A case study for homoclinic chaos in an autonomous electronic circuit

PMSgA

Physica D 62 (1993) 230-253 North-Holland

A case study for homoclinic chaos in an autonomous electronic circuit A trip from Takens-Bogdanov

to Hopf-Sil'nikov

E. Freire, A.J. Rodrfguez-Luis, E. Gamero and E. Ponce Dept. Applied Mathematics, E.T.S.I.L, University of Sevilla, Avda Reina Mercedes s/n, 41012 Sevilla, Spain

The dynamics of an autonomous electronic oscillator is analysed. Both theoretical study- with methods of local bifurcation theory - and numerical simulation - by using continuation methods to detect global behaviours - are carried out. Comment is made on the presence of homoclinic connections which organise the periodic and nonperiodic behaviour. The richness and complexity of the periodic oscillations are described and the presence of isolated branches of odd-period orbits are emphasized. Further, attention is paid to the chaotic attractor coexisting with Sil'nikov hornoclinicity, even in the presence of a Hopf degeneracy: the Hopf-Sil'nikov singularity.

1. Introduction

In this work, the dynamics and bifurcation b e h a v i o u r of an electronic oscillator, genealogically related to the classical Rayleigh and Van der Pol oscillators, are presented. On one hand, we aim at summarizing its periodic behaviour: existence, n u m b e r and different shapes of periodic orbits as well as their appearance, disa p p e a r a n c e and stability changes according to circuit p a r a m e t e r variations. As will be seen, the richness and complexity of such behaviour is manifested in the tridimensional state space modelling the dynamics of the a u t o n o m o u s oscillator. Also, we will take account of the existence of nonperiodic oscillations (i.e. chaotic attractors) with emphasis on those related to the presence of certain homoclinic orbits (the so called Sil'nikov attractors). Bifurcation studies focus on p a r a m e t e r values for which structural stability is lost; these raise b o t h local and global problems. So, we will firstly investigate, in section 2, the possible loss of hyperbolicity of the system equilibria (linear degeneracies), which gives rise to local bifurca-

tion p h e n o m e n a . A n example of second kind of p r o b l e m is the eventual structural instability of homoclinic orbits which, as shown in section 3, lead to different and complex global bifurcation phenomena. The scheme of the electronic system under study is shown in figure 1, and it represents a coupling, by means of a conductance G2, of two elemental circuits: a parallel RCL-circuit (a conductance G3, and inductance L and a capacity C) and a RC-circuit (a conductance G 1 and a capacity Co). Different selections for the conductances give rise to several oscillators previously investigated in recent years. So, when G~ is a negative nonlinear c o n d u c t a n c e - as in the classical Van der Pol oscillator, G 2 is a positive nonlinear conductance, and G 3 is strictly linear, the circuit is by no means different from the original prop o s e d by Shinriki et al. [25] as a r a n d o m wavef o r m generator; they did not present any theoretical analysis of state equations and the numerical and experimental study carried out is far from being complete (a more detailed analysis of the dynamics of such a circuit can be found in Freire et al. [4]).

0167-2789/93/$06.00 ~ 1993- Elsevier Science Publishers B.V. All rights reserved

E . Freire et al. / Case s t u d y in a u t o n o m o u s electronic circuit

231

The corresponding state equations of the circuit, taking the voltages on the capacitors and the current across the inductance as state variables, are

A similar choice for conductances Gj can be found in the work of Gomes and King [13], but now modeling Gt in a different way. They write (cf. (2))

Co dvl d---r"= __il(Vl) + i2(V 2 - -

il(vl) = P'0 +/xlvl + A3°~,

dv2

=

C dr

Vl),

--iL -- i2(v2 -- vl) - i 3 ( v 2 ) '

di L

L -87=

(1)

where i/ represents the current-voltage characteristics of conductance Gj, j = 1, 2, 3 respectively, which we assume are odd functions. In order to carry out both analytical and numerical studies, we model such characteristics by means of cubic polynomials as follows: 3 il(Vl) =/.tlV 1 + A3v 1 ,

i2(v 2 - 0 1 ) ~-- ].i,2(v 2 - V l ) --[-/~3(v2 - V l ) 3 ' 3 i 3 ( v 2 ) = P,3v2 + C 3 v 2 .

(2)

trying to take account of some imperfections in the circuit which, in spite of their small magnitude (/% is taken small), impose a loss of the odd character for the corresponding nonlinearity. They point out, for some aspects of the system dynamics, the consequences of the inherent Z2-symmetry breaking. Recently, an analogous configuration to that originally proposed by Shinriki et al. [25] has been analysed by Healey et al. [15] but eliminating G 3 and considering a small resistor R L in series with the inductance L, to take account of its resistive effect in the experimental circuit. Thus, the last state equation in (1) should be rewritten as di L

L ~

With the above hypothesis on i), it turns out that the system is invariant under the transformation ( v l , v2, i L ) - - ~ ( - - V l , --V2, - - i L ) , or, in other words, that the circuit is 71z-symmetric. Such property will affect the corresponding dynamics, and will be, in principle, beneficial to the observability and stability of periodic and chaotic behaviours. In the case of G 1 being a nonlinear negative conductance- again as in the Van der Pol oscill a t o r - but with G 2 a purely linear conductance and 6 3 = 0, the circuit studied by Matsumoto et al. [20] is obtained. It should be remarked that the quoted authors approximate the i - v characteristic for G 1 by means of a piecewise linear function; that is why their theoretical analysis of state equations cannot profit from many results of differentiable dynamics and, particularly, of bifurcation theory.

