Bifur ations, s aling laws and hysteresis
in singularly perturbed systems
1
N. Berglund
Weierstra-Institut fur Angewandte Analysis und Sto hastik
Mohrenstrae 39, D-10117 Berlin, Germany
Abstra t
Relations between the solutions of singularly perturbed di erential equations "x_ = f(x; t)
and bifur ations in the asso iated one-parameter families of di erential equations x_ = f(x; )
are examined. We dis uss bifur ations with single or double zero eigenvalue, as well as the Hopf
bifur ation.
We onsider singularly perturbed dynami al systems of the form
dx = f (x; t); x 2 R ; t 2 R ; 0 < " 1;
(1)
"
dt
where f is analyti in an open subset of R R . Our aim is to determine the qualitative
behaviour of solutions of (1), using properties of the asso iated family of dynami al systems
dx = f (x; );
(2)
ds
where is onsidered as a xed parameter. Equation (1) an be viewed as a version of (2) in
whi h the parameter = "s = t is made slowly time{dependent.
The existen e of a relation between solutions of (1) and (2) is on rmed by the following
result [1, 2, 3, 4℄: Assume that x () is a family of hyperboli equilibria of (2). Then (1) admits
a parti ular solution x(t) su h that kx(t) x (t)k 6 "; uniformly for in ompa t intervals.
If x () is asymptoti ally stable, x(t) attra ts nearby solutions exponentially fast. If x () has
both stable and unstable manifolds, one an asso iate time-dependent invariant manifolds with
x(t) on whi h the motion is either ontra ting or expanding.
This naturally raises the question of what e e t a bifur ation in (2) has on the dynami s of
(1). We will dis uss several types of lo al bifur ations. See [5, 6, 7℄ for related results.
d
d
?
?
?
1
?
Bifur ations with simple zero eigenvalue
Bifur ations with a simple zero eigenvalue have two main e e ts: Firstly, when approa hing the
bifur ation point, solutions tra king an equilibrium are no longer attra ted at an exponential
rate. This results in the distan e between solution and equilibrium s aling as another power of
" than 1. Se ondly, there may be several equilibrium bran hes emerging from the bifur ation
point, or no bran h at all. The solutions will then hoose between tra king one of the outgoing
bran hes, jumping on another attra tor, or diverging.
If the fun tion f depends periodi ally on time, the se ond me hanism may result in hysteresis
phenomena [8, 9℄, whi h may be haoti , even for arbitrarily small " [10℄. In this ase, the rst
me hanism produ es non-trivial s aling laws for the area of hysteresis y les. The s aling exponents an be obtained in a simple way from the Newton polygon asso iated with the bifur ation
1 Present Address:
S hool of Physi s, Georgia Te h, Atlanta GA 30332-0430, USA
point. Consider (1) for d = 1 and
X
f (x; t) =
nm
00 = 10 = 0:
x n tm ;
(3)
n;m
A well-known result of bifur ation theory states that if f (x; t) vanishes on a urve2 x = x ( ) j j , then q is ne essarily the slope of a segment of Newton's polygon, obtained by taking the
onvex envelope of the points (n; m) 2 N 2 su h that
6= 0. One an show that generi ally,
the fun tion a(t) = f (x (t); t) satis es
q (n 1) + m
6= 0 :
(4)
ja(t)j jtj ; p = >min
1 >0
?
q
nm
?
x
p
n
nm
;m
Graphi ally, p is the ordinate at 1 of the tangent to Newton's polygon with slope q.
Theorem 1 ([11, 9℄). Assume that for T 6 t < 0, f admits an equilibrium bran h x (t) jtj , satisfying (4), and su h that f (x (t) + y; t) < 0 for small positive y. Then (1) admits a
?
q
?
parti ular solution satisfying
x(t)
2
x? (t) (
"jtjq p 1
"q=p+1
if T 6 t 6 "1 +1
if "1 +1 6 t 6 0
=p
(5)
=p
Hopf Bifur ation
Neishtadt showed that the slow passage through a Hopf bifur ation results in the delayed appearan e of os illations. Assume that the linearization around the equilibrium x () has two
eigenvalues a() i !(), where a() has the same sign as , !(0) 6= 0, and all other eigenvalues
have a stri tly negative real part.
Theorem 2 ([12℄). All solutions of (1) starting in some neighbourhood of x (t0 ) at t0 < 0
tra k the equilibrium x (t) at a distan e of order " for t0 + Æ(") 6 t 6 t^ Æ("), where Æ(") goes
ontinuously to zero as " ! 0, and t^ = minf(t0 ); t+ g is onstru ted in the following way. For
t 2 C de ne
?
?
?
(t) =
Z
t
a(s) + i ! (s) ds:
(6)
0
Then (t0 ) is the smallest positive time su h that Re (t) = Re (t0 ). The bu er time t+ is the
largest positive time whi h an be onne ted to the negative real axis by a path in the omplex
plane with onstant Re , having ertain regularity properties stated in [12℄.
3
Bifur ations with double zero eigenvalue
The delayed appearan e of large amplitude os illations in the dynami Hopf bifur ation may
have disastrous onsequen es, if the equation des ribes a slowly ageing devi e whi h rea hes a
stability boundary, sin e it results in a sudden jump of some state variable. A way to prevent
this delay is to ontrol the system by a state feedba k of the form
dx = f (x; t) + b u(x; t);
(7)
"
dt
2 We write '("; t) "; t) if there exist
(
onstants
> 0 su
h that
' 6 6 + ' for small "; t.
where f (x; t) is the original ve tor eld undergoing Hopf bifur ation, b is a given ve tor in R ,
and u(x; t) is the s alar feedba k ontrol. Theorem 2 indi ates that in order to suppress the
bifur ation delay, one may try to de rease the bu er time t+ by produ ing a bifur ation with
double zero eigenvalue. In [13℄ we show that the ontrol u an be onstru ted in su h a way
that the dynami s of (7) is des ribed by the e e tive two-dimensional equation
"_ = (8)
"_ = (t) + 2a(t) + (t) 2 + Æ (t) 3 2 + "R(; ; t; ") + O (( 2 + 2 )2 );
where a(0) = (0) = (0) = Æ(0) = 0, and R(0; 0; t; 0) is related to the drift d x (t). The
asso iated system is a so- alled ubi Lienard equation, whi h is a odimension four unfolding
of the singular ve tor eld (; 3 2) [14℄.
Theorem 3 ([13℄). Assume that 0 (0) > 2a0 (0) > 0 and R(0; 0; 0; 0) 6= 0. Then solutions of
(8) starting suÆ iently lose to the origin at t0 < 0 will tra k, for t > 0, one of the equilibrium
bran hes of (8) given by = 0 and = t1 2 + O(t), at a distan e at most O("1 3 ). In other
n
t
=
?
=
words, the bifur ation delay is suppressed sin e solutions do not tra k the original equilibrium
represented by = = 0.
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