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Phase portraits of planar control systems
Nonlinear Ana/ysis,
Theory, Methods & Applications,
Vol. 27, No. IO, pp. 1177.1197, 1996 Copyright G 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+0.00
Pergamon 0362-546X(95)00129-8
PHASE PORTRAITS JAUME
OF PLANAR
LLIBREt
CONTROL
SYSTEMS
and JORGE SOTOMAYORS
t Departament de Mattmatiques, Universitat Autbnoma de Barcelona, Bellaterra, 08193 Barcelona, Spain; and $ Institute de Matematica e Estatistica, Departamento de Matematica Aplicada, Cidade Universitaria, Rua do Matao, 1010, 05508-900~Sao Paulo, Brazil (Received 7 June 1994; received for publication
6 July 1995)
Key words and phrases: Nonlinear control systems, phase portrait, bifurcation
diagram.
1. INTRODUCTION
Systems of the form x’ = Ax + cp(k * x)b,
(1)
where A is a n x n real matrix and x, k, b are in IR” and “. ” denotes the usual inner product, are of great importance in direct control [l]. The divergence from linearity of these systems results from the presence of the characteristic function v, of the control mechanism. The purpose of this mechanism is to improve the asymptotic stability behavior of the equilibrium located at the origin. The reader is referred to Lefschetz [l] and to Narendra and Taylor [2], for details in control theory. In this paper, we will focus our attention to the class 5s of two-dimensional systems (l), i.e. n = 2, such that (a) have the origin as an asymptotically stable equilibrium point; (b) the characteristic function cp has the form p(u) = -24,
for u 5 -24;
p(u) = u,
for -24 5 u 5 24;
p(u) = 24, for u 5 u,
where u is positive and fixed throughout this paper. Such systems will be referred to as fundamental systems (FS). They are symmetric with respect to the origin. That is, their solutions are exchanged under the mapping x + -x. Orbits which are invariant under this mapping will be called symmetric. The characteristic function rp induces a partition of lR2 into three open strips and two straight lines, as follows: S- = (x:x-k
< -u),
So = (x: --u < xv k < u),
I-- = {x :x - k = -u},
Therefore,
the FS splits into the following
S, = (x:x-k
> u)
I-+ = (x : x - k = u).
linear systems:
x’ = Ax - ub 3
in .5- UT
x’ = Ax + (k . x)b,
in I- Us,
x’ = Ax + ub ,
in S+ 1177
(2. -1
ix-+
7-q.
(2.0) G.+)
1178
J. LLIBRE
and J. SOTOMAYOR
Notice that fundamental systems are of class C’ on lR’\(lY UT+) but only of class Co on IR’. Nevertheless, they satisfy a Lipschitz condition (with L-constant smaller than or equal to IIAIj + [Ibk’jl) on the whole lR2. This fact makes applicable to these systems the classical theorems on existence, uniqueness and continuity on initial conditions and parameters. Their solution curves are in general C’, but not C2. The main goal of this paper is to describe all the phase portraits of FS, and to distinguish among their different qualitative patterns in terms of their basic parameters: t = Trace(A), T = Trace(B) < 0,
d = Determinant(A);
D = Determinant(B)
> 0,
where B = A + bk’.
(3.1) (3.2)
The inequalities express condition (a) in 5s above. An elementary result in this direction which originates in [3], proved in Section 2 for the sake of completeness, is the following classification of equilibria of FSs. 1. If d L 0, the origin is the unique equilibrium If d < 0, there are three equilibria for the FS:
PROPOSITION
of the FS.
0 in So, which is always an hyperbolic attractor, and e += -d-lb, in S+, and e- = uK’b, in S-, which are both saddle points. With this information in hand, the determination of the phase portraits of FSs will now depend on their periodic orbits, which must enclose the origin and therefore be always symmetric, and their saddle separatrices (present only when d < 0). Sometimes these separatrices are connected, in pairs due to the symmetry, forming a graph surrounding the origin, whose corners are the saddles e, and e- . These graphs will be called double saddle connections (DSC). The possibility of appearance of single saddle connections (homoclinic loops) with corners e, or e- is easily ruled out from the symmetry of the FS and the fact that both must surround the origin and be symmetric to each other. This establishes a second elementary property of FSs pointing out towards the determination of phase portraits of FSs. PROPOSITION 2. The unique simple closed curves invariant under the flow of a fundamental system are either periodic orbits or double saddle connections.
In this paper, we will be particularly concerned with the form and size of the basin of or stable manifold, of the origin, Q(O), of a FS. This element is of considerable interest in control theory [ 1,2]. Our main results will be summarized in what follows. attraction,
A. The following statements hold for a FS of the form (1): (a) if it has a periodic orbit or a double saddle connection, then t > 0. Moreover, situations cannot coexist on the same FS; (b) it has at most one periodic orbit, which moreover must be unstable. The space of all basic parameters of FSs will be denoted by a: THEOREM
@ = ((f, d; T, D) E R4 : with T < 0, D > 0).
both
Phase portraits of planar control systems THEOREM B. If the basic parameters
1179
are in the set
a = ((t, d; T, 0) E 63 : t I 0 and d 2 0), then the basin of attraction,
Q(O), is the whole plane.
THEOREM C. If the basic parameters are in
e = l(t,d;T,D)E63:t>OanddrO), then the FS has a unique periodic orbit which is unstable. It is the common boundary of n(O), the basin of attraction of the origin and Q(W), the basin of attraction of “infinity” formed by the orbits which tend to infinity as time grows. THEOREM D. In &3- = ((t, d; T, 0) E CR: d < 0), there is an analytic regular hypersurface, 3, contained in the half-space lt > O), whose boundary in 03 is &3,,,,= {t = 0, d = 01. If the basic parameters are on the hypersurface a>, then the FS has a double saddle connection which bounds Q(0). The phase portrait is illustrated in Fig. l.a>. The hypersurface D separates the open set Se c GJ-, whose boundary in 63 is contained in the half-hyperplane rZS, = (t > 0, d = O), from the open set @S c 03-, whose boundary on 63 is contained in the half-hyperplane @S, = (t < 0, d = 0). If the basic parameters are in QS, the phase portrait has an attractor and a pair of saddles, as illustrated in Fig. 1.024. If the basic parameters are in Se, the phase portrait has a repelling cycle and a pair of saddles, as illustrated in Fig. l.SC. d
t
\ Fig. 1. Bifurcation
1.D
diagram for fundamental systems in the basic parameter space.
1180
J. LLIBRE
and J. SOTOMAYOR
A word of explanation on the hypersurfaces a, as,, CS,, CE, = (t = 0, d > 0) in Fig. 1. They represent the codimension one bifurcations of FSs. That is, where the transition between the distinct structurally stable phase portraits happens. The notation suggests the qualitative change. On W?, the cycle in (9 disappears at infinity. Meanwhile, on @S, and (78, the saddles in G.8 and eS disappear at infinity. Explicit analytical expressions for 9 and for the regions @.S and SC, in terms of basic parameters of a FS, can be found in Section 6. Some aspects of the existence and bifurcation diagrams of periodic orbits established for two dimensions in this work, have been extended to higher dimensional control systems in 141. This work is organized as follows. Section 2 is devoted to establish some preliminaries including Jordan normal forms for FSs and the proof of proposition 1. In Section 3 we study periodic orbits of FSs and prove theorem A. Sections 4, 5 and 6 deal with the proofs of theorems B, C and D, respectively. 2. PRELIMINARIES
In this section we establish some basic properties of FSs for later use. 2.1. Normal forms Consider the fundamental system x’ = Ax + p(k . x)b. The linear change of variables x = My which reduces A to its real Jordan form J = M-‘AM, transforms the FS into y’ = Jy + p(Mfk.
y)M-‘b.
Renaming the transformed parameters M’k by k and M-lb variable y by x, the system can be written back as
by b, and the transformed
x’ = Jx + p(k * x)b, where J is one of the following
forms: if d = 0, t = 0; if d = 0, t # 0; if d > 0, t = 0; with sign(&) = sign(a) = sign(t), and p > 0, if d > 0, t # 0; with I, < 0 < &,
ifd
Notice that the null matrix has been omitted from this list. In fact, having a segment of the straight line k * x = 0 as equilibria, it is not compatible with definition (a) of FS.
1181
Phase portraits of planar control systems
2.2. Proof of proposition
1
See Section 1 for the statement and notation adopted. Case 1. Assume that d # 0. From (2.+) it follows that if e, E S+ it will be the unique equilibrium point of the FS on S+ U I+. Similar statements hold for e- in .S- U l--.
We have the following k * (-uJ-lb)
iff
d-‘(a22blk,
equivalences: e, E S+ and e- E S- iff > u and k * (uT1b) < -u, iff k * (J-lb) - alzbZkl
- a,,b,kT
+ a,,bzk2)
< -1,
< -1,
where J = (aij).
Now if d > 0, we get that e, E s+ and e- E S- iff d + (a22blkl
- a,,b,k,
- a,,b,k,
+ a,,bZk2) < 0.
On the other hand, if d < 0 then e, E S+ and e- E s- iff d + (azzblk,
- alzb,k,
- a,,blkz
+ a,,b2kJ
> 0.
But the left-hand member of this inequality is D. Therefore, 0 is the unique equilibrium point when d > 0, and 0, e, and e- are the unique equilibrium points when d < 0. The saddle character of e, and e- is obvious from the form of the FS outside S,,, Case 2. Assume that d = 0. We have only two possibilities for the Jordan form J
(:
i)
or
(“,
i),
A.0.
