Rev.Acad.Canar.Cienc., V (Núm. 1), 41-46 (1993)
An elementary exp licit example of unbounded limit behaviour
on the p lane
JOSÉ SABINA DE LIS
Departamento de Análisis Matemático, Universidad de La Laguna.
Abstract. An academic new example of two-dimensional planar dynamical system is constructed
to describe a very well-known fact. Namely, that unbounded semiorbits could genenate
nonconnected w-limit sets (see (H!J. page 48).
l. Introduction. The concept of w-limit set was introduced by .D. Birkhoff ( cf. (Bi]).
For a differential equation in R n
x' = f(x) (1)
the w-limit set of a certain solution or semiorbit "summarizes", roughly speaking, the
asymptotic behaviour of such a solution. If x = x(t) is a solution to (1) defined in t 2 t0 , a
z E Rn is said to be an w-limit point of x(t) (see (Bi], (Hl]) ifthere exists {tn}, tn-... oo, such
that x(tn) -... z. For a fixed solution x = x(t), or better, for the semiorbit·/' = ·{ x(t)/t 2'. t 0 }
attached to x = x(t), A+(/') usually designates the set of w-limit points of 1'· ·As a possible
reliable picture of a physical phenomenom it is clear that a major emphasis ~ust be put in
the study of A+(f') when /'is a bounded semiorbit to (1). In bounded regions of R2 with
finitely many critica! singularities of (1), the structure of A+(f') for semiorbits i liying in
such regions, is given by the celebrated Poincaré-Bendixon Theorem (see (Ha],(Hl]). In Rn
the more general information abo u t A+ (T'), for /' bounded, is that contained in the next
result (see for instance (Hl] page 47)
THEOREM .
If f' = {x(t)/t 2 t 0 } C Rn is a bounded semiorbit to (1) tben A+(T') is a nonempty,
invariant, compact and connected set.
Even in R 3 it is sometimes hardly possible to add a bit more.to the general asserts given
above about A-+(f') (see for instance the Lorenz's system in (GH]).
It is also well-known that the connectedness of A+(/') is a consequence of the boundedness
of 1'· Here we will focuss our attention in this precise fact. When A+(f') contains two
points Z¡ , z2, the semiorbit f' will meet infinitely many times every pair of arbitrarily small
neighbourhoods U1, U2 of Z¡ and z2 (respectively). Boundedness of /' will imply that Z¡
and z2 will be "connected" into A+(f'). The objective of this note is giving an explicit
example in R 2 that such connectedness is lost when /'is unbounded. Obvíously,_.this fact
is well-known since long (see for instance (Hl] page 48). Moreover, after thinking on it for a
while, it is not difficult to arrive to the conclusion that a picture of such an orbit "( shuold
be more or less as shown in figure l. .
What is presented in this work is a class of equations that make precise in an ·ex.pliéÍt
and analytic way this kind of behaviour.
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2.The results.
Let <po = 'Po(Y) E C1([0, +=)) such that cp0 (0) = O, cp0 (y) > O in y > O, and also
that limy-+oo cp0 (y) = O. Suppose, without loss of generality, that maxy;:::o lcp0 (y )1 < l.
Designate by cp = cp(y) the odd extension of cp0 and cal! a = maxy>o lcp0 (y)I. A simple
example of such a function is -
y
cp(y) = 1 + y2 .
figure 1
Consider now the following class of equations
{ x' = y(l - x 2 )
y'= cp(y)- X.
(2)
It will be shown that the w-limit set of every semiorbit 'Y starting at (x0 , y0 ) .¡. (O, O),
x~ < 1, exactly consits of the lines { x = 1} U { x = -1}. Let us first introduce the following
regions,
I = {(x, y)/y > O,cp(y) > x},
JI= {(x, y)/y > O,cp(y) < x},
III = {(x,y)/y < O,cp(y) < x},
IV= {(x,y)/y < O,cp(y) > x}.
JI
-.1
IV
figure 2
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The first fact to be proven below is the rotation around (O, O) of the solutions to (2)
lying in the strip (-1,1) x R.
LEMMA 2.1.
Let (x(t), y(t)) be a noncontinuable solution to (2) with initial position (x 0 , y0 ) E
(-1, 1) x R and maximal existence interval (a,w). Then w = +oo and (x(t),y(t)) pass
through the regions I, JI, III and IV -following this ordered sequence- arriving again in
finite time to I and repeating this sequence infinitely many times.