= v 2 - RLi L .

They carry out a theoretical study of some bifurcation behaviours, but their work mainly constitutes interesting experimental research. Our paper is organised as follows. Firstly, we investigate the possible local bifurcation behaviours for the equilibria of the electronic system of fig. 1 in its more general configuration, assuming only an odd character for the nonlinear i - v characteristics corresponding to the Gj. This approach will make evident the role played in the dynamics and how it is affected by each conductance. Therefore, in section 2, we look for the situations corresponding to higher degenerations for the equilibria of system (1), surveying some results already obtained in previous work. We firstly show the possibility of several linear degenerations (nonhyperbolicities) with codimension 1 (simple zero, Hopf), 2 (double

232

E. Freire et al. / Case study in a u t o n o m o u s electronic circuit

VI

V2

i ~t_

I _

Fig. 1. A schematic diagram of the electronic circuit, where three conductances (G1, G2 and G3) two capacitors (C O and C) and one inductance (L) appear. We take as state variables the voltages on the capacitors (Vl and o2) and the current across the inductance (iL).

zero, Hopf-zero) or 3 (triple zero eigenvalue), which will permit us to define, from the system parameters, a first bifurcation parameter set. We then carry out. a detailed study of the related bifurcation behaviours for some of those linear degenerations. This is done with the help of several symbolic computation algorithms previously developed by the authors and the use of analytic techniques such as centre manifold reductions and normal form investigations. Normal form computations provide information about nonlinear terms in the system which are essential to the bifurcation behaviour. At this point, we analyse the possible additional degeneracies in the nonlinear part in order to make evident how some new parameters, not yet considered as bifurcation parameters, can be recognized as such. We will show some of these degenerated situations and the corresponding multiparameter bifurcations (a degenerate Hopf bifurcation, a degenerate Takens-Bogdanov bifurcation) will be described. Degeneracies of equilibria normally act as organizing centres for the dynamics and local analysis provides interesting information about a great variety of each involucrate behaviours (periodic orbits and their bifurcations, homo-

clinic and heteroclinic connections,...). The validity of such analysis in section 2 should be, in principle, restricted to a neighbourhood of equilibria in the phase space and to an interval around the critical values in the parameter space. However, both from experimental observation and numerical simulation, it is frequently deduced that results from the local study remain valid far from the bifurcation and this makes local analysis a very important tool. The crux of the matter is that neighbourhoods predicted by the local theory have a concrete magnitude which is generally unknown a priori but can be of relevant size [26]. In section 3, we extend local results by means of numerical continuation methods. From the information gained in the study of local bifurcations, we obtain good starting points for the application of numerical techniques both in phase space approximations (for instance, initial conditions of periodic or homoclinic orbits) and in parameter space computations (approximations to bifurcation curves, e.g.). Thus, our objective is to obtain a global picture of the dynamics of system (1), both in its tridimensional state space and in the parameter space, where we restrict ourselves to biparametric slices. In this line of global results, we show the numerical continuation of a curve of homoclinic orbits, i n a certain parameter plane, which is born in a Takens-Bogdanov point and dies at its encounter with a Hopf bifurcation curve of the involucred equilibrium. In between, we find several kinds of homoclinic orbits and, in particular, repulsive homoclinic orbits corresponding to saddle-focus equilibria (also called Sil'nikov type). As the electronic system under study is one of the first examples of chaotic dynamics in the autonomous electronic oscillator's field, we plan, in section 4, to describe some of the chaotic attractors obtained by numerical simulation of system (1). To understand their dynamics, it is very relevant to relate them to the appearance of homoclinic orbits of the Sil'nikov type.