If the first case holds, we must have that bi # 0 and k2 # 0, otherwise the FS would have a segment of straight line of equilibria through the origin. It follows easily that 0 is the unique equilibrium of the FS. If the second possibility holds, similar arguments to those used in the previous case imply that b, # 0 and kz # 0 and therefore 0 is the unique equilibrium. This ends the proof of proposition 1. H 3. PERIODIC
ORBITS
OF FS
There are two main aspects in the study of periodic orbits of planar differential systems. The first one concerns their existence, stability, relative positions and cardinality. The other one considers their bifurcations, i.e. their dependence on parameters of the system, their occurrence and disappearance, as parameters change. The main purpose of this section is to prove theorem A. For doing so, both aspects will come into play. In proposition 2 it was established that, due to their symmetry, FSs do not admit homoclinic loops. Instead they admit double saddle connections (DSC) and periodic orbits. Their location in terms of the basic parameters is constrained by next proposition. PROPOSITION 3. If a fundamental system has a periodic orbit or a double saddle connection, then its basic parameter t = trace(A) is greater than 0.
1182
J. LLIBRE
and
J. SOTOMAYOR
Proof. Since on S = r+ U So U I?- the fundamental system has a linear attractor at the origin, the unique possible segments of straight lines invariant under the flow and contained in S must be on straight lines passing through the origin. Therefore, a segment on the lines r+ and l7cannot be part of an orbit of a fundamental system and, consequently, the integrals
ii 0
and
Xl~2--X2~,
ss E
(ax,/ax,
+ dX,/dX,)
dx, dxz
are well defined for a fundamental system whose components are X, and X,, o is a periodic orbit or double saddle connection and Z is the region bounded by CI. From Green’s formula, the above integrals are equal. Since X, dxz - X, dxr = 0 along cr, both integrals must vanish. So, from expression (3) we obtain
55 ~n(s-ns+)
tdX1dX, +
Tdqdq IS
= 0,
Ens,
or equivalently, t Area[E fI (S- U S,)] + TArea(C f~ S,) = 0. Then, since from (3.2) T
c
0, it follows that
t
> 0.
n
Remark. The above proof uses arguments motivated by those of the Bendixson-Dulac criterion for the absence of periodic orbits of vector fields on regions where the divergence has definite sign. See for instance [5, p. 1951. The knowledge of conditions which imply the uniqueness of limit cycles of planar differential system is still very limited. Our next result shows that fundamental systems with periodic orbits can be written in the form of LiCnard systems, for which a practical criterion for uniqueness will be established. PROPOSITION
4. Assume that a fundamental
system
x’ = Jx + q$k * x)b
(4)
has a periodic orbit or a double saddle connection (DSC) and let d, t, D, T be its basic parameters. Then after a linear change from the variables x = (x1, x2) to the variables (x, y) and a reversing of the time variable, the FS (4) can be written in Lienard form Y’ = -g(x),
x’ = y - F(x),
where g(x) = dx + d-9@
- d)
and
F(x) = tx + q$x)(T - t).
Proof. By proposition 3, t > 0. There are three cases to consider, according to the real Jordan form J of the matrix A of the FS.
Phase
portraits
of planar
control
systems
1183
Case 1. J = Diag(l,, A,), A, 2 A2 and A, + A2 = t. In this case, kr and k2 are both different from 0. Otherwise, if, say, kl = 0, then xi = A++ + cp(k2x2)b2. Therefore, the straight line x2 = 0 is invariant under the flow of the FS. Since the lines I?- and I + are parallel to it, we have a contradiction with the fact that the fundamental system has a periodic orbit or a DSC. We also have that A, # A,. Otherwise, the matrix B would also be a diagonal multiple of the identity and all the segments of straight lines passing through the origin and contained in S0 would be invariant under the flow of the FS, which therefore could not have a periodic orbit or a DSC. Thus the matrix
A,k;’ M
=
1411
-
A,)
-A
-k;’
k-l 2
2
1
k-l 2
is well defined and invertible. Now, the change of variables
x=M
* 0Y
(5)
sends FS (4) to
(“y>’ = M-VM(Y~)
+ p(k.M(;))M-lb
which amounts to
x’ = tx - y + q?(x)(T- t),
y’ = dx + p(x)@ - d).
(6)
Changing the sign of the time variable ends the proof in case 1.
Case2. Suppose that for 2o = t and /I > 0, J= We write
the FS in complex
a P -p a [ 1*
notation
as follows.
For
z = x1 + ix2, 1 = (Y + iB,
k = k, + ik2, b = bl + ib,, it becomes: z’ = Iz + &z + kZ)b. Now we consider the complex differential
system:
z’ = jz + q(kz + kZ)b, (7)
2’ = AZ + yl(kz + k?)b.
Notice that this system essentially consists of two copies of the initial FS. Now we can follow the same steps as in case 1. Since e + k$ # 0, it follows that k and consequently k are nonzero. Here clearly, A f i, and the matrix
I
J. LLIBRE
1184
is well defined and invertible.
and J. SOTOMAYOR
Again, performing
the change of variables
where x and y are complex, sends system (7) to (6). Now it is easy to check that if we choose the real part of the solutions x(t) and y(t) of (5) and go back through the change of variables, we obtain the solutions of the initial FS. So we can consider that the variables x and y are real in system (6), and that such system is equivalent to the fundamental system (4). Finally, reversing the time variable concludes the proof in case 2. Case 3. Assume that
with t = 2A. Then k2 # 0; otherwise the x,-axis would be invariant under the FS, since Iand r+ are parallel to such axis, the FS could not have a periodic orbit or a DSC. Therefore, the matrix
-4k 1
is well defined and invertible. system (6). H
1
After the change of variables (5) the FS (4) goes over to
Using the Lienard form in proposition 4, we will show that any fundamental system has at most one periodic orbit. Due to problems related either to the low differentiability class or to the restricted domain of definition of the functions F and g, we cannot apply directly the standard criteria for uniqueness of periodic orbits in these systems. See for instance [5, Chapter IV, Section 41. The next theorem shows that, nevertheless, the general ideas of this analysis, which are essentially due to Levinson and Smith [6] and go back to Lienard [7], apply well to the present case. THEOREM
5. Consider the Lienard system
Y’ = -g(x),
x’ = y - F(x),
(8)
defined either in the whole lR2 or in the open strip S = ((x, v) E lR2 : -x, - E < x < x1 + E], for some x1 > 0 and E > 0. I. Suppose that system (8) is defined in IR2 and satisfies the following assumptions: (i) F and g satisfy a Lipschitz condition on any bounded interval; (ii) g is an odd function such that xg(x) > 0 if x # 0; (iii) F is an odd function such that there exists x0 > 0 for which F(x) < 0, if 0 < x < x,, , and F’(x) > 0, if x 2 x,, .
Phase
portraits
of planar
control
systems
1185
II. Suppose that system (8) is defined in S and that it satisfies the following assumptions: (i’) F and g satisfy a Lipschitz condition on the interval -x1 - E c x < x1 + E; (ii’) g is an odd function such that xg(x) > 0 in (0, xi) and g(x,) = 0; (iii’) F is an odd function such that there exists x0 > 0, with 0 < x0 < x1, for which F(x)O,ifx~x,,; (iv’) the two equilibria e, = (xi, F(xl)), e- = (-xi, F(-x,)) have index - 1 and the origin has index 1. Then in both cases I and II (8) has at most one periodic orbit. Proof. I. Assume that the system is defined in ll? due to the symmetry, it holds that if (x(t), y(t)) is a solution then (-x(t), -y(t)) is also a solution. If both coincide, we say that the solution is symmetric. Since xg(x) > 0, for x # 0, the unique equilibrium of the system is the origin. So any periodic orbit must surround it. Therefore, by the uniqueness of solutions, it follows that any periodic orbit is symmetric. In particular, any periodic orbit intersect the y-axis in two points: (0, yO) and (0, -y,J. Thus, to find a periodic orbit of the system it amounts to find an orbit with symmetric y-intercepts. For showing that there is at most one such orbit we study the change of intercepts by regarding their intersections with the closed level curves of the x-symmetric function -x V(x, y) = y2/2 + g(r) dr. !0
Clearly, if an orbit intersects the y-axis in two points one positive and one negative, on V takes the same value, then the orbit is periodic. From (8), for x > 0 the orbits have negative (resp. positive) slope when y > F(x) y < F(x)). The slope being infinite when y = F(x). Thus we may assume that the arcs A’C’B’ and A”C”B” of Fig. 2 correspond to orbits of system (8). Suppose that the orbit ACB is contained in the strip 0 < x < x0, on which F(x) < dy < 0. Thus F(x) dy > 0 along such arc, and consequently B
V(B) - V(A) =
which (resp. ACB,
0 and
B
F(x) dy > 0.
dV=
sA
“A r
Therefore, OB - OA > 0.
(9)
Suppose that the arcs of orbits A’C’B’ and A”C”B” are not contained in the strip 0 < x < x0. Since -F(x) > 0 there and since y - F(x), along A”G, is greater than it is along A’E, we have G
V(G) - V(A “) =
G
dV=
i A”
s
s A”
-F’O~W(Y - F(x))1d-x
E
<
-P’O~WY
- R-91CLX
A’
E
dV = V(E) - V(A’).
= s A’
(10)
J. LLIBRE
Fig.
2. Three
and J. 8OTOMAYOR
typical
orbits
of system
(8).
Since F(X) > 0 along GH and du < 0 for x > 0, we obtain w
F(x) dy < 0.
V(H) - V(G) =
(11)
G
Now, since for the same y, F(x) along HI exceeds F(X) along EF, it follows that
Along IJ, as in the study made along GH, it holds that V(J) - V(1) < 0.
(13)
V(B”) - V(J) < V(B ‘) - V(F).
(14)
As in (lo), it is obtained that
Then adding relations from (10) to (14), we obtain V(B”) - V(A “) < V(B’) - V(A’); or equivalently, OB” - OA” < OB - OA’.
(15)
1187
Phase portraits of planar control systems
x
= -x
I
x
=
-x
0
x=-u
Fig. 3. The functions
.X=l(
F
and g when
X=X
0
x
= x,
d < 0.
