oscillations may occur. Since no synchronization
is possible for ß > 1 one might expect that a 0 would tend to infinity as ß -> 1
but Hayes has shown that
( (3 cos2 0O + 1)4 - 2 cos 0O)* < a 0 < 2 sin \ 0O,
where ß = sin 0O. Like the other writers on this subject he puts y = 0 and uses
the phase plane; he also replaces 0 by O+TC — 0O. His method depends on the
use of a comparison function
V = 2 [(Ja 2 + cos 0O)* - a / (0)] sin | 0 ,
where / (0) = 0 and / (0) is antisymmetric about 0 = n. The function g (0)
defined by
dy
y_
dy
+
ay _ y
_
a
y
=
gin 10 g(0)
can be calculated when / (0) is known and he shows that if g (0) remains positive
or negative throughout the interval 0 ^ 0 < 2 n then a > a0 or a < a 0 as the
case may be. Thus a remarkably precise quantitative result is obtained without
any use of approximations other than the comparison function y.
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Turning now to the second problem, that of the rapidity of convergence
we observe that whereas nearly linear systems converge slowly, solutions of
equations which are not nearly linear may converge with great rapidity. For
instance if a solution of
x — k (1 — x2) x + x = 0, k large,
starts at t = 0 with x = x0, x = y0 > 3/AI have shown that the solution reaches
x = 1 with x > y0 + % k — M (log k)\k, and therefore rises to a height
h > 3* given by
\y0 +
k(h-^k^)\<M(iogk)lk
where M is an absolute constant. From the maximum h the solution slowly
descends to x = 1 from which point its behaviour differs little from that of the
periodic solution.
In servomechanisms it is often possible to construct the element giving
the non-linear effect in such a way that it corresponds to a specified function
of x and x and some attempts have been made to improve response by intentional non-linearities. What seems to be needed here is a thorough grasp of the
real variable methods and principles which are used in existence theorems and in
work such as I have just described on relaxation oscillations and synchronous
motors, and although such methods may be difficult in the abstract they
correspond directly to physical principles which are often familiar to the engineer. Two simple examples of intentional non-linearities have been given by
McDonald; the equations representing the first are
x = — T for x + L x > 0,
= T for x + L x < 0.
The solutions move rapidly towards the origin but tend to overshoot it. The
second system cures the tendency to overshoot by making the switch along the
track of the solution through the origin so that
x = — Tior \x\x +
2Tx>0,
=
T for \x\ x + 2Tx < 0.
Unfortunately the physical system suffers from a tendency to other types of
instability especially if the switch does not occur precisely at the point intended,
so that some of the difficulties are engineering problems which I am not competent to discuss.
Many servomechanisms correspond to third order equations and then instead of the x, x phase plane we have to consider the x,x, x phase space.
Switching effects such as the above would probably require non linear functions
depending on x as well as x and x which may be impossible, and even the basic
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theory of oscillations in third order systems has hardly been touched. Van der
Pol, Levinson, Rauch and others have discussed particular equations, but lack
of characteristic common features in the third order equations of the engineers
seems to have discouraged the creation of a general theory. One may however
observe that, if a1 a2, a3 are positive constants, solutions of the equation
x *+ ax x + a2 x + a3 x = 0
all converge to x = 0 provided that
(5)
ax, a2 > as
This condition may be interpreted topologically by writing
x = y,
y = z,
z = — a3x — a2y — axz,
and using the theory of matrices to express this system in the form
i = ^ f,
t = A3 f,
where Xx, X2, A3 are the roots of
X3 + a1X2 + a2X + a3 = 0
By (5) the real parts of all three roots are negative; one A3, say, is real and the
other two may be conjugate complex numbers in which case writing | = re%q>,
r] = r~tq> we reduce the system to
r = - i (Xx + A2) r, i = A3 f
and an equation in cp. From which it follows that solutions cross all similar
ellipses of a certain shape with centre at the origin inwards. It is natural to ask
whether this is still the case when av a2, az are functions of x and x provided that
(6)
min (ax a2) > max a3
for the range considered, and in particular whether this is so for equations which
occur in connection with servomechanisms in which ax is constant and a2, a3
are functions of x only when (6) holds.
The theory of non-linear oscillations represented by equations of the
fourth and higher orders is even less well established from the point of view of
our two problems, and those few results which I have discussed for second and
third order equations (and others which I have omitted either from ignorance or
lack of time) are far from satisfying the demand of some engineers for a com75
plete theory comparable with the linear theory including Nyquist's criterion for
stability. That demand, in view of the extraordinary behaviour of some nonlinear systems, seems to me like asking for the moon, but I believe that there
are still many openings for the application of real variable and topological
methods especially for anyone who can appreciate the physical problems and
physical insight of the engineers.
REFERENCES
M. L. CARTWRIGHT, Journal Inst. Elee. Eng. (Radio Section) 95 (1948) 88—96.
M. L. CARTWRIGHT, Contributions to the theory of non-linear oscillations 2. Annals of
Maths. Studies 29 (1952) 3—18.
M. L. CARTWRIGHT and J. E. LITTLEWOOD, Annals of Maths. 48 (1947) 472—494
A. W. GILLIES, Quarterly Journal of Mech. and Applied Maths., 7 (1954) 152—167.
W. D. HAYES, Zeitschrift f. angewandte Math, und Physik (Zürich) 4 (1953.) 398—401.
D. C. MCDONALD, Proc. of the symposium on non-linear circuit analysis (New York 1953)
402—411.
ADDENDUM
Since the Congress I have found t h a t the problem of third order equations satisfying
(6) has to a large extent been solved by E. A. Barbashin, On the stability of the solution of
a certain non-linear third order equation, Akad. Nauk. S.S.S.R. Priklad. Mat. i Mekhan.
(1952) 16(5) 629 — 632 (See Math. Reviews 14 (1952) 376. An English translation has been
made by the Department of Scientific and Industrial Research, C.T.S. No. 57, Technical
Information Service, Charles House, 5—11 Regent St., London, S.W.I.) Further papers by
Malkin and Erugin on related topics are contained in the same volume of the periodical.
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