Remark.i.
a) In Lemma 2.1 the initial data (x 0 , y 0 ) could be taken in either of the regions JI, JI I or
IV. Then, the orbit starting at (x 0 , y0 ) will cross trough the remaining regions in a cyclic
and ordered way.
b )For v = (V¡, v2), V¡ f= O, v~ + v~ = 1 !et us define
where A designates any of the regions introduced above. r v can be ordered in the natural
way regarding s. Lemma 2.1 allows us defining the Poincaré map 7r (see [Ha], [Hl])
over r v· In fact , call (x(t, to, Xo, Yo), y(t, to, Xo, Yo)) the (unique) solution to (2) satisfying
(x(O),y(O)) = (x 0 ,y0 ). If (x 0 ,y0 ) E I'v then there exists r(x0 ,y0 ) = min{t >
O/(x(t,t0 ,x0 ,y0 ),y(t,t0 ,x0 , y0 )) E fv}· From Lemma 2.1 it follows that r(x 0 ,y0 ) is defined
and positive whatever the choice of (xo, Yo) E r V be. For (xo, Yo) E r V define s by
the equality (x 0 ,y0 ) = sv E I'v, write r(s) = r(x0 ,y0 ) and define u= u(s) by means of
uv = (x(r(s), x0 , y0 ), y(r(s), x0 , y0 )). lt is well-known that
II: I'v-+I'v
sv-+ u(s)v
(the Poincaré 's mapping) is an increasing C1 mapping whose (possible) fixed points give
rise periodic solutions to (2). In this case it also possible to show that II is also a C1
difeomorfism. Notice that s belongs to the interval (O, lv\ 1) provided ~ < O, meanwhile
I'v could consist on severa! connected pieces if ~>O. Designate by (a(v), 1.,1, 1) that one
the most separated from (O, O). For simplicity let us also set a( v) = O when ~ is negative.
We can state the following result,
LEMMA 2.2.
For each V E R 2 , !vi = 1, V¡ t= o, define the interval Iv = (a(v), ¡.,1,¡) and the C 1
difeomorfism u : Iv -+ (Iv) . Then, far each s E Iv, s < u( s ). Moreover,
S-+US
a) Equation (2) does not exhibit limit cycles into (-1, 1) x R .
b) Far each s E Iv, limun(s) = 1.,1,¡
The next result is the objective of this note and is a straightforward consequence of
Lemma 2.2.
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THEOREM 2.3.
Far each ( x o, Yo) lying in the strip ( -1, 1) X R the unique semiorbit 'Y passing through
(xo, Yo),
goes off the origin (O, O) turning around this point infinitely many times and satisfying
A+('Y) = {(x,y)/x = lor x = -1}.
3. The proofs.
3.1. The proof of Lemma 2.1.
First notice that the strip (-1, 1) x R is invariant. Indeed, x = 1 and x = -1 are orbits
of (2). Assume that (x 0 , y0 ) E I and !et (x(t), y(t)) the solution starting at that point,
with maximal existence interval (a,w), then (x(t) , y(t) leaves I at finite time. Otherwise,
(x(t),y(t)) E I and x'(t) >O, y'(t) >O for each t E (a,w). However, since x(t) <a for
each t then lim1-w y(t) = +oo. Indeed, y(t) can not be bounded. Otherwise, w = +oo,
which implies lim1-+oo x'(t) = lim1-+oo y'(t) = O what contradicts x(t) 2 x0 , y(t) 2 y0
for each t. Therefore, limt-w y(t) = +oo. But limt-w y(t) = +oo entails w = +oo since,
y'= <p(y) - x(t)
S <p(y) - Xo
s <p(y) - l.
and the solutions to z' = <p(z) - 1 do not blow up since J,+00 -'!JL_(di) = +oo. Therefore
Yo 'I' Y -
we must conclude that lim1-+oo y(t) = +oo. However w = +oo and the fact x(t) < a for
t > t0 imply lim1-+00 x'(t) =O. On the other hand
x'(t) = y(t)(l - x2) 2 y(t)(l - a 2 )
for t > t0 ; and lim1-+00 y(t)(l - a 2 ) = +oo . Thus (x(t), y(t)) must leave I at finite
time. In other words, there exits ti E [O,w) sucli that x(ti) = <p(y(ti)) whicli implies
(x(t),y(t)) E JI for t =ti+€, and certain positive small enough €. In fact, x'(ti) =
y(t¡)(l - x2(ti)) = y(ti)(l - <p(y(ti))2) is positive and the function h(t) = x(t) - <p(y(t))
vanishes at t =ti with h'(ti) = x'(t1)- ~(y(t 1 ))y'(t 1 ) = x'(ti) >O. Thus, the existence
of € is proveo.