In other words, from (9) it follows that the arcs of orbits which project into 0 < x I x0 satisfy OB - OA > 0. Also, from (15), OB - OA is monotonically decreasing for the arcs of orbits not contained in 0 < x I x0, when their end point A increase in y-coordinate. Therefore, OA = OB at most once. This is equivalent to the unicity of the periodic orbit, when it exists. II. Now assume that system (8) is defined in the strip S. Taking into account the behavior of the vector field on the line x = tx, (see Fig. 3), it follows that any periodic orbit must be contained in the strip -x1 < x < x r ; otherwise the bounded region Vlimited by the periodic orbit would contain the equilibrium points e- and e, simultaneously (due to the symmetry of the periodic orbits with respect to the origin), and consequently the sum of the indices of equilibrium points contained in V would not be 1, an obvious contradiction (see for instance 15, P- 1481). Since any periodic orbit of system (8) must be contained in the strip -x1 < x < xi, we can apply there the arguments used in part I for the whole plane. The outcome would be that the arcs of orbits which project onto 0 < x I x0, satisfy OB - OA > 0, and that OB - OA is monotonically decreasing in the y-coordinate of A for the arcs of orbits which are not contained in 0 < x 5 x0 but are contained in 0 < x < x1 and have their initial end point A in the positive y-axis and go to the point B of the negative y-axis coordinate. Therefore, OA = OB at most once. This is equivalent to the unicity of the periodic orbit, when it exists. n COROLLARY
6. Every FS has at most one periodic orbit.
Proof. Let d, t, D and T be the basic parameters of a given FS (4). By proposition 4, system (4) can be written as the Lienard system (8), which for d 1 0 satisfies the assumptions (i)-(iii) of theorem 5, part I if d < 0, the LiCnard system satisfies the assumptions (i’)-(iv’) of theorem 5, part II. Therefore, the corollary follows from theorem 5. n PROPOSITION 7. If a fundamental
system has a periodic orbit it must be unstable.
1188
J. LLIBRE
and
J. SOTOMAYOR
Fig. 4. Return map of a DSC.
Proof. Clearly, periodic orbits of fundamental systems must be internally unstable, since they enclose the origin which is an attractor. If some periodic orbit were semistable, a small rotation of the FS would produce another FS with a pair of periodic orbits, one unstable and one stable. See [5, Chapter V]. This would contradict proposition 6. n LEMMA 8. If a fundamental
system has a double saddle connection, then it must be internally
unstable. Proof. From propositions 1 and 3, the basic parameters of the FS must verify d < 0 and t > 0. The study of the return map T for the DSC gives the following: T splits into the composition of 4 mappings Tl, T2, T3, T4, as illustrated in Fig. 4. The mappings T, and c have the form d2 = a(d,/a)‘, with t9 < 1, which comes from 6’ = -h-/A+, with t = I- + 1, > 0. The mappings T, and & are smooth diffeomorphisms which have the form d + c(d)d, wiih a positive continuous c. The composition of these four mappings gives T as a strong expansion of the form d + f (d)d’ , for a positive continuous f. H PROPOSITION 9. If a fundamental system has a double saddle connection (DSC), then it does not have periodic orbits and the DSC must bound the basin of attraction of the origin.
Proof. Clearly the DSC must enclose any periodic orbit which necessarily surrounds the origin. By proposition 7 and lemma 8 this would lead to a contradiction since both the DSC and the periodic orbit are unstable. n
Notice that from proposition and (b) of theorem A.
3, corollary 6 and propositions
7 and 9, follow statements (a)
1189
Phase portraits of planar control systems 4. PHASE
PORTRAITS
FOR d 2 0 AND
t5 0
In this section consider a FS x’ = Jx + yl(k * x)b, and denote by d, t, D and T its basic parameters. Next lemma is the standard integral representation of solutions of differential systems. It will be used several times in what follows. LEMMA
10. If x(s) is a solution of a fundamental x(s) = eJSx(0) +
Proof.
Immediate by differentiation.
system, then
’ eJcS-“cp(k - x(r))b dr.
0
n
PROPOSITION11. If d > 0 and t < 0, then Q(0) = R2. Proof. In this case there are positive constants L, K and y such that l(eJ”]l 5 L emys and ((o(k * x)b( I K. Taking norms in the equality of the previous lemma, we get:
eJcswr)111 q(k - x(r)) b 1dr.
IW 5 lleJsll . b@)I + Therefore,
= L e-Y”(lx(0)l - K/y) + LK/y. Hence, every solution of the FS has its o-limit
set contained in the ball of center 0 and radius
LK/y.
From propositions 1 and 3, the system has the origin as unique equilibrium and does not have periodic orbits. Poincare-Bendixson theorem implies the o-limit set is the origin. That is Q(0) = R?.
n
Remark. The proof of this proposition completeness. See also [3].
is standard;
it is included here for the sake of
PROPOSITION12. If d = 0 and t < 0, then Q(O) = R2. Proof. We can assume that the matrix J is in Jordan form: J=
[ 00’1 A-
with
’
t
= A- < 0.
Let x(s) = (x1(s), x&r)) be a solution of the fundamental Ix,(s)\ I lex-s 1x,(O)1 +
system. From lemma 10, we have that
’ lex-@-‘) 1 tp(k * x(r))&1 dr.
0
Clearly, there exists a K > 0 such that Icp(k * x)b,l I K.
1190
J. LLIBRE
and J. SOTOMAYOR
Therefore,
Hence, as s tends to infinity, limlx,(s)I I -K/A-. It follows that the solutions of the FS have their w-limit sets on the vertical strip S bounded by the straight lines x, = +K/A _ , which turns out to be positively invariant under the flow. Since D = I _ b2 k2 > 0, k2 # 0. Then there exists h,, > 0 such that, for h > ho, the horizontal sides of the rectangle S, = (x E S, Ix,1 I h) do not intersect the parallel straight lines r+ and r- , for all h 2 ho. Therefore, on the horizontal sides of S,, the second component of the FS is constant and equal to f ub2, having opposite signs on opposite sides. We claim that the rectangle is positively invariant for all h 1 h,. In fact, it is sufficient to verify that on the horizontal sides, the FS point inward. Otherwise, the a-limit set of the orbit through (0, h) would be contained in S,. This is in contradiction with the fact that the origin is a local attractor and with the absence of periodic orbits, by proposition 3. Now since S,, is positively invariant for large h and the w-limit set of any orbit is contained in S, it follows by the Poincare-Bendixson theorem that n(O) = IR2. n PROPOSITION
13. If d > 0 and t = 0, then n(O) = lR2.
Proof. The matrix J must be of the form:
0 P -p [ 01’
with /3 > 0.
As in the proof of case 2 in proof of proposition Lienard form: Y’ = -g(x),
x’ = y - F(x),
4, we can write the FS in the following
with g(x) = dx + qo(x)(D - d), F(x) = q$x)T.
Notice that, due to the time reversal, the origin on this system is a repellor. Furthermore, it satisfies the assumptions (i) and (ii) of theorem 5 and, instead of (iii), it verifies that F(x) is an odd function with xF(x) < 0. Repetition of the arguments of the proof of theorem 5 for this system, leads to OB - OA > 0 for all arcs of orbits on x > 0. Then, by the symmetry with respect to the origin, all the solutions tend to infinity spiraling around the origin. Due to the time reversal, the origin of the FS is a global attractor, that is Q(O) = iE2. n Remarks. Arguments similar to those in the last proof can also be used to obtain propositions 11 and 12. Notice that proposition 13 does not follow from the work of Meyer [I, p. 55-561. In fact his conditions (9.2) imply in the present situation that T = 0, which is not the case here. PROPOSITION
14. If d = t = 0, then Q(O) = lR2.
Phase
portraits
of planar
control
1191
systems
Proof. The matrix J must be of the form
0 0 [ 1 0I * The FS can be written as x; =x1 + q(k*x)&.
x; = cp(k * x)b,,
Consider the following
Liapunov
function: k.x
V(Xl,X2) = 1/2(-&/&)x,2
+
r0
909 dr.
Clearly, V > 0 for x # 0, since -kz/bl = D(b,)-2 > 0. Also, differentiation with respect to the FS gives: I/’ = Tp2(k * x) < 0, if k * x f 0. Therefore, the origin is globally asymptotically stable, i.e. a(O) = lR2. n The idea of using the above Liapunov [l, p. 53-541.
Remark.
From propositions
function
in this case comes from Meyer
11-14 it follows the proof of theorem B. 5. PHASE
PORTRAITS
FOR
d 2 0 AND
t > 0.
In this section we will prove theorem C. PROPOSITION
topologically
15. If d > 0 and t > 0, then the phase portrait equivalent to that of C, Fig. 1.
of the fundamental
Proof. From Section 2, J must be equal to one of the following
(^b’
12)y
(:l
il)
Or
(-;
where the As, (Y and fi are positive. There are positive constants L, K and y such that ]le-JSlj I L e-” If x(s) is a solution of a fundamental system, then s emJsx(s) = x(0) + emJrq(k - x(r) b dr. .r0 Taking norms we get s Le+(x(s)l 1 Ix(O)/ - KL e-?‘dr. i0 Consequently, Ix(s)l 2 L-’ eY”(lx(0)l - KL/y)
system is
matrices: aa>> and l&k * x)b] I K.
+ K/y.
Hence, every solution whose initial condition is outside the ball B(LK/y), of center 0 and radius tends to infinity in forward time. Therefore, since the origin which is the unique equilibrium is an attractor, there must be a periodic orbit, by Poincare-Bendixson theorem. Furthermore, n(O) is contained inside the ball B(Lk/y). By theorem A, the proposition follows. n
LK/y,
1192
J. LLIBRE
PROPOSITION
topologically Proof.
and J. SOTOMAYOR
16. If d = 0 and t > 0, then the phase portrait equivalent to that of Fig. 1.e.
of the fundamental
system is
From Section 2, J must be
where the A = t > 0. If x(s) is a solution of a fundamental
x1(s) =
e""x,(O) +
s s
system, then
eLcrms)yl(k* x(r)b, dr.
0
If K > 0 is such that (p(k +x)b, 1 I K, then taking norms get Ixl(s)l 5 exs(Ix,(O)1 - K /Ie-‘rdr)
= ex”(lxl(0)l - K/A) + K/A.