233

E. Freire et al. / Case study in autonomous electronic circuit

2. Local bifurcations

~3

In this section we first analyse degeneracies in the linear parts corresponding to the equilibria of the system (I) in the framework of local bifurcation theory. We find several linear degeneracies of codimension 1, 2 and 3. Next, in order to determine the corresponding bifurcation behaviour, we compute and analyse the corresponding normal forms. From this analysis, additional degeneracies in the nonlinear part arise. Obviously, the origin (0, 0,0) is always an equilibrium for the system (1). If we write the characteristic polynomial of the jacobian matrix at this equilibrium /3(A) = A3 +/71A 2 "~-/72/~"~ P 3 ' then 1 /71 = -oW

1

+

+

(m +

1

I..Ol.. 1

2

1

1

+

We consider the cases when the origin is a nonhyperbolic equilibrium: Case 1. Zero is a simple eigenvalue and the other eigenvalues have nonzero real parts. Here,/73 = 0, /71 # 0, /72 ~ 0, or equivalently /z1 + ~ = 0, tx2 + it 3 # O, tt 2 # ~r-~o/L. From the hypothesis (2) the nontrivial equilibria of system (1) are easily obtained: (v~, v2, zL), (01-, v~, i~), where

A 3 q_ ~ 3

v2 = 0 ,

...'"

i i i •

I [

~,. S ' S

° ~' ° ~ ' ~ S ° |

Pl s " Fig. 2. Bifurcation set of system (1) where several bifurcations of codimension 1 (PI, pitchfork), 2 (TB~, TB z, TakensBogdanov, HZ, Hopf-zero) and 3 (TZ~, TZ z, triple-zero) appear.

a pitchfork bifurcation takes place (recall the ~2-symmetry).

CoC tx2 + L C '

G =GcL

. , 1 1 pl

,

i~=glv~+/}3v~.

It is deduced (under generic conditions on A3, J~3) that in the plane PI - / - h + ~ = 0 (see fig. 2)

Case 2. There exists a pair of pure imaginary eigenvalues ( A 1 = % i , A 2 = - % i ) and A3#0. Now,/~1[~2 = / 7 3 ' 0.)2 = / 7 2 > 0 , /~3 : --/71 # 0 , leading to an algebraic surface in the /.h-p~-/x3 parameter space where a possible H o p f bifurcation takes place. In the above two cases, we deal with linear degeneracies of codimension one; in the next two cases codimension two situations will appear. Case 3. Zero is a double eigenvalue (A1 = A2 = 0) and the third one is nonzero (As # 0). For this,/72 =/73 = 0, As = -/71 # 0, corresponding to Ix~ + P-2 = O, lx~ = Co/L and /.t2 + / z 3 ~ 0. So, we may have a Takens-Bogdanov bifurcation on the straight lines TB~ --=/x~ = - / % = V ~ o / L and TB2 ~ / x 1 = - / % = - CV'-C~o/L, /z32 # Co/L (see fig.

2). Case 4. There appears a pair of pure imaginary

E. Freire et al. / Case study in a u t o n o m o u s electronic circuit

234

eigenvalues as in case 2 but the third one is zero (h 3 = 0). Then, /~3 = J~l = 0 , t O o= 2 iO2 > 0, which imply /xI + / x 2 = 0, /*2 +/*3 = 0, l/x2{< V ~ o / L . So, we find a Hopf-zero (or more precisely Hopf-pitchfork) bifurcation on the segment (see fig. 2)

F r o m these expressions, we deduce that r > 0, A 3 >- 0, B 3 ~ 0. The system (1) becomes

H Z ~ {/,1 = _/x2 _=/x3, _ , ~ - ~ < / z 2 < C~-~}.

2 = y.

Case 3. Zero is a triple eigenvalue. So, ~1 = /~2 =/~3 = 0, which correspond to /x1 + ~ = 0, 2 ~2 + lz3 = O, tz 2 = Co/L. It is deduced that the geometric multiplicity of this triple zero eigenvalue is one and so we find a linear degeneracy of codimension three at the endpoints, TZ 1 = (V~o/L, -V~0/L, vC-Co/L) and T Z 2 = ( - C-~o/L , x/-Coo/L , - V ~ o / L ), of the segment H Z (see fig. 2).

From now on, we focus our attention on the bifurcation behaviour of the system (3) in the v - / 3 parameter plane. The Jacobian matrix at the equilibrium in the origin is

As the parameters & , tt 2, tt 3 correspond to the linear approximations to the i - v characteristics of the conductances Gx, G 2, G 3 and their variation is easily handled in practice, they constitute a natural choice for bifurcation parameters as expressed by the relations above. Preliminary results about the Hopf-pitchfork bifurcation (case 4) will appear in Gamero et al. [6]; the triple zero situation (case 5) is currently under investigation. Next we report the Hopf and Takens-Bogdanov bifurcations (cases 2, 3) summarizing previous works. For the sake of simplicity, we make some additional assumptions on system (1). So, we limit ourselves to the configuration most widely considered [13,15,20] eliminating conductance G3 (i3(v2)=0, and so /z3 = 0), and supposing A3 >-0, B 3 >--0. Now, we rescale the time, variables and parameters of the system according to t = tOr ,

iL

A3

=

- -

tOC'

-

A3 tOC '

= - / 3 ( y - x) - z - B3(Y

x = vl ,

CO

r= B3

__

(-(v A(v,/3)=

+ /3)/r /3 0

-

/,,=

~3 tOC

.

tOC'

