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Unformatted text preview: Chapter 6 Nonlinear conservative
systems The goal of this chapter is to formally introduce you to some adavnced
concepts in the theory of nonlinear systems, especially the study of
nonlinear systems that depend on a parameter. This will naturally
lead us to the notion of catastrophe and bifurcation. 6. 1 Local properties The simplest conservative system may be written as the secondorder
equation 5: 2 f0”): (6'1) where :1: usually is the position of, say, a mass and f is the sum of forces
acting on this mass. Studying the trajectories of this system is made
especially easy by multiplying the equation (6.1) by (i: to get 1d '2 — i /f(a) dd. (6.2) 551; __ dt 0
Using the canonical coordinates m1 = 3:, 1:2 2 :r, and noting V(:1:) =
— f0” f(cr)da, we can integrate (6.2) to obtain) the integral equation 2
E 83 84 CHAPTER 6. NONLINEAR CONSERVATIVE SYSTEMS where h is an arbitrary constant. In fact, in most cases the term as; / 2
may be intrepreted as a kinetic energy, whereas the term V(:1:1) is inter
preted as a potential energy. 17. is then the total energy for the system
and it stays constant throughout the motion. Therefore, any trajec tory has to lie on an equienergy curve deﬁned by (6.3). The equilibria
of conservative systems correspond to the points where :02 = 0 and'
f(a:1) = 0. Thus, equilibria correspond to the stationary values of the
potential energy V(m). There are only three possibilities, which are
for V to either be a minimum, a maximum or an inﬂection point with
horizontal tangent. Thus, the singular points of a conservative system
may be classiﬁed in terms of the extremal properties of the potential
energy at the singular points. In order to construct the phase portrait for conservative systems, one may immediately notice that (6.3) implies
the phase portrait needs to be symmetric with respect to the mlaxis. Assume that the potential energy of the system, V(:c) is given. Then, writing 2:2 33 = h — V(:z:1), (6.4)
it is possible to construct the equienergy curve corresponding to a ﬁxed
value of h, and therefore the phase portrait. For each type of singularity,
it is possible to build the corresponding shape of the equienergy curves
in the vicinity of that equilibrium. Starting with minima, we see that the integral curves built in the
vicinity of a minimum of the potential energy are closed curves, centered
around the stable equilibrium of the system as shown in Fig. 6.1. These
curves each correspond to limit cycles for speciﬁc values of the energy
h. To determine the shape of these curves, one can rely on expansions
of f and V. Assume that f can be expanded about an equilibrium point with (:10). We then can write
f(m)=a1(m—i)+% m—5)2+—(a:—5:)3+~~ V(m) may therefore be expanded as 6.1. LOCAL PROPERTIES 85 Figure 6.1: Stable equilibrium for a conservative system such that locally around an equilibrium, the equienergy curves need
to satisfy the equation 2
2:2 (11 _ 2 02 _ 3 a3 — 4
— h ——:1: —:1: —:1: —:1: +—a: —:1: —...=h.
2+02!(1 ) 3!1 ) 411)
If the equilibrium is stable, then the first nonzero coefﬁcient a. must be
with odd index and negative. If the ﬁrst nonzero coefﬁcient is (11, then
we see that an approximation of the equi—energy curves near equilibrium IS
2 E2E+ho—%(cc1—i)2=h.
Whenever nonempty, these curves are ellipsoids, and the corresponding
motion is an harmonic motion. The period of the motion is then inde
pendent from its amplitude, and the system locally looks like a linear
massspring system. If a1 is 0 and the equilibrium is stable, then we
know a2 = 0 and we look for a3. a3 nonzero and negative is illustrated
by the massspring system shown in Fig. 4.10. In general, the equations
characterizing the equienergy curves for a stable equilibrium are of the 86 CHAPTER 6. NONLINEAR CONSERVATIVE SYSTEMS Figure 6.2: Unstable equilibrium for a conservative system form
(cg 2
3+K3(21—§})p=h—ho, and it is easily seen that, as p tends towards inﬁnity, these curves look
more and more like sausages elongated along the zlaxis. Also, the pe
riod of these closed trajectories increases as their amplitude decreases. When the equilibirum is unstable, then the equienergy curves may
again be computed using the equation (6.4), as shown in Fig 6.2. We
now see that depending on the energy level h, the equienergy curves
in the vicinity of 0 do actually change shape. The separating curves
are the ones that go through the origin. Using the expansion (6.5) of
V around its maximum, we see that an unstable equilibrium can take 6.1. LOCAL PROPERTIES 87 place only if the ﬁrst nonzero term of the expansion must be with odd
index and positive. The general form of the equations for the equi
energy curves around the equilibrium is 2
3’3 2 — Ic(m1 — (12—1)”) 2 h — ho. when p = 1, this corresponds to a family of hyperbolas whose focuses
are either on the 3:1 or $2 axis. In general, the separatrices curves are
then the two lines satisfying 2
ﬂ 2 —K,(:I:1——52p=0, which in the case p = 1 gives two lines with opposite slopes passing through the equilibrium point. When p > 1, this family is still made of hyperbolalike curves. However, the shape of the separatrices is now given by '
1'2 = i2I€($1 — (2)1’, and hence the two separatrices are tangent to the wlaxis.