Hence, every solution whose initial condition x(0) is outside the vertical strip S bounded by the lines xi = *K/A, tends to infinity. Also, if Ix,(O)1 = K/A, for positive time, s, it holds that Ixi(s)l z K/A, as follows from the above inequality. Therefore, the strip S is negatively invariant under the flow of the FS. Since D = A&k, > 0, kZ # 0. Then there exists H > 0 such that, for h > H, the horizontal sides of the rectangle S, = ((xi, x2) E S : Ix,1 5 h] are disjoint from the lines L and lY+. Therefore, on the horizontal sides of S, the vertical component of the FS is constant, equal to +u&, and with opposite signs on different sides. We will verify that S, is negatively invariant for all h z H. Since D = Ab2k, > 0 and T = 1 + bl k, + b2 k2 < 0, we have that bzkz > 0 and bl kl < 0. There are four cases to consider: (1) kl > 0, k2 > 0; b, < 0, b2 > 0; (2) kl < 0, k2 > 0; b, > 0, b2 > 0; (3) kl > 0, kz < 0; 6, < 0, b2 < 0; (4) kl < 0, k2 < 0; bl > 0, b2 < 0. It is sufficient to analyze the first case, since the others are similar. In fact, on the upper horizontal side of the rectangle S,, , xi = ub, > 0 and consequently xi = -ub, on the lower side. This concludes our verification in this case. Now, since S, is negatively invariant for all h 1 H, and the solutions starting outside S escape to infinity for positive time, it follows that all solutions outside the rectangle S,, escape to infinity. Under these circumstances, since the origin is an attractor, Poincare-Bendixson theorem implies the existence of periodic orbits inside S,, . From theorem A, there is a unique unstable periodic orbit there. n Remark. The existence of periodic orbits under the assumptions of propositions
also be found in [3], where no uniqueness is mentioned. From propositions
15 and 16 follows the proof of theorem C.
15 and 16 can
Phase portraits of planar control systems 6. PHASE
PORTRAITS
1193
FOR d < 0
In this section we will prove theorem D. To this end, explicit analytical expressions will be given for the hypersurface 6) and for the regions C&Sand SC! in U3- = (d < 01. The expressions are given in Table 2, in terms of functions F and G which themselves are defined in Table 1 from the basic parameters of a FS. For D > 0, T < 0, d < 0, let A,
= (-T
+ (T2 - 4D)“‘)/2,
A+ = (-t
y(A,) = (1 - A,/A+)/(A,/L
k (t2 - 4d)15’)/2,
- 1).
Define F(A-) and G(A-) in Table 1. In terms of the functions F and G, the phase portraits of the FS in 63-, with the notation in Fig. 1, are given in Table 2. This amounts to a reformulation of the conclusions of theorem D. For d < 0, the Jordan form of the FS is J = diag@+, A-),
with A+ > 0 > A-.
We can write the FS in the form x’ = Jx + yl(k - x)b.
Assume that k = (k,, &) is an eigenvector of J. That is, either ki = 0 or k2 = 0. Take kz = 0, since the other case can be studied in the same way. The straight lines IY and I+ are xi = -u/k, and xi = u/k,, respectively. Without loss of generality assume that ki > 0. Since D = L (A+ + b, k,) > 0, it follows that b,k, < 0, and therefore bi > 0. Now we assume that b, > 0; in fact, for b2 I 0 would give the same phase portrait. The separatrices of the saddle points look as in Fig. 5. Table 1. Definition of F(L) T2 - 40
F(L)
and G(k)
for FSs in a3W-)
>o
Y(A-1
=o
YW1 + 1 Ir(A-)l
y(b)“*- y(A-)I’*+ y(AJ - exp(l/2(y(A-) - l/y(A-)) (Im y(AJ)/Re v(A-)) - tan(log(/(A_)~‘m(Y”~))‘Re(y”-))))
Table 2. Analytical characterization of phase portraits in @-
F(L) >l 51
W-l >o =o
Phase portrait l.SC? l.a> 1.&S 1.&S
1194
J. LLIBRE
and J. SOTOMAYOR
i,
7-
I-
3
0
c
et
L
I'
e-
I -.I Fig.
/' I'
I’
r-
i2>0
5
rt
5. Case d < 0, k an eigenvector
Notice that the rectangle S with vertices e, and e- and positively invariant, must contain the unstable separatrices existence of limit cycles (which we know must be unstable). portrait must be as that in Fig. 1.&S. Now we assume that k = (k,, &) is not an eigenvector of lines I- and I+ intersect both coordinate axes. There are four cases to consider: (1) k, > 0, k2 > 0;
of J.
sides parallel to the axes, being of the saddles. This prevents the Therefore, in this case the phase J. Therefore,
the parallel straight
(2) k, > 0, kz < 0; (3) kl < 0, k, > 0; (4) kl > 0, k2 < 0.
It is sufficient to analyze the first case, since the others reduce to it by changing the orientation on one or both axes. In fact, the saddle separatrices look as in Fig. 6. Due to the symmetry with respect to the origin, only the three possibilities (I), (2) and (3) for the positions of the separatrices y!! and y”, arise, as illustrated in Fig. 6. From the results in theorem A, it follows that each of these possibilities correspond, respectively, to the phase portraits SC, D and Q.S of Fig. 1. Now we will see that the three phase portraits are realizable for FSs. In fact, 0% exists when k is an eigenvector of J, for t < 0, because of proposition 3, from which there are no periodic orbits nor DSC. To show that 9 exists we will compute explicitly the analytic relations to be satisfied by the basic parameters of a FS to present a saddle connection. Once this is done and verified that it is nontrivially satisfied by FSs, a rotation argument shows that there are FSs of types @S and Se. Now we proceed to compute the analytic expressions and verify that, in fact, there are FSs fulfilling them.
Phase
Fig.
According to proposition can be written in the form
portraits
of planar
control
systems
1195
6. Case d < 0, k, > 0, k, > 0.
4, after reversing the time variable, a FS with a saddle connection x’ = y - F(x)
Y’ = -g(x),
with g(x) = dx + yl(x)(D - d), F(x) = tx + &x)(T - t), where d < 0, t > 0, D > 0, T < 0. So the FS splits into the following three linear systems: y’ = -dx
+ u(D - d), y’ = -Dx,
y’ = -dx
x’ = y - tx + u(T - t), x’ = y - TX,
- u(D - d),
ifx5
-u;
if -u 5 x 5 u;
x’ = y - tx + u(T - t),
ifx2
(16)
u.
Then, e, = -e- = (-u(D - d)/d, -u(Dt - Td)/d) are the saddles of this system. The numbers L, = (-t + (t2 - 4d)““)22, at the beginning of this section, are the eigenvalues at the saddle points of (16). Notice that due to the change in sign for writing the Lienard form above, A+ above corresponds to I- in the original FS and vice versa. The same remark applies to A, = (-T f (T2 - 4D)“2)/2, the eigenvalues of the attractor at the origin for (16). A straight computation shows that the stable separatrix of e- that intersects the line lY_ does it at the point pm = u(-1,
-T-
DA-/d).
1196
J. LLIBRE
Fig.
7. Geometry
and J. SOTOMAYOR
of conditions
for a DSC.
On the other hand, the unstable separatrix of e, that intersects r+ does it at the point p+ = ~(1, T + DA+/d). See Fig. 7 for an illustration. In order to get a saddle connection, we must find the values of d, D, t, T for which the solution of (16) that passes through p+ also passes through p- . There are two cases to consider.
Case 1. T2 - 40 # 0. If A, = (-T be written as
f (T2 - 40)““)22,
then the general solution of (16) can
(x(.s),y(s)) = (aeSA- + beSA+, -aA+ es*- - bh-
es*+).
In order that (x(O), y(O)) = p+ , it must hold that a = a* = uA+(AJL
- l)/(A-
- A+),
b = b* = -uA-(A+/L
- l)/(A-
- A+).
Denote by (x*(s),y*(s)) the solution of (16) with a = a* and b = b*. Now we will analyze the possibility of existing some positive S, for which (x*&J, y*(s,)) = p- . For the case T2 - 40 > 0, calculation shows that, with the notation in Table 1, v(A) = (1 - AJL+)/(A-/L - 1) > 1 amounts to the existence of the time s0 for the solution to reach the line IY. The condition to actually reach the point p- is written as y(AJ”*-
- y(A+)“*+
= 0.
For the case T2 - 40 < 0, use the real form of the complex general solution above and the notation in Table 1. Calculation shows that Iv(A-)l > 1 amounts to the existence of the time so for the solution to reach the line A-. The actual encounter with the point p- is expressed as (Im y(A-))/(Re
y(A))
- tan(log(~y(AJ~‘m(y~A-))‘R”(y~“-))) = 0.
Phase
portraits
of planar
control
systems
1197
Case 2. T2 - 40 = 0. The general solution of (16) is (x(s), y(s)) = ((a + sb) e-sT’2, (a + b(s + 2/T))T
e- sT’2/2).
In order that (x(O),y(O)) = p+, it must hold that a = a* = u, b = b* = uT(1 + T/2L)/2. As above, denote by (x*(s), y*(s)) the solution of (16) with a = a* and b = b*. Now proceed to analyze the possibility of existing some positive s,, for which (x*(&J, y*(s,)) = p- . Using the notation in Table 1, calculation shows that y(A) > 0 amounts to the existence of s,, to reach I-- ; the actual encounter with p- happens only when y(k)
= exp((y(A-)
- l/#-)/2).
