Blowout bifurcations of codimension two

20 July 1998

ELSEVIER

PHYSICS LETTERS A

Physics Letters A 244 (1998) 261-270

Blowout bifurcations of codimension two Peter Ashwin, Philip J. Aston

Department of Mathematics and Statistics, University of Surrey, Guildford GV2 SXH, UK

Received 4 February 1998; revised manuscript received 9 April 1998; accepted for publication 21 April 1998 Communicated by A.P. Fordy

Abstract

We consider examples of loss of stability of chaotic attractors in invariant subspaces (blowouts) that occur on varying two parameters, i.e. codimension-two blowout bifurcations. Such bifurcations act as organising centres for nearby codimension- one behaviour, analogous to the case for codimension-two bifurcations of equilibria. We consider examples of blowout bifurcations showing change of criticality, blowouts that occur into two different invariant subspaces and interact, blowouts that occur with onset of hyperchaos, interaction of blowout and symmetry increasing bifurcations and collision of blowout bifurcations. As in the case of bifurcation of equilibria, there are many cases in which one can infer the presence and form of secondary bifurcations and associated branches of attractors. There is presently no generic theory of such higher codimension blowouts (there is not even such a theory for codimension-one blowouts). We want to present a number of examples that would need to be covered by such a theory. @ 1998 Elsevier Science B.V.

1. Introduction

Bifurcations of arbitrary invariant sets of dynamical systems on varying a parameter in the systems seem to show an endless number of possibilities. This, coupled with the fact that chaotic invariant sets in real systems are only rarely structurally stable makes a classification of them appear hopeless.

It probably is a hopeless task to try to get an analogous theory to the classification of bifurcations of equilibria by codimension [ l] and so it is a welcome surprise that in some cases there does seem to be progress possible; namely in the case of dynamics that preserves an invariant subspace. Dynamically invariant subspaces can be forced by symmetries in the system, or for other reasons. Rather than trying to capture all the dynamics, we can try to gain a description of attractors and their containment in invariant subspaces.

Ott and Sommerer [ 21 made a crucial step in this

direction by identifying what they called “blowout” bifurcations of attractors contained within invariant subspaces; although these had been noticed for some years before [3,4] it was Ott and Sommerer that pointed out that basin riddling [5] and on-off intermittency [ 61 are two aspects of the same loss of stability of an attractor within an invariant subspace.

Suppose that we have a dynamical system generated by iterating a smoothly parametrised family of smooth mappings fV : Rm --t Rim, with parameter v E Iw, such that a linear subspace N c R” is invariant under the map. Suppose that A is an attractor contained in N for Y < 0. If A ceases to be an attractor for Y > 0 we say it undergoes a blowout bifurcation at Y = 0. Ott and Sommerer identified two scenarios,

- Hysteretic scenario (subcritical blowout): There are no nearby attractors for v > 0. The basin of A is riddled for Y < 0.

- Nonhysteretic scenario (supercritical blowout) :

0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00334-X

262 I? Ashwin. RJ. Aston/Physics Letters A 244 (1998) 261-270

There is a branch of on-off intermittent attractors that

branch from A on increasing v through 0.

In Ref. [7] we show that for a particular family of

mappings, one can find both scenarios and prove the existence of branches of attractors and their scaling properties. That paper conjectures that in some sense, these two scenarios are “generic codimension-one sce-

narios”.

The purpose of this paper is to look at what new

transitions may happen if there are two parameters, v E Iw*, and observe examples of what we believe are “generic codimension-two” scenarios.

2. Blowout with change of criticality

A blowout bifurcation occurs when the largest normal Lyapunov exponent changes sign. Such a bifur-

cation is subcritical if there is no attractor close to the

invariant subspace after the blowout and is supercritical if there is an attractor which is “stuck on” to the

invariant subspace [ 81. It is possible to have a situa-

tion in which the blowout bifurcation is supercritical

and in which, at another parameter value, the bifur-

cated attractor undergoes some form of crisis and then disappears; it may be that almost all points near the

invariant subspace are in the basin of an attractor that is bounded away from the invariant subspace. Such a

bifurcation has been studied in Ref. [9] but where one

parameter is not normal. We present an example where blowout occurs on

varying one parameter and the second parameter can

be used to change the scenario from subcritical to supercritical. The two-parameter bifurcation diagram for

such a scenario is shown in Fig. 1. The map we consider is a modification of an example in Ref. [ 81,

x’ = q&x(x* - l),

y’ = Yn [ V,e-*2-?.2 +i(l -e-!‘*)+v2r*]. (2.1)