Equienergy curves for a saddle point have been drawn in Fig. 6.3.
Whenever the potential function V(a:) can be expanded about a saddle
point, it is easy to see that its ﬁrst nonzero coéfﬁcient must be with
even order. The separatrix then must satisfy an equation of the form 01' We can now state the two basic theorems relating stability of an
equilibrium and the related potential energy V. We ﬁrst have La
grange’s theorem, which says: If in a state of equilibrium, the potential energy is minimum, then
the equilibrium is stable. We then have Lyapunov’s converse theorem: If in a state of equilibrium the potential energy is not a minimum,
then the state of equilibrium is unstable 88 CHAPTER 6. NONLINEAR CONSERVATIVE SYSTEMS Figure 6.3: Equienergy curves for a. saddle point 6.2. GLOBAL PROPERTIES 89 6.2 Global properties 6.2.1 Separatrices We now are interested in the global behavior of the conservative system .17 = ﬂy), once again by basing our analysis on the ﬁrst integral of this equation, given by
 2 y? +V(y) = h. Depending on the relative positions of the curves 2 = V(a:) and z = h,
several types of motion may occur for different values of the potential
function as shown in Fig. 6.4. In particular, let us see what happens as
we start from a high value of the total energy It and progressively let
it go down. Here, we make the (not so bad) assumption that nothing
really interesting happens outside the ﬁgure. 1. The line 2 = h never intersects the curve 2 = V(:c). if the line
2 = h constantly is below the line 2 = V(:I:), then no motion can
take place. If it lies above, then the resulting motion never has
zero speed, and the two possible trajectories are symmetric with
respect to the mlaxis. These are the dashed surves in Fig. 6.4.
As time tends to +00 or —00, such motions go to inﬁnity as well,
and they are named runaway motions. It is easily seen that the
nature of such motions does not change with small perturbations of h. 2. The line 2 = h tends towards the curve 2 = V(:z:) as 1: tends to
inﬁnity. The resulting motion is once again inﬁnite; however, as
the position tends towards +00, speed goes down to 0. These are
the fat dashed curves in Fig. 6.4. Note that this situation is not
generic: if h moves up, then the motion changes (speed does not
tend to 0), if h moves down, then the motion stops being inﬁnite,
as we see next. 3. The line 2 = h intersects the curve 2 = V(a:) on one point: We
now see that the corresponding trajectory is bounded on the right, 90 CHAPTER 6. NONLINEAR CONSERVATIVE SYSTEMS Figure 6.4: Building separatrices for conservative systems 6.2. GLOBAL PROPERTIES 91 while it is inﬁnite on the left: states come from inﬁnity on the
left with positive speed and then return to inﬁnity to the left,
as shown by the continuous trajectories in Fig. 6.4. Note this
situation is generic, that is the trajectories stay the same under
small perurbations of h. 4. The line 2 = h becomes tangent to one point of V(z). Another
set of trajectories now occurs: Previously unique trajectories now
split into 4 distinct trajectories, shown in fat continuous curves.