/3/r -/3 1

0)0 --1 ,

(4)

which has P(A) = A3 + p l A 2 + p2 A + P3 as characteristic polynomial, where Pl =

v +/3(1 + r) r

~/3+r

P2--

m

P3 ~

- -

,

r

v+/3 r

and, consequently, the pitchfork bifurcation of the origin occurs on the straight line PI = { v + /3 = 0} (see fig. 3). We consider firstly the Hopf bifurcation (cf. case 2 above), where A ( v , / 3 ) has the eigenvalues +--to0i, A0#0, which correspond to PiP2 =P3, 2 tOO = P 2 > 0 or f l [ v / 3 ( l + r) + r 2 + v 2] = 0 , 2

r--l)

2

r

A0=-#0.v

Y = V2 ,

Id,1

C '

- x) 3 ,

(3)

tOo= r(l+r------~>0,

1 tO = X/-L-t~ '

Z

+ / 3 ( y - x) - A 3 x3 + B3(Y - x) 3 ,

rye = - v x

Ate /3=

- -

tOC

,

So, we have different Hopf bifurcation curves in the u - 3 plane. The case 3 = 0 is easily characterized as a supercritical Hopf bifurcation and, in section 3, we will return to it. Here we analyse in detail the other case. If we denote /3c(v)=

E. Freire et al. / Case study in autonomous electronic circuit

235

"'.,RI

R~. .......

\

..'".. DI"".

B

H

D2"., # u~ 0

HP

0

I

/

i

"'.~'~

'

'

0

-0.5

-I /,/

1

8

4

6

7

8

9

Fig. 3. Partial bifurcation set in the ~,-fl parameter plane for r = 0 . 6 , A 3 = 0.328578, B3 =0.933578. Three codimension-2 bifurcation points (TB, Takens-Bogdanov; DH, degenerate Hopf bifurcation of the origin; HS, Hopf-Sil'nikov) and several codimension-1 bifurcation curves (PI, pitchfork of equilibria; H, Hopf of the origin; h, Hopf of the non-trivial equilibria; HP, homoclinic orbit; SN and sn, saddle-node bifurcations of periodic orbits) are present. The dotted line RI separates the regions where the three eigenvalues of the origin are real or a complex conjugate pair appears. The other dotted lines mark 8 = 1 (D1) and 8 = ½ (D2), i.e., the Sil'nikov region. Configurations of equilibria and periodic orbits are sketched below. Solid curves indicate stable periodic orbits and broken lines non-stable periodic orbits.

E. Freire et al. / Case study in autonomous electronic circuit

236

- ( r 2 + v2)/v(1 + r), for 0 < v 2 < r, then the possible H o p f bifurcation occurs on the curve

H ~ {/3 =/3c(V): - x / ? < v < 0 } U {/3 = & ( . ) : 0 < ~ < W } and /3 is chosen as the bifurcation parameter, hence v is thought of as a constant. We pay attention to the branch of H corresponding to - x / ? < v < 0 (see fig. 3), because A0 = r/v < 0 and then stable p h e n o m e n a appear. That restriction implies a nonlinear negative conductance for G~ (v < 0) and a nonlinear positive conductance for G 2 (/3 > 0 ) , which is the situation most usually considered. For /3 near /3c, the eigenvalues are a(/3)+ito(/3), h(/3), with a ( / 3 c ) = 0 , to(tic ) =too and A(/3~) = )t0. The transversality condition a '(/3c) # 0 is easily verified and so, to characterize the H o p f bifurcation, it suffices to compute the normal form for the critical value/3~. Applying some recursive symbolic computation algorithms developed by Freire et al. [2,3] we carry out a center manifold reduction and obtain the H o p f normal form up to 5-order, whose formulation in polar coordinates is t~ = a l p 3 + a 2 p 5, O= too+ bxp 2+ b2P 4. Expressions for the coefficients a 1, a2, b l, b 2 a r e presented in G a m e r o et al. [7]; in particular, we have 3

A 3 ( v 4 - r 4) + B31-'4(1 + r) 2

a l = 8-r

(r 2 + v z ) ( r 2 + r - v

2)

A n analogous expression is obtained by Gomes and King [13] for the case B 3 = 0. So, a 1 = 0 for A3r 4