The change of criticality occurs when the upper boundary of the stuck-on attractor is determined by the unstable manifold of a saddle fixed point. If this saddle fixed point collides with another fixed point at a saddle-node bifurcation, then iterates will be able to escape (in our case to an attractor at infinity) and the stuck-on attractor is lost (see Fig. 2).

p2

III

I /(1.429,0.1954)

b I II I I > VI

Fig. 1. Two-parameter bifurcation diagram for the iterated map (2.1). At VI = 1.429 there is a blowout bifurcation that is supercritical for 13 < 0.1954 and subcritical for y > 0.1954. The regions are as follows, I: Invariant subspace stable, II: “stuck-on” attractor, III: no attractor near to the invariant subspace. The var- ious line styles denote: supercritical blowout bifurcation (- - -), subcritical blowout bifurcation (. . .), crisis for “stuck-on” attractor

(--).

1.5

y--J ._:<,;i?;~~i .~ ;__ _________I .-..._ _--- _=:.~=c=

=_---_=i =z:

Y

0.19 0.19 0.20 0.21 0.22 0.23

” 2

Fig. 2. A blowout bifurcation of (2.1) on varying ~1 gives rise to a stuck-on attractor at rq = 1.46 bounded by a saddle fixed point which is lost on varying v2 beyond this point. The dii shows the saddle node of fixed points (above) and the y-component of lo3 points of a trajectory after a transient of length 106 has been allowed to decay. Note the apparent prolongation of stability due to the very long transient times just after the crisis. By increasing the number of iterates in the transient the crisis appears to occur closer to the saddle node.

Clearly the n-axis is invariant for the map, and the dynamics on the x-axis are chaotic with an ab- solutely continuous invariant measure p. The normal Lyapunov exponent for p is

(+1=log., - J x2 dp = log zq - 0.3575.

Thus the blowout bifurcation, which is defined by

P: Ashwin, F?J, Aston/Physics Letters A 244 (1998) 261-270 263

Fig. 3. Model of the dynamics to unfold a change of criticality

caused by a local saddle-node bifurcation occurring on the bound-

ary of an attractor. The coordinates 5 and 7 are local to a saddle

node, see (2.2) and the invariant subspace is located far below

this diagram.

CTI = 0, occurs at I/] = I .4298. The blown-out attrac-

tor contains a saddle fixed point on the y-axis which is repelling in the x-direction and attracting in the y-

direction. The unstable manifold of this saddle is ob-

served to form the upper boundary of the stuck-on at-

tractor for the parameter values investigated. By varying ~2, another fixed point on the y-axis can be made

to coincide with the saddle fixed point in a saddle-node bifurcation. This occurs when vi = i +y2/e’-V2/2. Af-

ter the saddle-node bifurcation has eliminated both the fixed points on the y-axis, then iterates which come close to the axis can escape to infinity and so the stuck- on attractor has been destroyed by the saddle-node bifurcation. There is a codimension-two point when the saddle-node bifurcation occurs at the same point as

the blowout bifurcation and this occurs at (vt,v2) = ( 1.429,O. 1954). When VI = 1.429, the blowout bifur-

cation is subcritical for ~2 > 0.1954 and supercritical for ~2 < 0.1954. An interesting feature of this bifur-

cation is the apparent prolongation of the region of supercriticality due to the presence of very long transients.

2.1. Long transients near change of criticality

Fig. 3 shows a detail near a generic saddle-node bifurcation that has just occurred. One region is rein- jetted into what was the attractor whereas another region escapes to another attractor. We wish to find the scaling of e with an unfolding parameter for the

saddle-node bifurcation (note that the saddle node is

assumed to take place far from any invariant manifold). We model the dynamics near this region by the normal form flow for a saddle node at ~2 = 0,

(=t2+v2, ?j=?,l, (2.2)

where we have scaled time and the coordinates to en-

sure the coefficients are as above. We assume that a trajectory starting at ( - 1, E) will hit (0, 1) after a

time T; this is assumed to be the separatrix for the

trajectories. Solving the above we have

v(t) =ee’ * E = emT,

and so for v2 > 0 we havet = &tan[&(t -

T) ] implying that

T= tan-‘(l/$5)

Jy2 .