Next, designate by (x1, y¡)= (x(ti + €), y(ti + €)). We are going to show the existence
of t2 > t1 + € sucli that O< x(t2) < 1, y(t2) = Y2 =O and so that (x(t), y(t)) E JI for eacli
t1 S t < t2. To see this, set X = {y S Yi} n I J. Then, if necessary taking a smaller €,
(x(t), y(t)) E X for t¡ S t S t¡ + €. We claim that a t2 >ti exist so that (x(t2), y(t2)) E
ax. Otherwise w = +oo and we wolud have lim1-+oo(x(t),y(t)) = (x2,y2) with X2 >X¡.
However, monotonicity of both x(t) and y(t) would imply lim1-+00(x'(t), y'(t)) = (O, O),
what is not possible. On the other hand, if t2 = min{t/t > ti,(x(t) ,y(t)) E . ax} then
(x(t2),y(t2)) = (x2,0) with o < X2 < l. In fact, observe that a part of ax con~ist of
y = y1, other one consist of a piece of the graph x = <p(y) meanwhile a third part consist
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of a piecc of the orbit x = l. Because of the direction field of (2) the only piece of fJK
where the ':.hit points of (x(t), y(t)) can be located is just y= O, as claimed (see figure 3).
Finally, since ·y'(t2) = -x2 <O there exist t2 < t3 < w such that (x(t3),y(t3)) E III.
Thus, it can be asserted that every solution to (2) starting in an arbitrary ( x 0 , y0 ) E I reachs
the region I II in a finite period of time t, after passing through the regions I , I J. Therefore,
it is straightforward to show that every semiorbit ¡ to (2) starting in ( X3, y3) E I II also
reachs the region I -after passing through II I and IV- in a finite period of time. In fact,
'f! = 'f!(Y) was chosen in order to (2) were symmetric with respect to (O, O). Specifically, the
orbit 'Y starting at (x 0 , y0 ) = (-x3, -y3) E I is the symmetric of ¡ with respect to (O, O).
This is dueto the following fact "(x(t ), y(t)), t E (a,w), is a noncontinuable solution to
(2) if and only if (x1(t), y¡(t)) = (-x(t), -y(t)), t E (a,w) is a noncontinuable solution to
(2)" . So, by the uniqueness of solutions,
x(t, -x0 , -yo)= -x(t, Xo, Yo )
y(t, -x0 ,-y0 ) = -y(t,x0 ,y0 ), t E (a,w).
figure 3
Thus, every semiorbit ¡ starting in (x0 , y0 ) E I finally arrive again in I after crossing
I ,II,III,IV in finite t ime, as was claimed.
3.2. The proof of Lemma 2.2.
Firstly, it should be remarked that, in the cases where ~ < O the orbits to (2) reach in
finite time the connected piece of r v most separated from (O, O). To see t his, it suffices with
empluying the argument to be developed below, and concerning the Lyapunov function
V(x , y)= y2 - log(l - x2).
Now observe that for each v E R 2,v1 -# O,lvl = 1, there exists s0 E(a(v), 1.,1,¡) such
that s0 < <7(s 0 ). To see this Jet us observe that by replacing the solution to (2) (x(t), y(t))
instead of ( x , y) in the (Lyapunov) function
V(x, y)= y2 - log (l - x2) (3)
we get, after derivation with regard t,
d dt (V(x(t ), y(t)) = 2y'f!(y).
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On the other hand, observe that V ::'.:'. O and is convex in the strip (-1, 1) x R, V= O if
and only if y = O. In additíon, the restriction of V to every segment r v is an increasing
function of s, provlded v2 # O. Therefore, the existence of a s 0 E (a( v ), 1.,1,¡) such that
s0 < a(s0 ) follows, in the case V2 # O. By continuity, and using again the behaviour of
V the same holds true for the segments r v, v2 = O. Own existence of V also entails that
a(s) # s for each s E (a( v ), ¡f.¡) and for each v E R 2 , lv11 # O, lvl = l. This fact avoids the
possible existence of infinitely many closed orbits to (2), /n, whose intersection points with
the open segment (O, 1) x {O}, say (xn, O), could converge towards the point (x, y) = (1, O).