~1/4

vc= - A3 + B3(l + r)2/ and

al>0

(vc, 0). In other words, we have a subcritical H o p f bifurcation for v ~ ( - # ? , vc), a supercritical one for v E (vc, 0), and a degenerate H o p f bifurcation at the point D H - - ( v c,/3~(v¢)) (see fig. 3). T o determine the order of degeneracy at the point D H it is necessary to analyse whether the coefficient a 2 may be annihilated (for v = v~). Such analysis is carried out by G a m e r o et al. [7] for different cases in relation to the parameters r, A3, B 3. In the simplest situation (a 2 # 0) a codimension two bifurcation is found, characterized by the local existence of a curve SN of saddle-node bifurcations of periodic orbits in the v-/3 plane leaving the point D H (see fig. 3). For the a 2 = 0 case (see also [7]), degenerate H o p f bifurcations of codimension 3 are considered and the resulting cusp bifurcations of periodic orbits are pointed out. Although not theoretically studied by Healey et al. [15], the appearance of such a cusp behaviour in a neighb o u r h o o d of D H is confirmed in their experimental results. We now analyse the T a k e n s - B o g d a n o v bifurcation (cf. case 3 above) when A(v,/3) has a double eigenvalue and the other one h 0 # 0 , which corresponds to P2 = P3 = 0, A 0 = - P l # 0, or v + / 3 = 0 , v 2 = r , )t0 = v # 0 . As stated above, two T a k e n s - B o g d a n o v points are present in the bifurcation set. H e r e we consider one of them: TB = (v0,/30) = ( - x / T , x/T) because in this case A0 = v 0 < 0 and so we can expect stable p h e n o m e n a (the point TB, intersection of curves PI and H , is shown in fig. 3). Setting ~ = v - v0,/3 =/3 -/30 and, after some linear transformations and the application in system (3) of a symbolic algorithm for center manifold computations (see Freire et al. [2]), we achieve the two-dimensional reduced system

(x) (0 =

e 1 e2

)7 + N L T ,

E(-x/-f,O)

if u E ( - v ' - f , v ~ ) ,

al<0

for v ~

where N L T denotes nonlinear terms in (£, 37, if, /3) and

E. Freire et al. / Case study in autonomous electronic circuit

e, = - ( r ' + f l ) l r 3/z ,

e z = [ i f ( l - r) +/~(1 + r ) l / r z . From this we check the transversality condition O(el, e2)

2

a05, /~)

- -T/Tr # 0

at ff =/3 = 0 (geometrically, the curves PI and H meet transversally in TB). As a first consequence, we have ~, ~ (or alternatively ~1, e2) as unfolding parameters of the double zero degeneracy. Moreover, it is enough to compute the normal form for ~ =/3 = 0, what is carried out by means of a recursive computer algebra procedure described in Gamero et al. [5], thereby obtaining

+

(

'o)(y)

o

3,3x 3 + 63xZy + 3,5x 5 + 6sx4y

)

(5)

where T3 = - ( A 3

of these curves which, using continuation methods (see e.g. [18, 23]), can be extended, so obtaining the partial bifurcation set shown in fig. 3. So far, we have assumed A 3 > 0, B 3 > 0 (which implies 3,9<0) and supposed r E ( 0 , 1) to warrant that 63 # 0. Allowing r >-1, 69 may be annihilated, giving rise to an additional nonlinear degeneracy and a higher codimension bifurcation. In this way, 63 = 0 if A 3 ( 1 - r) + B3(1 + r) = 0 and selecting then r as a new bifurcation parameter we have a degenerated Takens-Bogdanov point at (~'0, /30, r0), where r o = ( A 3 + B 3 ) / ( A 3 - B3) > 1 provided that A 3 > B 3. We n o t e that for r = r o, vc = u0 = -v'-~0 is implied and so the degenerate Hopf point D H and the double-zero point TB coalesce, giving rise to a more degenerate situation. In this case the fifth-order normal form is needed: ;?=y, = r3 x3 + 3,5xs + 65x4y,

+ B3)/r5/2 ,

237

(6)

where 3'3 < 0, 65 < 0 at the critical values (t 0,/30, r0).

69 = 3[A3(1 - r) + B3(1 + r ) ] / r 3 (lengthy expressions for coefficients 3'5, 65 can be found in Gamero et al. [7]). In the hypothesis A 3 > 0, B 3 > 0, it is clear that 3'9 < 0. If moreover 0 < r < 1, then 63 > 0. So, we have a codimension-2 Takens-Bogdanov bifurcation (taking into account the 77z-symmetry) at point TB, which is determined by the unfolding of the normal form (5) truncated to third order. As a result, we can assure (cf. [14]) the appearance of three new bifurcation curves, locally at TB (see fig. 3): a curve h of Hopf bifurcations of nontrivial equilibria, a curve HP of homoclinic connections and a curve sn of saddle-nodes of periodic orbits. The local analysis provides us with the slopes

The bifurcation analysis of the 3-parameter system that unfolds (6), it=y, = e t x + e z y - x 3 + e3xZy + Kx4y,

( K = - + 1 ) is carried out in [21,22]. A s a main feature of the bifurcation set, the existence of six codimension two bifurcation curves arising from the origin is shown, corresponding to: a degenerate Hopf bifurcation of the origin; a degenerate H o p f bifurcation of nontrivial equilibria; two cases of nondegenerate Takens-Bogdanov bifurcation; a homoclinic connection with zero trace; and a cusp of saddle-node of periodic orbits. In fig. 4 the bifurcation set in the v-/3 plane

238

E. Freire et al. / Case study in autonomous electronic circuit

'13 4

sn2

PI

3 j. / / I/ I ~~H H ~

1 1

V

\

4

1

3

4

6

5

.o-........ 4_

( Q. Q ) ",~

........... I"