(Trajectories from ( - 1, r]) never reach (0,l) for v:! < 0.) If we consider small ~2 then tan-’ (l/

6) N in. and so to lowest order in ~2 > 0 we have

E=e -n/2fi or more generally, just

~ N e-cu,‘l’,

for some constant C > 0. This means that there is an exponentially small gap

through which trajectories can escape when v2 is very small, and so an attractor may seem to persist after it may have in fact lost stability. We expect that escapes will scale with rq and ~2 jointly as

E N zqe _ I-“; ‘1’

where zq is the transverse Lyapunov exponent. The

average transient time T to escape (averaged over an invariant measure just before the crisis) will scale as

T w E-’ . Fig. 4 shows that the transients in (2.1) are well modelled by this scaling in ~2 for transient lengths up to 5 x 1 O6 iterates. The figure was produced

by choosing a random initial condition on the line y = 0.1 and iterating until escape. The above relationship

would predict a linear relationship between the average l/ log(T) and Jv;! - 0.19139; this is evident in the numerical results. Note that numerically escapes are not observed for ~2 close to 0.19139 consistent with the results in Fig. 2.

264 P Ashwin, PJ. AstodPhysics Letters A 244 (1998) 261-270

E g 0.15 -

0.10

0.10 0.15 0.20 0.25

(~~-0 19139) In

Fig. 4. Scaling of l/log(T) of the average escape time T against square root difference of ~2 from the saddle node at 0.19139 for A = 1.46 in (2. I ). The circles show calculated transient lengths while the straight line is a least-squares tit that shows good

agreement with T - e c/m_ Transients of length up to 5 x IO6 were calculated for this figure.

3. Interaction of blowouts

We consider two examples of how blowout bifurcations into different invariant subspaces can interact. The first example is a piecewise linear mapping where we can prove a number of results about the bifurcat- ing attractors (analogous to Ref. [7]). The second

example is a smooth mapping.

3.1. Interaction of blowouts in a piecewise linear mapping

One interesting class of problems are mode interactions where two paths of bifurcations intersect at a point in the two-parameter plane [ lo]. In the unfolding of a mode interaction, secondary bifurcations from the primary branches can be shown to occur. We consider a corresponding scenario where instead of steady solutions we consider chaotic solutions in the mapping

x’ = f(x), y’=g(x,y,z),

z’=h(x,y,z), (3.1)

with (x,y,z) E [0, 1) x lR*. Let 0 = cyo < (~1 < (~2 < cq < cy4 = 1. We define the four intervals Ji = [a;_l,ai],i= 1,2,3,4,Thefunctionf isapiecewise linear map that is linear on each of these intervals and is given by

0 Y I

Fig. 5. Pictorial representation of the normal dynamics assuming tha~Ky>l~dKz<l.

f(X) = X - Ofi-

y X E Ji. Lyi - Lyi-1

We define AI = 2al - ~2 and A2 = 2~3 - cy2 - 1. The parameter A1 measures the difference between the lengths of the intervals 51 and 52. Thus, JI and 52 are of the same length if and only if A1 = 0. The functions g and h are defined as follows,

-if/y1 6 land/z] < 1,thengandharedefinedon the four subintervals (JI, Jz, J3,Jh) by g(x,y,z) = (Y,YJyY9~‘y1Y), MGY,Z) = (Y,z,Y,‘z,z,z);

- if IyI > 1 and 121 d 1, then g(x,y, z) = 1 +

k&(1 -Y)I h(&y,Z) =KzZ; - if [yI 6 1 and lzl > 1, then g(x,y,z) = boy,

h(n,y,z) = 1 +&cl -z);

- if (yI > 1 and Izl > 1, then g(x, y, z) = y;‘y, h(x,y,z) =y,‘z.

We assume that yr, yZ > 1 and so there are regions of expansion and contraction in both y and z near to the x-axis. The parameters &, & > 1 are chosen so that iterates are reinjected back towards the x-axis. The choice of the parameters K~, K, > 0 will be discussed later. The action of this map is shown in Fig. 5.