Since the points p+ depend analytically on the basic parameters, it follows that the hypersurface 33 corresponding to double saddle connections is analytic in these parameters, independently on the conditions on the sign of T2 - 40 used in its definition. This ends the proof of theorem D. W Acknowledgements-The
first author had initial discussions on fundamental systems with R. Suarez. He is partially supported by CIRIT and DGICYT grants. This paper was written in part during his visit to IMPA in 1992, supported by CNPq, Brazil. The second author wrote part of this paper in IMPA, in 1992 and completed it while visiting the Centre de Recerca Matematica, Barcelona, in 1993. REFERENCES 1. LEFSCHETZ 2. NARENDRA New York 3. ALVAREZ University 4. LLIBRE Barcelona 5. ZHANG Vol. 101. 6. LEVINSON 7. LIENARD
S., Stability of Non-linear Control Systems. Academic Press, New York (1965). K. S & TAYLOR J. M., Frequency Domain Criteria for Absolute Stability. Academic Press, (1973). J., SUAREZ R. & ALVAREZ J., Planar Linear Systems with Single Saturated Feedback (preprint). Autbnoma Metrop., Mexico (1992). J. & PONCE E.. Periodic Orbits in Nonlinear Control Systems (preprint). University Autbnoma. (1993). ZHIFEN et al., Qualitative Theory of DifferentialEquations, Translation of Mathematical Monographs, American Mathematical Society (1992). N. & SMITH 0. K., A general equation for relaxation oscillations, Duke math. J. 9, 382-403 (1942). A., Etudes des oscillations entretenues, Rev. Gen. d’E/ectricite’ XXIII, 901-946 (1928).
Theory, Methods & Applications,
Vol. 27, No. IO, pp. 1177.1197, 1996 Copyright G 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+0.00
Pergamon 0362-546X(95)00129-8
PHASE PORTRAITS JAUME
OF PLANAR
LLIBREt
CONTROL
SYSTEMS
and JORGE SOTOMAYORS
t Departament de Mattmatiques, Universitat Autbnoma de Barcelona, Bellaterra, 08193 Barcelona, Spain; and $ Institute de Matematica e Estatistica, Departamento de Matematica Aplicada, Cidade Universitaria, Rua do Matao, 1010, 05508-900~Sao Paulo, Brazil (Received 7 June 1994; received for publication
6 July 1995)
Key words and phrases: Nonlinear control systems, phase portrait, bifurcation
diagram.
1. INTRODUCTION
Systems of the form x’ = Ax + cp(k * x)b,
(1)
where A is a n x n real matrix and x, k, b are in IR” and “. ” denotes the usual inner product, are of great importance in direct control [l]. The divergence from linearity of these systems results from the presence of the characteristic function v, of the control mechanism. The purpose of this mechanism is to improve the asymptotic stability behavior of the equilibrium located at the origin. The reader is referred to Lefschetz [l] and to Narendra and Taylor [2], for details in control theory. In this paper, we will focus our attention to the class 5s of two-dimensional systems (l), i.e. n = 2, such that (a) have the origin as an asymptotically stable equilibrium point; (b) the characteristic function cp has the form p(u) = -24,
for u 5 -24;
p(u) = u,
for -24 5 u 5 24;
p(u) = 24, for u 5 u,
where u is positive and fixed throughout this paper. Such systems will be referred to as fundamental systems (FS). They are symmetric with respect to the origin. That is, their solutions are exchanged under the mapping x + -x. Orbits which are invariant under this mapping will be called symmetric. The characteristic function rp induces a partition of lR2 into three open strips and two straight lines, as follows: S- = (x:x-k
< -u),
So = (x: --u < xv k < u),
I-- = {x :x - k = -u},
Therefore,
the FS splits into the following
S, = (x:x-k
> u)
I-+ = (x : x - k = u).
linear systems:
x’ = Ax - ub 3
in .5- UT
x’ = Ax + (k . x)b,
in I- Us,
x’ = Ax + ub ,
in S+ 1177
(2. -1
ix-+
7-q.
(2.0) G.+)
1178
J. LLIBRE
and J. SOTOMAYOR
Notice that fundamental systems are of class C’ on lR’\(lY UT+) but only of class Co on IR’. Nevertheless, they satisfy a Lipschitz condition (with L-constant smaller than or equal to IIAIj + [Ibk’jl) on the whole lR2. This fact makes applicable to these systems the classical theorems on existence, uniqueness and continuity on initial conditions and parameters. Their solution curves are in general C’, but not C2. The main goal of this paper is to describe all the phase portraits of FS, and to distinguish among their different qualitative patterns in terms of their basic parameters: t = Trace(A), T = Trace(B) < 0,
d = Determinant(A);
D = Determinant(B)
> 0,
where B = A + bk’.
(3.1) (3.2)
The inequalities express condition (a) in 5s above. An elementary result in this direction which originates in [3], proved in Section 2 for the sake of completeness, is the following classification of equilibria of FSs. 1. If d L 0, the origin is the unique equilibrium If d < 0, there are three equilibria for the FS:
PROPOSITION
of the FS.
0 in So, which is always an hyperbolic attractor, and e += -d-lb, in S+, and e- = uK’b, in S-, which are both saddle points. With this information in hand, the determination of the phase portraits of FSs will now depend on their periodic orbits, which must enclose the origin and therefore be always symmetric, and their saddle separatrices (present only when d < 0). Sometimes these separatrices are connected, in pairs due to the symmetry, forming a graph surrounding the origin, whose corners are the saddles e, and e- . These graphs will be called double saddle connections (DSC). The possibility of appearance of single saddle connections (homoclinic loops) with corners e, or e- is easily ruled out from the symmetry of the FS and the fact that both must surround the origin and be symmetric to each other. This establishes a second elementary property of FSs pointing out towards the determination of phase portraits of FSs. PROPOSITION 2. The unique simple closed curves invariant under the flow of a fundamental system are either periodic orbits or double saddle connections.
In this paper, we will be particularly concerned with the form and size of the basin of or stable manifold, of the origin, Q(O), of a FS. This element is of considerable interest in control theory [ 1,2]. Our main results will be summarized in what follows. attraction,
A. The following statements hold for a FS of the form (1): (a) if it has a periodic orbit or a double saddle connection, then t > 0. Moreover, situations cannot coexist on the same FS; (b) it has at most one periodic orbit, which moreover must be unstable. The space of all basic parameters of FSs will be denoted by a: THEOREM
@ = ((f, d; T, D) E R4 : with T < 0, D > 0).
both
Phase portraits of planar control systems THEOREM B. If the basic parameters
1179
are in the set
a = ((t, d; T, 0) E 63 : t I 0 and d 2 0), then the basin of attraction,
Q(O), is the whole plane.
THEOREM C. If the basic parameters are in
e = l(t,d;T,D)E63:t>OanddrO), then the FS has a unique periodic orbit which is unstable. It is the common boundary of n(O), the basin of attraction of the origin and Q(W), the basin of attraction of “infinity” formed by the orbits which tend to infinity as time grows. THEOREM D. In &3- = ((t, d; T, 0) E CR: d < 0), there is an analytic regular hypersurface, 3, contained in the half-space lt > O), whose boundary in 03 is &3,,,,= {t = 0, d = 01. If the basic parameters are on the hypersurface a>, then the FS has a double saddle connection which bounds Q(0). The phase portrait is illustrated in Fig. l.a>. The hypersurface D separates the open set Se c GJ-, whose boundary in 63 is contained in the half-hyperplane rZS, = (t > 0, d = O), from the open set @S c 03-, whose boundary on 63 is contained in the half-hyperplane @S, = (t < 0, d = 0). If the basic parameters are in QS, the phase portrait has an attractor and a pair of saddles, as illustrated in Fig. 1.024. If the basic parameters are in Se, the phase portrait has a repelling cycle and a pair of saddles, as illustrated in Fig. l.SC. d
t
\ Fig. 1. Bifurcation
1.D
diagram for fundamental systems in the basic parameter space.
1180
J. LLIBRE
and J. SOTOMAYOR
A word of explanation on the hypersurfaces a, as,, CS,, CE, = (t = 0, d > 0) in Fig. 1. They represent the codimension one bifurcations of FSs. That is, where the transition between the distinct structurally stable phase portraits happens. The notation suggests the qualitative change. On W?, the cycle in (9 disappears at infinity. Meanwhile, on @S, and (78, the saddles in G.8 and eS disappear at infinity. Explicit analytical expressions for 9 and for the regions @.S and SC, in terms of basic parameters of a FS, can be found in Section 6. Some aspects of the existence and bifurcation diagrams of periodic orbits established for two dimensions in this work, have been extended to higher dimensional control systems in 141. This work is organized as follows. Section 2 is devoted to establish some preliminaries including Jordan normal forms for FSs and the proof of proposition 1. In Section 3 we study periodic orbits of FSs and prove theorem A. Sections 4, 5 and 6 deal with the proofs of theorems B, C and D, respectively. 2. PRELIMINARIES
In this section we establish some basic properties of FSs for later use. 2.1. Normal forms Consider the fundamental system x’ = Ax + p(k . x)b. The linear change of variables x = My which reduces A to its real Jordan form J = M-‘AM, transforms the FS into y’ = Jy + p(Mfk.
y)M-‘b.
Renaming the transformed parameters M’k by k and M-lb variable y by x, the system can be written back as
by b, and the transformed
x’ = Jx + p(k * x)b, where J is one of the following
forms: if d = 0, t = 0; if d = 0, t # 0; if d > 0, t = 0; with sign(&) = sign(a) = sign(t), and p > 0, if d > 0, t # 0; with I, < 0 < &,
ifd
Notice that the null matrix has been omitted from this list. In fact, having a segment of the straight line k * x = 0 as equilibria, it is not compatible with definition (a) of FS.
1181
Phase portraits of planar control systems
2.2. Proof of proposition
1
See Section 1 for the statement and notation adopted. Case 1. Assume that d # 0. From (2.+) it follows that if e, E S+ it will be the unique equilibrium point of the FS on S+ U I+. Similar statements hold for e- in .S- U l--.
We have the following k * (-uJ-lb)
iff
d-‘(a22blk,
equivalences: e, E S+ and e- E S- iff > u and k * (uT1b) < -u, iff k * (J-lb) - alzbZkl
- a,,b,kT
+ a,,bzk2)
< -1,
< -1,
where J = (aij).
Now if d > 0, we get that e, E s+ and e- E S- iff d + (a22blkl
- a,,b,k,
- a,,b,k,
+ a,,bZk2) < 0.
On the other hand, if d < 0 then e, E S+ and e- E s- iff d + (azzblk,
- alzb,k,
- a,,blkz
+ a,,b2kJ
> 0.