More precisely: one possible trajectory is the equilibrium point
sitting on the local maximum of the potential function. The
trajectory coming from —00 to the left with positive speed dies
against that equilibrium. And the symmetric trajectory origi
nates from the same equilibrium to go to ~00. Finally, there is
a closed trajectory initiating from the equilibrium and returning
to it. Note that this situation is nangeneric: it does not keep
holding under small perturbations of h. 5. The line 2 = h intersects the curve 2 = V(a:) in three points. The
set of possible trajectories now reduces to two, as shown by the
dotted curves in Fig. 6.4: We have one inﬁnite trajectory to the
left, and then one limit cycle. This situation is generic. 6. The line 2 = h becomes tangent to the local minimum of the
potential curve: there, once again only two trajectories may exist:
There is an inﬁnite trajectory to the left, and there also is a single
equilibrium trajectory (a point). This situation is nongenem'c. 7. The line 2 = h cuts the curve 2 = V(a:) one one point and there is
only one possible trajectory shown by the leftmost dashed curve. This detailed analysis of the behavior of the conservative system shows
that several phenomena occur: as h varies, the possible trajectories
usually do not change nature, except at those special levels where z = h
becomes tangent to the potential function. Note these events are hard
to observe, since they happen only at speciﬁc values of h. However, it
sufﬁces to know what the trajectories are at those points to be able to
“interpolate” what all the other trajectories should be. These special 92 CHAPTER 6. NONLINEAR CONSERVATIVE SYSTEMS trajectories form the “squeleton” of the whole phaseplane portrait.
The trajectories emanating from the equilibria are named separatrices.
Separatrices and equilibria are enough to build the rest of the phase
portrait. They operate as “dividing” curves which separate regions
with paths of different types. 6.2.2 Introduction to bifurcations: Dependence
on a parameter Let us now look at the previous problem from another angle: as h was
varied from high to low values, it crossed certain very speciﬁc values
(corresponding to equilibria and separatrices), where the aspect of the
trajectories fundamentally changed nature: such events are generically
named catastrophes, and they are directly related to the problem of
computing the roots of V(z) = h as h varies. Each time the number
of real roots to that equation changes, the nature of the corresponding
trajectories changes as well. Note the similarity of this problem with
the rootlocus problem. The problem is then to study when the roots
of 1 + KG cross the jwaxis. Let us see now how this notion of catastrophe may be extended
to study families of conservative systems: in reallife applications, it
is often that dynamical systems are uncertain, because some parame
ters that enter them are unknown. Thus it is important to know how
the system should behave for all possible values of the parameter. In
particular, we may expect to see qualitative changes in the phase por
trait at some speciﬁc values of the parameter, while for most values of
the same parameter, we will only encounter quantitative changes of the
phase portrait. These speciﬁc values of the parameter will be named
bifurcation or branch values of the parameter. We now show how to
perform the bifurcation analysis of conservative systems. However, the
bifurcation concept extends to nonconservative systems as well, at the
expense of many more computations. For purposes of simplicity, we
limit ourselves to studying the case of a dynamical system that de
pends on a single parameter only: 51': = f(a:,/\), 6.2. GLOBAL PROPERTIES 93 Figure 6.5: Generic bifurcation diagram and we ask how the phaseplane portrait of the system changes as A
is varied. From the preceding developments, we know that the phase
portrait of a conservative system is entirely determined by the position
of the maxima and minima of V, that is, by the zeros of f. Thus, a
qualitative change in the phase portrait should occur for those value of
A for which the pattern of the zeros of f changes. A plot that shows
the position of the zeros of f vs. the value of the parameter A is named
a bzfurcation diagram. Assume for example the bifurcation diagram
shown in Fig. 6.5. Then to obtain the number of zeros of f, we just
need to count the number of intersections of a vertical line A = A0.
Bifurcations occur when the number of roots for f changes. in this
case, bifurcations occur at points A, B, and C. In order to characterize analytically how the roots of f vary with
the parameter A, we can differentiate the equation f(:n, A) = 0
to obtain (if d df
a:
aa+n=°
Thus, we have
if _ _fi(9=,/\)
dA " f;(:v, ,\ ‘ So, if for a given value A = AD, the system of equations f(m,A) = 0,
f;(m,A) = 0 has no real solutions for 1:, then in a. neighborhood of 94 CHAPTER 6. NONLINEAR CONSERVATlVE SYSTEMS A : A0, the positions of all equilibria are continuous, differentiable
functions of A and A0 is not a bifurcation point. Assume now that at
a given value of A and :1: we have f(:c,A) = 0 and also f;(a:,A) = 0.