The system clearly has 22 x Z2 symmetry generated by the transformations y + -y and z + -z. The following fixed point spaces are invariant for the dynamics,

Sl := Fix(& x Z2) = {(n,O,O) : x E [0, l)},

SZ :=Fix(Z$‘)) = {(x,y,O) :x E [O,l),y E Iw},

S3 :=Fix(Zi’)) = {(x,O,z) : x E [O,l),z E W}.

(3.2)

P Ashwin, M Aston/Physi& Letters A 244 (1998) 261-270 265

We show that invariant densities on the attractors can be determined explicitly, not just for the dynamics in Si but also for bifurcated attractors in S2 or Ss (for par-

ticular parameter values). Using these invariant densities, the Lyapunov exponents normal to these fixed point spaces can be calculated explicitly.

The iteration in the one-dimensional invariant subspace Si is chaotic with uniform invariant density. The Lyapunov exponent for this map is

3 (T, = -

c (cu;+i - cu,) log(cy;+i - Ly;) > 0. (3.3) i=o

The linearisation of Eqs. (3.1) on Si is diagonal and is precisely the map defined on the region y < 1 and

z < 1. Thus, the two normal Lyapunov exponents indicate whether or not S, is stable with respect to perturbations in the y- and the z-directions. Since the invariant density of the attractor in Si is uniform, we

can easily find these transverse Lyapunov exponents,

ff; = CQlog 1 + (Ly3 - Ly?_) logy!. + (1 - a3) logy,’

= A2logy!.,

a; = Al logy;.

Interaction of blowouts. We have seen that a blowout bifurcation from Si into S2 occurs when

A2 = 0 and a blowout bifurcation from Si into S:, occurs when Ai = 0. These are the two primary bifurcations and clearly there is a mode interaction when

A, = A2 = 0. We restrict attention to the invariant subspace S2

and note that similar considerations apply for Ss.

We define the strips I,, = [0, 1) x (y;‘+‘, y_yn), n = -1,0,1,2,3 ,... The height of strip Z, is h, =

-‘-’ (y - 1) and clearly 1z,, 4 0 as n -+ 00. Choose

&=(Y?+l)/y:and I <~~<i(l+fi) (which ensures that pY > I ). With this choice of Pr the strip

1-i is mapped onto II U IO. Every other strip I,, is mapped onto I,,+, U I, U f,,_-l, n = 0, 1,2,. . . Thus each strip is mapped onto the union of a set of strips and so the density on each strip will be uniform.

Suppose that the measure of strip I,, is given by

d,. Since the density associated with the map f is uniform, the probability of transition between strips can be expressed in terms of the ai. Using similar methods as in Ref. [ 71 the densities can be computed and are given by

n= 1,2 9...,

with c a constant given by

A2[ho(l --as)+h(l-CY2)1

'= (1-(Y3)(LY3--2)(A2(hO+h,) +ha+2h,)’

(We also get expressions for do and d-1.) Setting

hu=~y’(~~-l),ht =yy2(yr-l)andcus=$(A2+

a2+1) givesc= [2/(1--~~)]A2+O(Ag).Thetwo Lyapunov exponents are ~7~ defined by (3.3) and (T?, which can be found from gv (n, y, 0) as

The coefficient of A2 in u,, is positive and so the bifurcated attractor has two positive Lyapunov exponents.

Secondary blowout bifurcations. For the attractor in S2 there is only one normal direction ( z ) and so there

is one normal Lyapunov exponent. Since we have the

density of the attractor, we can also determine this

Lyapunov exponent which is given by

c+z=hllogy,+Az lOgK, +...,

to first order in At and AZ. Now the chaotic attractor in S2 will undergo a blowout bifurcation into the full three-dimensional space when (T, = 0 and clearly this occurs to a first approximation when

This secondary blowout bifurcation can only occur after the first blowout bifurcation which occurs when

AZ = 0. Thus, we require that A2 > 0. The sign of At is determined by the sign of log K, . If K~ > 1, then iterates far from the x-axis are pushed away from S2 and so the blowout from SZ will occur before the blowout from SI into Ss. However, if K~ < 1, then iterates far from the x-axis are attracted towards S2

and so the secondary blowout bifurcation will occur

after the primary blowout bifurcation into Ss. A similar analysis can be performed for the attrac-

tor in Ss provided that /3, = ( yz + 1) /rZ and 1 <

yz < $ ( 1 + &). In this case, the normal Lyapunov exponent is

266 F! Ashwin, PJ. AstodPhysics Letters A 244 (1998) 261-270

hz II

I

II

/

III

4 III

\

X IV

Fig. 6. Two-pammeter bifurcation diagram for ~~ > 1, K~ < i. I: x-axis stable, II: n-y attractor stable, 111: attractor in full space, IV: n-z attractor stable.