7

8

10

9

11

Fig. 4. A bifurcation set, in the v-fl parameter plane, for a value of r > 1. As a consequence of a nonlinear degeneration in the Takens-Bogdanov bifurcation (compare with fig. 3) new codimension-2 points (CU, cusp of saddle-node of periodic orbits; DHP, degenerate homoclinic orbit) and eodimension-1 bifurcation curves (SN2 and sn2, saddle-node of periodic orbits) appear. Configurations of equilibria and periodic orbits are sketched below.

for a case with r > 1 is sketched, W e remark on four c o d i m e n s i o n - 2 points. Besides T B and D H , a cusp point C U , w h e r e the saddle-node curves o f periodic orbits sn and S N join together, appears; in this manner the points T B and D H are

c o n n e c t e d . T h e fourth codimension-2 point is D H P , w h e r e the homoclinic orbit on the curve H P changes from attractive (negative trace) to repulsive (positive trace), giving rise to a degenerated homoclinic bifurcation [17]: a pair of

E. Freire et al. / Case study in autonomous electronic circuit

saddle-node bifurcations curves appear (SN2 and sn2). In this case, the homoclinic orbit stays outside the Sil'nikov region. For other parameter values, the Hopf bifurcation of nontrivial equilibria is degenerate and five codimension-2 points will appear in the bifurcation set as a consequence of the degeneration in the TakensBogdanov bifurcation. In [15] and [13] the double-zero degeneracy at the point TB is also analysed. Only Gomes and King [13] point out the possibility of additional nonlinear degeneracies but the consequent bifurcation behaviours are not studied.

3. Global bifurcations

Returning to fig. 3 some comments are in order. Above the lower branch of the curve RI (when v + / 3 < 0 ) the three eigenvalues of A 0 ' , / 3 ) (see (4)) are real. Below it, two of them form a complex conjugate pair -p--+ toi, p > 0, w > 0 and the third one A > 0. Putting 8 = p/A, then 6 = 1 on the curve D1, 8 > 1 above it and 8 < 1 below it. On D2, 8 = 1. The shown bifurcation set is only partial. Near TB the dynamics is essentially planar and corresponds to the local Takens-Bogdanov bifurcation (section 2). As we go far from the point TB (the bifurcation curves leaving that point have been numerically continuated) the tridimensional dynamics is to be developed, giving rise to richer periodic and aperiodic behaviour which will be described in this section. Thus we show in fig. 5 different shapes of homoclinic orbits along the curve HP: it arises from the double-zero point TB being essentially planar (5a); after that, once the dotted curve RI is crossed, it is of attractive (8 > 1) saddle-focus type (5b); next it becomes a repulsive one, 6 < 1, (Sil'nikov type) when the curve D1 is gone over (5c, 5d, 5e); finally (50, the curve HP ends at the intersection with the straight line /3 = 0 (corresponding to a supercritical Hopf bifurcation of the origin) in the point HS (see fig. 3). At

239

this point the origin is a nonhyperbolic saddlefocus equilibrium with attracting weak focus, exhibiting an homoclinic orbit. So, the term Hopf-Sil'nikov point has been attached to it [9]. As a consequence of the presence of a Sil'nikov homoclinic orbit, a rich variety of periodic and aperiodic motions will appear. We describe below the achieved results. The bifurcation diagrams we show for all kinds of analysed periodic orbits correspond to horizontal paths (/3 = constant) through HP in fig. 3.

3.1. Principal asymmetric periodic orbits A stable asymmetric periodic orbit appears around each nontrivial equilibrium when crossing the curve h (see from the right side to the left, fig. 3). The evolution of the asymmetric branch P is pointed out in fig. 6 (similar bifurcation diagrams were found experimentally by Healey et al. [15]). In the case /3--0.7 (6a), which corresponds to 8 > 1, the single pair of asymmetric limit cycles becomes homoclinic at the value H P indicated with a dotted line, and then the period of these cycles against v result in a monotonically increasing graph which diverges at H P (as predicted by Holmes [16] and Glendinning [11]). When/3 = 0.6, 6 ~ 1 (6b), a perioddoubling bifurcation originates the presence of the branch D that disappears in a new flip bifurcation. After that, the branch P approaches the homoclinic value HP wiggling. Finally, the Sil'nikov wiggle, around the value HP, is evident when/3 = 0.4, 8 < 1 (6c). In this case, the asymmetric pair undergoes a period-doubling cascade. We sketch the branches D (of double-period orbits) and C (of four-period orbits). As the branch D wiggles around the value HS, we suspect the existence of a secondary homoclinic o r b i t - a double-pulse saddle connection-(see [8,11]) that is detected and drawn in fig. 7. For other values of the parameters, we show a fourpulse homoclinic orbit organizes the wiggle of a four-period orbit branch. A partial bifurcation set corresponding to the

E. Freire et al. / Case study in autonomous electronic circuit

240 (a)

'

(b)

i

.