But the left-hand member of this inequality is D. Therefore, 0 is the unique equilibrium point when d > 0, and 0, e, and e- are the unique equilibrium points when d < 0. The saddle character of e, and e- is obvious from the form of the FS outside S,,, Case 2. Assume that d = 0. We have only two possibilities for the Jordan form J
(:
i)
or
(“,
i),
A.0.
If the first case holds, we must have that bi # 0 and k2 # 0, otherwise the FS would have a segment of straight line of equilibria through the origin. It follows easily that 0 is the unique equilibrium of the FS. If the second possibility holds, similar arguments to those used in the previous case imply that b, # 0 and kz # 0 and therefore 0 is the unique equilibrium. This ends the proof of proposition 1. H 3. PERIODIC
ORBITS
OF FS
There are two main aspects in the study of periodic orbits of planar differential systems. The first one concerns their existence, stability, relative positions and cardinality. The other one considers their bifurcations, i.e. their dependence on parameters of the system, their occurrence and disappearance, as parameters change. The main purpose of this section is to prove theorem A. For doing so, both aspects will come into play. In proposition 2 it was established that, due to their symmetry, FSs do not admit homoclinic loops. Instead they admit double saddle connections (DSC) and periodic orbits. Their location in terms of the basic parameters is constrained by next proposition. PROPOSITION 3. If a fundamental system has a periodic orbit or a double saddle connection, then its basic parameter t = trace(A) is greater than 0.
1182
J. LLIBRE
and
J. SOTOMAYOR
Proof. Since on S = r+ U So U I?- the fundamental system has a linear attractor at the origin, the unique possible segments of straight lines invariant under the flow and contained in S must be on straight lines passing through the origin. Therefore, a segment on the lines r+ and l7cannot be part of an orbit of a fundamental system and, consequently, the integrals
ii 0
and
Xl~2--X2~,
ss E
(ax,/ax,
+ dX,/dX,)
dx, dxz
are well defined for a fundamental system whose components are X, and X,, o is a periodic orbit or double saddle connection and Z is the region bounded by CI. From Green’s formula, the above integrals are equal. Since X, dxz - X, dxr = 0 along cr, both integrals must vanish. So, from expression (3) we obtain
55 ~n(s-ns+)
tdX1dX, +
Tdqdq IS
= 0,
Ens,
or equivalently, t Area[E fI (S- U S,)] + TArea(C f~ S,) = 0. Then, since from (3.2) T
c
0, it follows that
t
> 0.
n
Remark. The above proof uses arguments motivated by those of the Bendixson-Dulac criterion for the absence of periodic orbits of vector fields on regions where the divergence has definite sign. See for instance [5, p. 1951. The knowledge of conditions which imply the uniqueness of limit cycles of planar differential system is still very limited. Our next result shows that fundamental systems with periodic orbits can be written in the form of LiCnard systems, for which a practical criterion for uniqueness will be established. PROPOSITION
4. Assume that a fundamental
system
x’ = Jx + q$k * x)b
(4)
has a periodic orbit or a double saddle connection (DSC) and let d, t, D, T be its basic parameters. Then after a linear change from the variables x = (x1, x2) to the variables (x, y) and a reversing of the time variable, the FS (4) can be written in Lienard form Y’ = -g(x),
x’ = y - F(x),
where g(x) = dx + d-9@
- d)
and
F(x) = tx + q$x)(T - t).
Proof. By proposition 3, t > 0. There are three cases to consider, according to the real Jordan form J of the matrix A of the FS.
Phase
portraits
of planar
control
systems
1183
Case 1. J = Diag(l,, A,), A, 2 A2 and A, + A2 = t. In this case, kr and k2 are both different from 0. Otherwise, if, say, kl = 0, then xi = A++ + cp(k2x2)b2. Therefore, the straight line x2 = 0 is invariant under the flow of the FS. Since the lines I?- and I + are parallel to it, we have a contradiction with the fact that the fundamental system has a periodic orbit or a DSC. We also have that A, # A,. Otherwise, the matrix B would also be a diagonal multiple of the identity and all the segments of straight lines passing through the origin and contained in S0 would be invariant under the flow of the FS, which therefore could not have a periodic orbit or a DSC. Thus the matrix
A,k;’ M
=
1411
-
A,)
-A
-k;’
k-l 2
2
1
k-l 2
is well defined and invertible. Now, the change of variables
x=M
* 0Y
(5)
sends FS (4) to
(“y>’ = M-VM(Y~)
+ p(k.M(;))M-lb
which amounts to
x’ = tx - y + q?(x)(T- t),
y’ = dx + p(x)@ - d).
(6)
Changing the sign of the time variable ends the proof in case 1.
Case2. Suppose that for 2o = t and /I > 0, J= We write
the FS in complex
a P -p a [ 1*
notation
as follows.
For
z = x1 + ix2, 1 = (Y + iB,
k = k, + ik2, b = bl + ib,, it becomes: z’ = Iz + &z + kZ)b. Now we consider the complex differential
system:
z’ = jz + q(kz + kZ)b, (7)
2’ = AZ + yl(kz + k?)b.
Notice that this system essentially consists of two copies of the initial FS. Now we can follow the same steps as in case 1. Since e + k$ # 0, it follows that k and consequently k are nonzero. Here clearly, A f i, and the matrix
I
J. LLIBRE
1184
is well defined and invertible.
and J. SOTOMAYOR
Again, performing
the change of variables
where x and y are complex, sends system (7) to (6). Now it is easy to check that if we choose the real part of the solutions x(t) and y(t) of (5) and go back through the change of variables, we obtain the solutions of the initial FS. So we can consider that the variables x and y are real in system (6), and that such system is equivalent to the fundamental system (4). Finally, reversing the time variable concludes the proof in case 2. Case 3. Assume that
with t = 2A. Then k2 # 0; otherwise the x,-axis would be invariant under the FS, since Iand r+ are parallel to such axis, the FS could not have a periodic orbit or a DSC. Therefore, the matrix
-4k 1
is well defined and invertible. system (6). H
1
After the change of variables (5) the FS (4) goes over to
Using the Lienard form in proposition 4, we will show that any fundamental system has at most one periodic orbit. Due to problems related either to the low differentiability class or to the restricted domain of definition of the functions F and g, we cannot apply directly the standard criteria for uniqueness of periodic orbits in these systems. See for instance [5, Chapter IV, Section 41. The next theorem shows that, nevertheless, the general ideas of this analysis, which are essentially due to Levinson and Smith [6] and go back to Lienard [7], apply well to the present case. THEOREM
5. Consider the Lienard system
Y’ = -g(x),
x’ = y - F(x),
(8)
defined either in the whole lR2 or in the open strip S = ((x, v) E lR2 : -x, - E < x < x1 + E], for some x1 > 0 and E > 0. I. Suppose that system (8) is defined in IR2 and satisfies the following assumptions: (i) F and g satisfy a Lipschitz condition on any bounded interval; (ii) g is an odd function such that xg(x) > 0 if x # 0; (iii) F is an odd function such that there exists x0 > 0 for which F(x) < 0, if 0 < x < x,, , and F’(x) > 0, if x 2 x,, .
Phase
portraits
of planar
control
systems
1185
II. Suppose that system (8) is defined in S and that it satisfies the following assumptions: (i’) F and g satisfy a Lipschitz condition on the interval -x1 - E c x < x1 + E; (ii’) g is an odd function such that xg(x) > 0 in (0, xi) and g(x,) = 0; (iii’) F is an odd function such that there exists x0 > 0, with 0 < x0 < x1, for which F(x)
Clearly, if an orbit intersects the y-axis in two points one positive and one negative, on V takes the same value, then the orbit is periodic. From (8), for x > 0 the orbits have negative (resp. positive) slope when y > F(x) y < F(x)). The slope being infinite when y = F(x). Thus we may assume that the arcs A’C’B’ and A”C”B” of Fig. 2 correspond to orbits of system (8). Suppose that the orbit ACB is contained in the strip 0 < x < x0, on which F(x) < dy < 0. Thus F(x) dy > 0 along such arc, and consequently B
V(B) - V(A) =
which (resp. ACB,
0 and
B
F(x) dy > 0.
dV=
sA
“A r
Therefore, OB - OA > 0.
(9)
Suppose that the arcs of orbits A’C’B’ and A”C”B” are not contained in the strip 0 < x < x0. Since -F(x) > 0 there and since y - F(x), along A”G, is greater than it is along A’E, we have G
V(G) - V(A “) =
G
dV=
i A”
s
s A”
-F’O~W(Y - F(x))1d-x
E
<
-P’O~WY
- R-91CLX
A’
E
dV = V(E) - V(A’).
= s A’
(10)
J. LLIBRE
Fig.
2. Three
and J. 8OTOMAYOR
typical
orbits
of system
(8).
Since F(X) > 0 along GH and du < 0 for x > 0, we obtain w
F(x) dy < 0.
V(H) - V(G) =
(11)
G
Now, since for the same y, F(x) along HI exceeds F(X) along EF, it follows that
Along IJ, as in the study made along GH, it holds that V(J) - V(1) < 0.
(13)
V(B”) - V(J) < V(B ‘) - V(F).
(14)
As in (lo), it is obtained that
Then adding relations from (10) to (14), we obtain V(B”) - V(A “) < V(B’) - V(A’); or equivalently, OB” - OA” < OB - OA’.
(15)
1187
Phase portraits of planar control systems
x
= -x
I
x
=
-x
0
x=-u
Fig. 3. The functions
.X=l(
F
and g when
X=X
0
x
= x,
d < 0.