(Ty = A2 logy, + Al

and so there is a secondary blowout bifurcation from

Sj into the full three-dimensional space when

In this case, we must have At > 0 so that the blowout into S3 has occurred and then the sign of A2 is deter-

mined by the sign of log K~.

The quadrants of the two-parameter plane which contain the paths of secondary bifurcation points are determined by the sign of the two quantities log K? and log K~. These paths are shown in Fig. 6 for the case K? > 1, ~~ < 1. Choosing the parameter values at

the point indicated by a cross in Fig. 6, the bifurcated attractor projected onto the normal directions is shown

in Fig. 7. Note that the iterates are clustered near to the

z-axis since the attractor in S3 has only just lost normal stability. However, there are very few near to the y-

axis since there has not been a blowout bifurcation into S2 at this point.

Remark. For steady state mode interactions, it is possible to have a tertiary Hopf bifurcation from anon- symmetric solution branch [ lo]. Hopf bifurcations can be defined in the chaotic context [ 1 l] but these are also associated with bifurcation out of an invariant

Fig. 7. The normal dynamics of an attractor with no symmetry when rY = yr = ~~ = K~ -’ = 1.25 at the point in the parameter plane indicated by a cross in Fig. 6.

subspace. As the chaotic attractor with no symmetry in this case is not in an invariant subspace, we would

not expect to find a tertiary Hopf bifurcation. One needs of course to regard the relevance of the

above example to smooth maps with some suspicion because the map is not even continuous, let alone dif- ferentiable. However, the next example shows remark- ably similar behaviour.

3.2. Interaction of blowouts in a smooth mapping

Consider a mapping on R3 defined by

x’=4x(l-X),

y’ = Arye-YZ-22(e-x + aty2 + ptz2), r,’ = A2Ze-Yz-Zz (e-” + a2y2 + P2z2), (3.4)

with six parameters Ai, /Ii, (pi for i = 1,2 and Z2 x Z2 symmetry as in the previous example. The map has the structure of a skew product over the x-dynamics. The invariant subspaces defined in (3.2) are also invariant for this map and all the parameters are normal with respect to St. The dynamics in Sr is simply the dynamics of the logistic map and so the Lyapunov exponent is UX = log2. The invariant density for this map is given by Y(X) = [x(1 - x)]-‘/~. Thus, the two normal Lyapunov exponents can be found from linearising the second and third equations at y = z = 0, giving

R Ashwin, i?J. Aston/Ph.vsics Letters A 244 (1998) 261-270 267

I

go = ? J’ log(A,e-“) v(x) d.r=logA, - ;.

0

Similarly,

U; = logA?. - 4.

There is a blowout bifurcation from the invariant sub-

space Si into the invariant subspace Sz when hi = e’/*. Similarly there is a blowout bifurcation from Si

to & when A’ = e . ‘I2 The invariant subspace Si is

stable with respect to all normal perturbations when

Ai, AZ < e’12. The main difference between this example and the

previous one is that in this case the parameter Ai is a

normal parameter with respect to Ss but not with re-

spect to S2. Similarly, A2 is normal with respect to 52 but not Ss. Thus, once the blowout into S2 has occurred

and Ai > 0, then the one normal Lyapunov exponent

for this attractor will vary smoothly with A2 but discontinuously in general with Al [ 131. Thus, a secondary

bifurcation point can be defined by allowing only A2 to vary and determining the point at which it changes

sign. However, if a small change is made to Ai and the process repeated, then the secondary bifurcation

point can in principal occur at a very different value of A?. Thus, the path of secondary bifurcation points

will usually be discontinuous although it may appear

to be continuous on a large measure subset of parame-

ters. Fig. 8 shows the numerically determined paths of secondary bifurcation near interaction of blowouts for

(3.4). The inset shows the projection of an attractor onto the (y, z) plane for a parameter value in region III; this is stuck on to invariant subspaces S2 and Ss.