.

.

.

.

.

.

i

. . . .

i

. . . .

J

,

,

7

o

>~ o

7 i

-0.05

0

-0.2

0.Q5

-0.1

0

0,1

0.2

X

(d) °"

d

-,

,

i

.

I

i

.

.

.

i

.

I

L

.

.

.

i

,

I

i

,

(c)

>- 0

c~ d I I

i

i

~

-0.4

i

L

I

-0.2

i

0

I

i

0.2

0.4

i

=

L

,

-0.5

X

(~

(e)

tQ

f

(t) o

i

h

i

b

[

l

-0.5

h

t

,

,

i

0

i

i

0.5

X

}

-1

i

0

i

i

i

,

i

i

=

0.5

L

,

i

1

X

~

.

.

.

.

i

.

i

/1

,

,

i

i

.

.

i

,

t

-0.5

.

.

i

.

i

.

i

0

.

,

.

.

,

,

L

. . . .

t

. . . .

0.5

i

,

,

:

,

,

1

X

Fig. 5. Evolution of the homoclinic orbit along the curve HP of fig. 3: (a) /3 = 0.77; (b) /3 = 0.7; (c) /3 = 0.6; (d) /3 = 0.4; (e) /3 = 0.2; (f) /3 = O.

asymmetric periodic orbits is drawn in The curve F corresponds to the first doubling bifurcation undergone by such periodic orbits and the curve H2 to the pulse homoclinic connection previously

fig. 8a. periodkind of doubleshown.

3.2. Principal symmetric periodic orbits We will now describe the behaviour of these principal limit cycles that appear from periodic orbits born in the supercritical Hopf bifurcation

241

E. Freire et al, / Case study in autonomous electronic circuit

(b)

(a)

go 'E 0

"E

HP ,I

J

IHP i

I

i

i

-0.74

,

i

I

,

i

-0,735

i

i

I

P

i: -0.72

i

- 0 . 7,3

,

I --0.7

,

J --0. 6 8

__J

o

(0)

i

HS

i

D

i ItP

) i

-0.75

-0.7 v

Fig. 6. Bifurcation diagrams of the principal asymmetric periodic orbits: (a)/3 = 0.7; (b) fl = 0.6; (c)/3 = 0.4. The branches are labelled as follows: P indicates the branch of such principal asymmetric periodic orbits, D of double-period orbits and C of four-period. The principal homoclinic orbit exists for the value denoted by the dotted line HP. The dotted line HS indicates a double-pulse homoclinic orbit. A Hopf bifurcation of nontrivial equilibria is marked as h.

of the origin H that undergo two consecutive saddle-node bifurcations (SN and sn in fig. 3). The evolution of the symmetric branch S is shown in fig. 9. When fl = 0.7 (9a) this symmetric periodic orbit approaches homoclinicity as

pointed out by Holmes [16] and Glendinning [11], that is with its period increasing monotonically. In the case fl =0.6, a rich bifurcation behaviour is already present. First, this symmetric periodic orbit exhibits a symmetry-breaking (bl

d >- o

>-

e,l

?

I

-0.6

-0,4

-0.2

0 X

0.2

0.4

0.6

-0.5

0

0.5

X

Fig. 7. (a) A double-pulse homoclinic orbit (~ ~ - 0 . 6 8 3 5 , fl = 0.4). (b) A quadruple-pulse homoclinic orbit (u ~ - 0 . 6 9 9 3 , /3 = 0.5462).

242

E. Freire et al. / Case study in autonomous electronic circuit

a3 i

0

(a)

(b) :......

i~ ........

r--.

d

" .............

(o

"

r~

.

.

.

.

R~ ................ ".." DI"'-.

h

~"d

..... 132 u')

d

d

"t. -0.8

°0.9

-0.7

-0.6

o-0.9

-0.7

-0.8

-0.6

t,.'

Fig. 8. Partial bifurcation sets for: (a) principal asymmetric periodic orbits; (b) principal symmetric periodic orbits. In enlargement of the region of fig. 3, below the Takens-Bogdanov point TB, several new bifurcation curves appear: in (a) F, period-doubling bifurcation of the principal asymmetric periodic orbits; HS, double-pulse homoclinic orbit; in (b) symmetry-breaking bifurcation; HAS, homoclinic orbit that organizes the branches of asymmetric periodic orbits born in

bifurcation and a branch of asymmetric orbits A1 appears. This one will undergo a period-doubling cascade as conjectured by Glendinning [11]. The branch FA corresponds to the orbits born in the first flip bifurcation. Finally, the symmetric orbit becomes stable after a new symmetry-breaking bifurcation from which a new branch A2 of asymmetric orbits starts. These orbits also undergo a Feigenbaum cascade. The shape of the symmetric and asymmetric periodic orbits appears in fig. 10: a pair of asymmetric stable cycles (10a) and two different unstable symmetric orbits (also a larger amplitude

stable periodic orbit is present in this configuration). In the same picture (10b, 10c) the first subharmonics of the asymmetric limit 'cycles are also drawn. It is clear that the branches of asymmetric orbits A1 and A2 (see fig. 9b) wiggle respectively around two values of the parameter v, indicated with the dotted lines HA1 and HA2. We have found at these values a pair of homoclinic connections as sketched in fig. 11. Finally, the partial bifurcation set corresponding to these symmetric periodic orbits is sketched in fig. 8b. For clarity we have only indicated the . . . . .