In other words, from (9) it follows that the arcs of orbits which project into 0 < x I x0 satisfy OB - OA > 0. Also, from (15), OB - OA is monotonically decreasing for the arcs of orbits not contained in 0 < x I x0, when their end point A increase in y-coordinate. Therefore, OA = OB at most once. This is equivalent to the unicity of the periodic orbit, when it exists. II. Now assume that system (8) is defined in the strip S. Taking into account the behavior of the vector field on the line x = tx, (see Fig. 3), it follows that any periodic orbit must be contained in the strip -x1 < x < x r ; otherwise the bounded region Vlimited by the periodic orbit would contain the equilibrium points e- and e, simultaneously (due to the symmetry of the periodic orbits with respect to the origin), and consequently the sum of the indices of equilibrium points contained in V would not be 1, an obvious contradiction (see for instance 15, P- 1481). Since any periodic orbit of system (8) must be contained in the strip -x1 < x < xi, we can apply there the arguments used in part I for the whole plane. The outcome would be that the arcs of orbits which project onto 0 < x I x0, satisfy OB - OA > 0, and that OB - OA is monotonically decreasing in the y-coordinate of A for the arcs of orbits which are not contained in 0 < x 5 x0 but are contained in 0 < x < x1 and have their initial end point A in the positive y-axis and go to the point B of the negative y-axis coordinate. Therefore, OA = OB at most once. This is equivalent to the unicity of the periodic orbit, when it exists. n COROLLARY
6. Every FS has at most one periodic orbit.
Proof. Let d, t, D and T be the basic parameters of a given FS (4). By proposition 4, system (4) can be written as the Lienard system (8), which for d 1 0 satisfies the assumptions (i)-(iii) of theorem 5, part I if d < 0, the LiCnard system satisfies the assumptions (i’)-(iv’) of theorem 5, part II. Therefore, the corollary follows from theorem 5. n PROPOSITION 7. If a fundamental
system has a periodic orbit it must be unstable.
1188
J. LLIBRE
and
J. SOTOMAYOR
Fig. 4. Return map of a DSC.
Proof. Clearly, periodic orbits of fundamental systems must be internally unstable, since they enclose the origin which is an attractor. If some periodic orbit were semistable, a small rotation of the FS would produce another FS with a pair of periodic orbits, one unstable and one stable. See [5, Chapter V]. This would contradict proposition 6. n LEMMA 8. If a fundamental
system has a double saddle connection, then it must be internally
unstable. Proof. From propositions 1 and 3, the basic parameters of the FS must verify d < 0 and t > 0. The study of the return map T for the DSC gives the following: T splits into the composition of 4 mappings Tl, T2, T3, T4, as illustrated in Fig. 4. The mappings T, and c have the form d2 = a(d,/a)‘, with t9 < 1, which comes from 6’ = -h-/A+, with t = I- + 1, > 0. The mappings T, and & are smooth diffeomorphisms which have the form d + c(d)d, wiih a positive continuous c. The composition of these four mappings gives T as a strong expansion of the form d + f (d)d’ , for a positive continuous f. H PROPOSITION 9. If a fundamental system has a double saddle connection (DSC), then it does not have periodic orbits and the DSC must bound the basin of attraction of the origin.
Proof. Clearly the DSC must enclose any periodic orbit which necessarily surrounds the origin. By proposition 7 and lemma 8 this would lead to a contradiction since both the DSC and the periodic orbit are unstable. n
Notice that from proposition and (b) of theorem A.
3, corollary 6 and propositions
7 and 9, follow statements (a)
1189
Phase portraits of planar control systems 4. PHASE
PORTRAITS
FOR d 2 0 AND
t5 0
In this section consider a FS x’ = Jx + yl(k * x)b, and denote by d, t, D and T its basic parameters. Next lemma is the standard integral representation of solutions of differential systems. It will be used several times in what follows. LEMMA
10. If x(s) is a solution of a fundamental x(s) = eJSx(0) +
Proof.
Immediate by differentiation.
system, then
’ eJcS-“cp(k - x(r))b dr.
0
n
PROPOSITION11. If d > 0 and t < 0, then Q(0) = R2. Proof. In this case there are positive constants L, K and y such that l(eJ”]l 5 L emys and ((o(k * x)b( I K. Taking norms in the equality of the previous lemma, we get:
eJcswr)111 q(k - x(r)) b 1dr.
IW 5 lleJsll . b@)I + Therefore,
= L e-Y”(lx(0)l - K/y) + LK/y. Hence, every solution of the FS has its o-limit
set contained in the ball of center 0 and radius
LK/y.
From propositions 1 and 3, the system has the origin as unique equilibrium and does not have periodic orbits. Poincare-Bendixson theorem implies the o-limit set is the origin. That is Q(0) = R?.
n
Remark. The proof of this proposition completeness. See also [3].
is standard;
it is included here for the sake of
PROPOSITION12. If d = 0 and t < 0, then Q(O) = R2. Proof. We can assume that the matrix J is in Jordan form: J=
[ 00’1 A-
with
’
t
= A- < 0.
Let x(s) = (x1(s), x&r)) be a solution of the fundamental Ix,(s)\ I lex-s 1x,(O)1 +
system. From lemma 10, we have that
’ lex-@-‘) 1 tp(k * x(r))&1 dr.
0
Clearly, there exists a K > 0 such that Icp(k * x)b,l I K.
1190
J. LLIBRE
and J. SOTOMAYOR
Therefore,
Hence, as s tends to infinity, limlx,(s)I I -K/A-. It follows that the solutions of the FS have their w-limit sets on the vertical strip S bounded by the straight lines x, = +K/A _ , which turns out to be positively invariant under the flow. Since D = I _ b2 k2 > 0, k2 # 0. Then there exists h,, > 0 such that, for h > ho, the horizontal sides of the rectangle S, = (x E S, Ix,1 I h) do not intersect the parallel straight lines r+ and r- , for all h 2 ho. Therefore, on the horizontal sides of S,, the second component of the FS is constant and equal to f ub2, having opposite signs on opposite sides. We claim that the rectangle is positively invariant for all h 1 h,. In fact, it is sufficient to verify that on the horizontal sides, the FS point inward. Otherwise, the a-limit set of the orbit through (0, h) would be contained in S,. This is in contradiction with the fact that the origin is a local attractor and with the absence of periodic orbits, by proposition 3. Now since S,, is positively invariant for large h and the w-limit set of any orbit is contained in S, it follows by the Poincare-Bendixson theorem that n(O) = IR2. n PROPOSITION
13. If d > 0 and t = 0, then n(O) = lR2.
Proof. The matrix J must be of the form:
0 P -p [ 01’
with /3 > 0.
As in the proof of case 2 in proof of proposition Lienard form: Y’ = -g(x),
x’ = y - F(x),
4, we can write the FS in the following
with g(x) = dx + qo(x)(D - d), F(x) = q$x)T.
Notice that, due to the time reversal, the origin on this system is a repellor. Furthermore, it satisfies the assumptions (i) and (ii) of theorem 5 and, instead of (iii), it verifies that F(x) is an odd function with xF(x) < 0. Repetition of the arguments of the proof of theorem 5 for this system, leads to OB - OA > 0 for all arcs of orbits on x > 0. Then, by the symmetry with respect to the origin, all the solutions tend to infinity spiraling around the origin. Due to the time reversal, the origin of the FS is a global attractor, that is Q(O) = iE2. n Remarks. Arguments similar to those in the last proof can also be used to obtain propositions 11 and 12. Notice that proposition 13 does not follow from the work of Meyer [I, p. 55-561. In fact his conditions (9.2) imply in the present situation that T = 0, which is not the case here. PROPOSITION
14. If d = t = 0, then Q(O) = lR2.
Phase
portraits
of planar
control
1191
systems
Proof. The matrix J must be of the form
0 0 [ 1 0I * The FS can be written as x; =x1 + q(k*x)&.
x; = cp(k * x)b,,
Consider the following
Liapunov
function: k.x
V(Xl,X2) = 1/2(-&/&)x,2
+
r0
909 dr.
Clearly, V > 0 for x # 0, since -kz/bl = D(b,)-2 > 0. Also, differentiation with respect to the FS gives: I/’ = Tp2(k * x) < 0, if k * x f 0. Therefore, the origin is globally asymptotically stable, i.e. a(O) = lR2. n The idea of using the above Liapunov [l, p. 53-541.
Remark.
From propositions
function
in this case comes from Meyer
11-14 it follows the proof of theorem B. 5. PHASE
PORTRAITS
FOR
d 2 0 AND
t > 0.
In this section we will prove theorem C. PROPOSITION
topologically
15. If d > 0 and t > 0, then the phase portrait equivalent to that of C, Fig. 1.
of the fundamental
Proof. From Section 2, J must be equal to one of the following
(^b’
12)y
(:l
il)
Or
(-;
where the As, (Y and fi are positive. There are positive constants L, K and y such that ]le-JSlj I L e-” If x(s) is a solution of a fundamental system, then s emJsx(s) = x(0) + emJrq(k - x(r) b dr. .r0 Taking norms we get s Le+(x(s)l 1 Ix(O)/ - KL e-?‘dr. i0 Consequently, Ix(s)l 2 L-’ eY”(lx(0)l - KL/y)
system is
matrices: aa>> and l&k * x)b] I K.
+ K/y.
Hence, every solution whose initial condition is outside the ball B(LK/y), of center 0 and radius tends to infinity in forward time. Therefore, since the origin which is the unique equilibrium is an attractor, there must be a periodic orbit, by Poincare-Bendixson theorem. Furthermore, n(O) is contained inside the ball B(Lk/y). By theorem A, the proposition follows. n
LK/y,
1192
J. LLIBRE
PROPOSITION
topologically Proof.
and J. SOTOMAYOR
16. If d = 0 and t > 0, then the phase portrait equivalent to that of Fig. 1.e.
of the fundamental
system is
From Section 2, J must be
where the A = t > 0. If x(s) is a solution of a fundamental
x1(s) =
e""x,(O) +
s s
system, then
eLcrms)yl(k* x(r)b, dr.
0
If K > 0 is such that (p(k +x)b, 1 I K, then taking norms get Ixl(s)l 5 exs(Ix,(O)1 - K /Ie-‘rdr)
= ex”(lxl(0)l - K/A) + K/A.