4. Interaction of blowout and symmetry increasing bifurcations

Blowout bifurcations can also interact with many other bifurcations of attractors. We consider an example from a mapping on (x, z) E R X Cc,

x’ = (A + ,ux2)e-“.i-,

2’ = (a + Dx’ + cl: 1’): + idxz. (4.1)

This has six real parameters a, b, c, d, A and ,u. It commutes with the following (reducible) action of O(2) on R x @,

PS(X,Z) = (x,eiBz), KO(X,Z) = (-x,7>. (4.2)

Note that this action has an isotypic decomposition into a standard action on Cc and an action on R where

reflections act as x H -x. The map is a skew product over the map on x and we note that a, b, c, d are normal parameters for the invariant subspace consisting of the

x-axis.

We have performed simulations of the mapping (4.1) at the following parameter values, a = 1.1

c = -0.02, d = 0.1, A = 4.8 and varying b and ,u. For example, when b = -0.11, p = - 1.5 11 the attractor has symmetry SO(2) on average and is in the triv- ial stratum but is stuck on to the invariant subspace

Fix(S0(2)), i.e. the x-axis. On increasing b through -0.1124 we observe a blowout bifurcation of an at-

tractor with SO( 2) symmetry within Fix( SO( 2) ) to

an attractor with the same symmetry but in Fix(id) that is stuck on to Fix(SO(2)). On decreasing ,z

through - 1.511 there is a fusing of conjugate at-

tractors in a bifurcation within Fix(SO( 2)) from an attractor with SO( 2) symmetry to one with O(2)

symmetry at for example b = -0.11 and p = - 1.53. Thus we can infer the existence of an interaction

of two symmetric bifurcations of chaotic attractors at

a codimension-two point, as illustrated schematically

in Fig. 9. The primary bifurcations are of attractors

within the x-axis and labelled C and D on this diagram.

Their interaction gives rise to secondary bifurcations B, a fusion of conjugate attractors off the x-axis (but

stuck on to it) and A, a fusion of chaotic saddles that

are attractors for the map within the x-axis.

5. Collision of blowout bifurcations

Another codimension-two phenomenon that can

occur is when two blowout bifurcations collide. This is analogous to the (well-understood) situation when two bifurcation points collide [lo] and can be

thought of as a singularity in the parametrisation. The codimension-two point occurs at a turning point on the path of bifurcation points in the two-parameter

plane. There are two possibilities for the normal form when two pitchfork bifurcations collide and these are given by

x3 + cA*x + ,ux = 0, e=fl.

268 P. Ashwin. RJ. Aston/Physics Letters A 244 (1998) 261-270

1.80

1.80 k 1.60

si’ ‘3 S;’ s2 w= sp

Fig. 8. Location of the primary blowout bifurcations (from Sl ) and secondary blowout bifurcations (from Sz and S3) for the map (3.4)

in the (Al, AZ) parameter plane. The fixed parameters are (~1 = 0.65, aa = 0.53, PI = 0.53 and & = 0.2. The lines show the primary

bifurcations .S, -+ S2 an SI + Sj and secondary bifurcations S2 -+ and S3 + which signify blowouts from S2 and S3 to an attractor not

contained in any invariant subspace. There are six regions I-VI which can be distinguished by the presence or absence of an attractor in

the invariant subspaces S,. An attractor from region 111 is shown in the inset for Ai = 1.7 and A2 = 1.77. This has arisen via secondary

blowout bifurcations from chaotic attractors created in the Y- and z-axes at primary blowouts.

C

D

Fig. 9. Schematic bifurcation diagram of the map (4.1) in the

(p, h) plane near 0 = ( - I .5 I I, 0. I 124). A blowout bifurcation

C-C interacts with a symmetry increasing bifurcation of attractors in the x-axis D-O (and of chaotic saddles O-A) give a secondivy

branch of symmetry increasing bifurcations of attractors stuck on

to the x-axis O-B.