We should observe that (2) is a perturbation of the equation (4) below (see remark a)),
which exhibits a continuum of periodic orbits "filling" the strip (-1, 1) x R. Therefore, we
need to rule out the existence of such a family of closed orbits converging to the "sides"
x = ±1 of the strip. On the other hand, since the existence of a S¡ E (a( v ), 1.,1,¡) such
that S¡ < a(s1) would entail - together with the own existence of s0 - the existence of an
"intermediate" s2 so that a(s2) = s2, such kind of s1 can not exist. Thus, s < a(s) for each
s and so, the sequence { an( s)} is always increasing whatever the values of s E (a( v ), 1.,1,¡)
and v be. As the limit s = liman(s) exists it must necessarily be s = ¡+,¡, otherwise
a( s) = s, what is not possible.
Remarks
a) Equation (2) is a perturbation of the equation
{ x' = y(l - x 2 )
y'= -x.
(4)
Equation ( 4) exhibits a continuum of closed orbits filling the strip ( -1, 1) x R and surrounding
(O, O). This can be easyly checked by using the first integral V(x , y) = y2 - log(l - x 2 ).
b) If we cal! U(x , y) the right hand side of (3) -what is usually coined as the derivative of
V with regard to equation (2)- it is checked that the single point {(0,0)} is the )nvariant
maximal set contained into {U(x, y) = O}. Therefore, by reserving the time t in (2) and
using the La Salle's invarianze principie (see [HI], Theorem 1.3 page 316) we can complete
the picture of the phase portrait of (2) by asserting that
lim (x(t, X 0 , y0 ), y(t, X 0 , Yo))= (O, O),
t--CX>
for each (x 0 , y0 ) E (-1, 1) X R.
c) The assertion concerning A+(¡) in Theorem 2.3 is an inmediate consequence of the fact
lim O' n ( S) = ¡+,¡ On every segment r V.
REFERENCES
[Bi]. G. D. Birkhoff, Dynamical Systems, AMS Colloq. Publications, Providence (1927).
[GH]. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector
Fields, Springer Verlag, Berlin (1983).
[Ha] . P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).
[HI]. J. K. Hale, Ordinary Differential Equations, R. E. Krieger Pub. Co., New York (1980).
38271 La Laguna, Spain.
Recib i do : 15 de Junio de 1993
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Rev.Acad . Canar.Cienc. , V (Núm.!), 47 - 59 (1993)
TAMAÑO MUESTRAL MÍNIMO Y CONTRASTE DE DOS
PROPORCIONES BINOMIALES.
AUTORES: SÁNCHEZ GARCÍA, Miguel*
CUESTA AL V ARO, Pedro **
FELIPE ORTEGA, Ángel***
• Facultad de Medicina. Universidad Complutense.
•• Servicios Infonnáticos. Universidad Complutense.
••• Facultad de Matemáticas. Universidad Complutense.
PALABRAS CLAVE
CLASIFICACIÓN AMS
Sample Size, Binomial Model,
Test Hypothesis.
62F03.
(Este trabajo ha sido parcialmente desarrollado dentro del Proyecto CICYT INF91-74).
RESUMEN
En este trabajo diseñamos un algoritmo para calcular el mínimo tamaño muestra!
para el test de dos parámetros binomiales.
Se contrasta H0 = p, = p2 = Po frente Hª = p, = Po - .ó. y p2 = Po + .ó.,
O < .ó. < p0 s 1/2 con nivel de significación (Error del primer tipo) menor que a y
función de potencia mayor que 1-P (Error del segundo tipo menor que P). Se
proporcionan unas tablas que definen la función de decisión entre las dos hipótesis.
ABSTRACT
In this paper, we design an algorithm to calculate the minimum sample size for
the two parameters binomial test.
We test H0 =p1 = p2 =p0 against Ha= p1 =p~- .ó. and p2 = p0 +.ó. ,
O< .ó. < p0 s 1/2 with leve! of significance (Type I error) below a and power function
above 1-P (Type II error below P). We supply tables that define the decision function
between the two hyphotesis.
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