Hence, every solution whose initial condition x(0) is outside the vertical strip S bounded by the lines xi = *K/A, tends to infinity. Also, if Ix,(O)1 = K/A, for positive time, s, it holds that Ixi(s)l z K/A, as follows from the above inequality. Therefore, the strip S is negatively invariant under the flow of the FS. Since D = A&k, > 0, kZ # 0. Then there exists H > 0 such that, for h > H, the horizontal sides of the rectangle S, = ((xi, x2) E S : Ix,1 5 h] are disjoint from the lines L and lY+. Therefore, on the horizontal sides of S, the vertical component of the FS is constant, equal to +u&, and with opposite signs on different sides. We will verify that S, is negatively invariant for all h z H. Since D = Ab2k, > 0 and T = 1 + bl k, + b2 k2 < 0, we have that bzkz > 0 and bl kl < 0. There are four cases to consider: (1) kl > 0, k2 > 0; b, < 0, b2 > 0; (2) kl < 0, k2 > 0; b, > 0, b2 > 0; (3) kl > 0, kz < 0; 6, < 0, b2 < 0; (4) kl < 0, k2 < 0; bl > 0, b2 < 0. It is sufficient to analyze the first case, since the others are similar. In fact, on the upper horizontal side of the rectangle S,, , xi = ub, > 0 and consequently xi = -ub, on the lower side. This concludes our verification in this case. Now, since S, is negatively invariant for all h 1 H, and the solutions starting outside S escape to infinity for positive time, it follows that all solutions outside the rectangle S,, escape to infinity. Under these circumstances, since the origin is an attractor, Poincare-Bendixson theorem implies the existence of periodic orbits inside S,, . From theorem A, there is a unique unstable periodic orbit there. n Remark. The existence of periodic orbits under the assumptions of propositions
also be found in [3], where no uniqueness is mentioned. From propositions
15 and 16 follows the proof of theorem C.
15 and 16 can
Phase portraits of planar control systems 6. PHASE
PORTRAITS
1193
FOR d < 0
In this section we will prove theorem D. To this end, explicit analytical expressions will be given for the hypersurface 6) and for the regions C&Sand SC! in U3- = (d < 01. The expressions are given in Table 2, in terms of functions F and G which themselves are defined in Table 1 from the basic parameters of a FS. For D > 0, T < 0, d < 0, let A,
= (-T
+ (T2 - 4D)“‘)/2,
A+ = (-t
y(A,) = (1 - A,/A+)/(A,/L
k (t2 - 4d)15’)/2,
- 1).
Define F(A-) and G(A-) in Table 1. In terms of the functions F and G, the phase portraits of the FS in 63-, with the notation in Fig. 1, are given in Table 2. This amounts to a reformulation of the conclusions of theorem D. For d < 0, the Jordan form of the FS is J = diag@+, A-),
with A+ > 0 > A-.
We can write the FS in the form x’ = Jx + yl(k - x)b.
Assume that k = (k,, &) is an eigenvector of J. That is, either ki = 0 or k2 = 0. Take kz = 0, since the other case can be studied in the same way. The straight lines IY and I+ are xi = -u/k, and xi = u/k,, respectively. Without loss of generality assume that ki > 0. Since D = L (A+ + b, k,) > 0, it follows that b,k, < 0, and therefore bi > 0. Now we assume that b, > 0; in fact, for b2 I 0 would give the same phase portrait. The separatrices of the saddle points look as in Fig. 5. Table 1. Definition of F(L) T2 - 40
F(L)
and G(k)
for FSs in a3W-)
>o
Y(A-1
=o
YW1 + 1 Ir(A-)l
y(b)“*- y(A-)I’*+ y(AJ - exp(l/2(y(A-) - l/y(A-)) (Im y(AJ)/Re v(A-)) - tan(log(/(A_)~‘m(Y”~))‘Re(y”-))))
Table 2. Analytical characterization of phase portraits in @-
F(L) >l 51
W-l >o =o
Phase portrait l.SC? l.a> 1.&S 1.&S
1194
J. LLIBRE
and J. SOTOMAYOR
i,
7-
I-
3
0
c
et
L
I'
e-
I -.I Fig.
/' I'
I’
r-
i2>0
5
rt
5. Case d < 0, k an eigenvector
Notice that the rectangle S with vertices e, and e- and positively invariant, must contain the unstable separatrices existence of limit cycles (which we know must be unstable). portrait must be as that in Fig. 1.&S. Now we assume that k = (k,, &) is not an eigenvector of lines I- and I+ intersect both coordinate axes. There are four cases to consider: (1) k, > 0, k2 > 0;
of J.
sides parallel to the axes, being of the saddles. This prevents the Therefore, in this case the phase J. Therefore,
the parallel straight
(2) k, > 0, kz < 0; (3) kl < 0, k, > 0; (4) kl > 0, k2 < 0.
It is sufficient to analyze the first case, since the others reduce to it by changing the orientation on one or both axes. In fact, the saddle separatrices look as in Fig. 6. Due to the symmetry with respect to the origin, only the three possibilities (I), (2) and (3) for the positions of the separatrices y!! and y”, arise, as illustrated in Fig. 6. From the results in theorem A, it follows that each of these possibilities correspond, respectively, to the phase portraits SC, D and Q.S of Fig. 1. Now we will see that the three phase portraits are realizable for FSs. In fact, 0% exists when k is an eigenvector of J, for t < 0, because of proposition 3, from which there are no periodic orbits nor DSC. To show that 9 exists we will compute explicitly the analytic relations to be satisfied by the basic parameters of a FS to present a saddle connection. Once this is done and verified that it is nontrivially satisfied by FSs, a rotation argument shows that there are FSs of types @S and Se. Now we proceed to compute the analytic expressions and verify that, in fact, there are FSs fulfilling them.
Phase
Fig.
According to proposition can be written in the form
portraits
of planar
control
systems
1195
6. Case d < 0, k, > 0, k, > 0.
4, after reversing the time variable, a FS with a saddle connection x’ = y - F(x)
Y’ = -g(x),
with g(x) = dx + yl(x)(D - d), F(x) = tx + &x)(T - t), where d < 0, t > 0, D > 0, T < 0. So the FS splits into the following three linear systems: y’ = -dx
+ u(D - d), y’ = -Dx,
y’ = -dx
x’ = y - tx + u(T - t), x’ = y - TX,
- u(D - d),
ifx5
-u;
if -u 5 x 5 u;
x’ = y - tx + u(T - t),
ifx2
(16)
u.
Then, e, = -e- = (-u(D - d)/d, -u(Dt - Td)/d) are the saddles of this system. The numbers L, = (-t + (t2 - 4d)““)22, at the beginning of this section, are the eigenvalues at the saddle points of (16). Notice that due to the change in sign for writing the Lienard form above, A+ above corresponds to I- in the original FS and vice versa. The same remark applies to A, = (-T f (T2 - 4D)“2)/2, the eigenvalues of the attractor at the origin for (16). A straight computation shows that the stable separatrix of e- that intersects the line lY_ does it at the point pm = u(-1,
-T-
DA-/d).
1196
J. LLIBRE
Fig.
7. Geometry
and J. SOTOMAYOR
of conditions
for a DSC.
On the other hand, the unstable separatrix of e, that intersects r+ does it at the point p+ = ~(1, T + DA+/d). See Fig. 7 for an illustration. In order to get a saddle connection, we must find the values of d, D, t, T for which the solution of (16) that passes through p+ also passes through p- . There are two cases to consider.
Case 1. T2 - 40 # 0. If A, = (-T be written as
f (T2 - 40)““)22,
then the general solution of (16) can
(x(.s),y(s)) = (aeSA- + beSA+, -aA+ es*- - bh-
es*+).
In order that (x(O), y(O)) = p+ , it must hold that a = a* = uA+(AJL
- l)/(A-
- A+),
b = b* = -uA-(A+/L
- l)/(A-
- A+).
Denote by (x*(s),y*(s)) the solution of (16) with a = a* and b = b*. Now we will analyze the possibility of existing some positive S, for which (x*&J, y*(s,)) = p- . For the case T2 - 40 > 0, calculation shows that, with the notation in Table 1, v(A) = (1 - AJL+)/(A-/L - 1) > 1 amounts to the existence of the time s0 for the solution to reach the line IY. The condition to actually reach the point p- is written as y(AJ”*-
- y(A+)“*+
= 0.
For the case T2 - 40 < 0, use the real form of the complex general solution above and the notation in Table 1. Calculation shows that Iv(A-)l > 1 amounts to the existence of the time so for the solution to reach the line A-. The actual encounter with the point p- is expressed as (Im y(A-))/(Re
y(A))
- tan(log(~y(AJ~‘m(y~A-))‘R”(y~“-))) = 0.
Phase
portraits
of planar
control
systems
1197
Case 2. T2 - 40 = 0. The general solution of (16) is (x(s), y(s)) = ((a + sb) e-sT’2, (a + b(s + 2/T))T
e- sT’2/2).
In order that (x(O),y(O)) = p+, it must hold that a = a* = u, b = b* = uT(1 + T/2L)/2. As above, denote by (x*(s), y*(s)) the solution of (16) with a = a* and b = b*. Now proceed to analyze the possibility of existing some positive s,, for which (x*(&J, y*(s,)) = p- . Using the notation in Table 1, calculation shows that y(A) > 0 amounts to the existence of s,, to reach I-- ; the actual encounter with p- happens only when y(k)
= exp((y(A-)
- l/#-)/2).
Since the points p+ depend analytically on the basic parameters, it follows that the hypersurface 33 corresponding to double saddle connections is analytic in these parameters, independently on the conditions on the sign of T2 - 40 used in its definition. This ends the proof of theorem D. W Acknowledgements-The
first author had initial discussions on fundamental systems with R. Suarez. He is partially supported by CIRIT and DGICYT grants. This paper was written in part during his visit to IMPA in 1992, supported by CNPq, Brazil. The second author wrote part of this paper in IMPA, in 1992 and completed it while visiting the Centre de Recerca Matematica, Barcelona, in 1993. REFERENCES 1. LEFSCHETZ 2. NARENDRA New York 3. ALVAREZ University 4. LLIBRE Barcelona 5. ZHANG Vol. 101. 6. LEVINSON 7. LIENARD
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