If f;(m,A) = 0 and ff\(:c,A) 75 0, then the curve has at this point a
vertical tangent. This corresponds to two values of zero curve merging
and then becoming complex. If f,’\(a:, A) also vanishes, then we are in 
the situation shown at point A and we also have a bifurcation. Note
the similarity with rootlocus theory for linear systems! In order to determine the stability of a point of equilibrium, we need
to check the value of the second derivative of V, or, in other terms, the
ﬁrst derivative of f with respect to m. f;(m,A) > 0 corresponds to a
minimum of the potential energy, while f;(a:,A) < 0 corresponds to a
maximum. Determining graphically the stability of equilibria may be
done by shading those areas where f (at, A) is positive: every zero located
under such a region will be unstable, whereas every zero located above
such a region will be stable. Note that the bifurcation diagram does provide some, but not all
bifurcations fOr dynamical systems. Rather than going through an
exhaustive'list, we will present how the other bifurcations, which relate
to the behavior of the separatrices occur in a practical example. 6.2.3 Example: Motion of a mass along a circle
that rotates about a vertical axis (watt reg ulator) We consider a mass m along a circle of radius a. The circle rotates
about its vertical diameter with constant angular velocity 9. The mass
is subject to gravity, such that, noting 6 the angular position of the
mass, the equation of motion for the mass is d29 .
metde 2 mflza2 sin 0 cos 0 — mga s1n (9.
Introducing the dimensionless parameter A = g/Qza and performing the
change of time tmw 2 (2t, we obtain the nondimensionalized equation of motion d+(A—cos0)sin0 =0. 6.2. GLOBAL PROPERTIES 95 Figure 6.6: Bifurcation diagram. Thus, the potential function is V(9) = —A cos 0 —— gsinz 0. To study the bifurcations of this system, we plot the zeros of (c0309) —
A)sin0 as a function of A (these solutions are periodic with period
27r and we will limit ourselves to the interval [—7r 7r]. It is obvious
that the solutions 0 = (—71", 0, 7r) always exist. Now for A S 1,
there also exist solutions 0 = :1: cos"1 A, where the function cos‘1 maps
the segment [—1 1] to the segment [—7r 7r]. The resulting bifurcation
diagram is shown in Fig. 6.6, and the shaded regions represent the
regions where the derivative of V is negative. From this diagram, we
may now immediately see that we have bifurcations for A = ——1 and z 1. Note that the variations of A may either be interpreted as the
variations of Q or g. In particular, A g 0 corresponds to either zero
or negative gravity: the pendulum is upside down. When A > 1, the
system has two equlibria: a. centre at 0 = 0, 9 = 0, and a saddle point
at 6 = :lzﬂ', 6 = 0. When —1 < 9 < 1, the system has four singular
points: there are two centres at 0 = :1: cos'1 A, 0 = 0 and two saddle points at 9 = 0, 0 = O and 0 = 21:7r, 0 = 0. Finally, when A < —1, 96 CHAPTER 6. NONLINEAR CONSERVATIVE SYSTEMS we again have two equilibria only: a centre at 9 = d:1r, 9 = 0 and a
saddle point at 9 = 0, 9 = 0. To determine the separatrices’ behavior, we will use the fact that
each separatrix passes through a saddle point, for which the corre
sponding value of the total energy h may easily be evaluated. For
A > 1, there is one saddle point 9 = i7r. The corresponding energy
level is then —Acos7r — 0.5 sin2 7r, that is, A. Therefore, the equation
characterizing the separatrix is 392 — £(sin2 9 + 2A cos 9) = A,
2 2
or, in other terms,
92 = sin2 9 + 2A(1+ cos 9). The phase portraits for all values of A are given in Fig. 6.7. When
A : 1, the phase portrait actually does not change. Writing down the
potential V = sin2 9 + 2(1+ cos 9), and expanding it about 0, we obtain
V = 2/4l94 + C(96); thus, we are in this situation where the system is
stable around 0, yet the oscillations are not harmonic. You can easily
check this is a case where the eigenvalues of the linearized model are
both zero. We see that, in addition to 1 and —1, A = 0 is also a branch value
for the phase portrait, corresponding to the change of relative sizes
between unstable equilibria. Eventhough it is hard to imagine, it suffices once again to know the
phase portraits for the branch values of A to be able to “interpolate”
the phase portraits for the intermediate values of A. //'\ 6.2. A>1 0<A<1 —1</\<0 A<~1 GLOBAL PROPERTIES 97 <‘.5——'———— O
—5 2 4 ’ Figure 6.7: Phase portrait as A changes. ...
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 Spring '04
 EricFeron

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