For blowout bifurcations, there is again a turning point on the two-parameter path of blowout bifurcations. In this case we describe the blowout bifurcation in terms of the average of the normal variable, which we call

ji. Now at a blowout bifurcation, it has been shown that jj varies linearly to lowest order in the bifurcation

parameter rather than as a square root as occurs for pitchfork bifurcations [7]. Thus, there are two possibilities for the collision of blowout bifurcations, as-

suming that they are always supercritical but with dif-

ferent orientation with respect to the parameter. The form of jj near to the codimension-two point is then

described to low order by the equation

whose solutions are only valid if 7 2 0. The solutions of this equation are shown in Fig. 10 for both signs of E. Note that when I_L = 0, a normal Lyapunov exponent is zero at A = 0 and also has zero derivative with

respect to A.

6. Discussion

In this paper we have shown examples of a few blowout bifurcations that occur in codimension-two settings. They give rise to a number of secondary bifurcation branches involving chaotic attractors, and or- ganise the nearby dynamics.

I! Ashwin. FY AstodPhysics Letters A 244 (1998) 261-270 269

CL<0 p=o iJ>o

Fig. IO. Unfoldings of a collision of blowout bifurcations. (i) e = I, (ii) l = -1.

There will be many more such blowouts, for ex-

ample a codimension-two blowout bifurcation that we infer must exist is a transition from a supercritical

blowout that produces a chaotic attractor to one that produces a hyperchaotic attractor; that was seen in

the map investigated in Ref. [ 71. This corresponds to

an interaction of a blowout bifurcation with an onset of hyperchaos in the blowout attractor. Near such a codimension-two point, there is a secondary path of bi-

furcations from chaotic to hyperchaotic blown-out attractors although such a bifurcation typically results in no observable change in the attractor. However, this is

of interest in the skew product case as it corresponds to

creation of attractors that are not graphs over the base transformation and thus the breakdown of the inertial

function described by Campbell and Davies [ 121.

6.1. Towards a classification of blowout bifurcations

Ideally one would like to have a “classification by codimension” as exists for steady bifurcations with symmetries. There are however a number of problems that need to be overcome. We list some of them here.

Non-normal parameters. One can only really dis-

cuss blowouts from a branch of attractors in N, the

invariant subspace. If parameters are not normal (and

generic parametrisations are such) this will only be the case if the attractor in N is hyperbolic in a very strong

sense. For the cases usually considered, one can only

expect at best a measure-theoretic sort of genericity, that a branch is continuous only on a Cantor set of parameter values; see Refs. [ 13,141 for a discussion of

blowout bifurcations without normal parameters. For example, it can be shown that there can be a set of blowout bifurcations in a one-parameter family that

has positive Lebesgue measure in parameter space. A simpler first step is certainly to fully understand

the case of normal parameters [ 151. Note that even in

the case of normal parameters, if N is of codimension greater than one in I[$” the transverse Lyapunov ex-

ponents can vary discontinuously. Nothing analogous to this occurs for steady state bifurcations. We note that the effects detailed in Ref. [9] can be thought of as an example of a blowout with change of criticality where one of the parameters is not normal and hence the two-parameter picture is not as simple.

Whichever setting is chosen, the following problems

270 I? Ashwin, l?J. AstodPhysics Letters A 244 (1998) 261-270

need to be addressed. The examples in this paper can

be thought of as a contribution to the characterisation problem.

Characterisation problem. One needs to compile

a list of possible generic blowouts and give ways of characterising them. This will depend greatly on the dimensions n, m, the presence of any group action and

other invariant subspaces in the phase space. Recognitiorl problem. This would be an algorithm

such that given any generic n-parameter blowout bifurcation and thus classify it according to codimension.

Unfolding problem. Once a blowout is classified by codimension, it would be of interest to classify generic

unfoldings of the blowout and hence find possible sec-

ondary bifurcations associated with generic unfoldings

of that bifurcation.

6.2. Local versus global codimension-two bifurcations

The examples we have presented suggest that it is helpful to classify all codimension-two blowout bifurcations as to whether they are local (i.e. local to the

dynamics on the invariant subspace) or global (any other case). Of the examples discussed here, only the

change of criticality and the onset of hyperchaos are

global bifurcations while the others are local to the

invariant subspaces.

Acknowledgement

Peter Ashwin is partially supported by a Nuffield

“Newly appointed science lecturer” grant and by EP-

SRC grant GR/K77365. We thank Matt Nicol for interesting and enlightening discussions concerning

these results and a referee for drawing our attention to Ref. [9].

References

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Blowout bifurcations of codimension two