(b)

(a)

this first AS, AS.

'.A2T fl .

.,]

. . .

As2~9:

ASlti

i.~'i

(3o

S s ,

-0.9

,

,

,

-0.8

,

,

m,

,

I

. . . .

-0.7

I

-0.6 l/

. . . .

I

-0.5

,

i -0.78

!

..

~

$

o I

i

i

i

i

--01.76

i

b

-01.74

i

fl

-0.72

i

-0.7

I/

Fig. 9. Bifurcation diagrams of the principal symmetric periodic orbits: (a) /3 = 0.7; (b)/3 = 0.6. The branches are labelled as follows: S, principal symmetric periodic orbits; AS1 and AS2, asymmetric orbits born in the symmetry-breaking bifurcation AS; FA, double-period asymmetric orbits. Dotted lines mark the presence of homoclinic orbits: HP (principal), HA1 and HA2 (secondary, corresponding to the curve HAS in fig. 8b). The saddle-node points sn and SN belong to the corresponding curves in fig. 3.

E. Freire et al. I Case study in autonomous electronic circuit

243

(a) Ib)

'

a

.

,

.

i

.

.

.

i

,

.

,

r

i

{5 ///// >-

o

III

//

I~l -0.4

~i //

//I

-0.2

//// I

0.2

0

(C)

i

,

,

,

,

i

i

-0.4

-0.2

-0.4

0.4

.

,

.

i

,

i

-0.2

t

,

,

,

,

,

i

0

•

.

•

i

,

0 X

0.2

0.4

,

0.4

0.2

X

Fig. 10. Unstable symmetric periodic orbits (broken curves in (a)) and asymmetric periodic orbits resulting from: (a) pitchfork 0 ' = - 0 . 7 4 4 9 , / 3 = 0.65); (b) and (c) flip bifurcations (v = - 0 . 7 5 1 3 , / 3 = 0.6 a n d v = - 0 . 7 6 1 1 , / 3 = 0.6 respectively).

hibited by the principal periodic orbits (symmetric and asymmetric as well as their subharmonics) and the one predicted in the analysis carried out by Gaspard et al. [10] and Glendinning and Sparrow [12] in a neighbourhood of the homoclinic connections. From now on we study several kinds of limit cycles and homoclinic orbits that are not present neither in such local analysis nor in their subsequent numerical simu-

curve AS where the symmetry-breaking bifurcation occurs and the curve HAS where the pair of saddle connections organizing the branches of asymmetric periodic orbits exists.

3.3. Odd-period orbits Up to here we have pointed out the correspondence between the bifurcation behaviour ex'

i

.

I

,

.

.

.

r

.

I

,

.

.

.

i

.

I

,

,

,

{5

>-

o

i

i

-0.5

i

i

,

0

,

,

i

0.5

X

Fig. 11. A pair of homoclinic orbits related to the asymmetric periodic orbits (p -~ -0.8147,/3 = 0.4).

244

E. Freire et al. / Case study in autonomous electronic circuit

lations. Specifically, we refer to odd-period orbits that were already reported in the experimental and numerical simulations of Freire et al. [4].

3.3.1. Asymmetric odd-period orbits We first focus our attention on three-period asymmetric orbits. These orbits, as the one shown in fig. 12a, appear in isolated branches organized around two pairs of triple pulse homoclinic orbits (see in 12b one of such saddle connections). We have sketched in fig. 12c the bifurcation diagram where the dotted lines HT1 and HT2 indicate the presence of a pair of triple pulse homoclinic orbits. We enhance two important features of these branches detected in our numerical study: the principal branch P3T corresponds to non-stable orbits and the stable orbits undergo period-doubling cascades. The partial bifurcation set corresponding to these (a)

i

i

3-period orbits is drawn in fig. 12d. On the curve SNT the principal branch of non-stable orbits exhibits a saddle-node bifurcation and on HT the triple pulse saddle connections shown above exist. It seems that the presence of these orbits is strongly related to the principal homoclinic orbit (curve HP). Furthermore they have been detected in a region where 8 is clearly less than 1, that is, inside the Sil'nikov region (the dotted line D2 indicates 6 = ½). Additionally, the curve SNT seems to be a good frontier limiting the region where these detected orbits exist.

3.3.2. Symmetric odd-period orbits We will concentrate on the case of 3T-orbits, although our comments about them will also be valid for the other subjects shown below (5T-, 7T- and 9T-orbits). The symmetric 3T-orbits, as the one sketched

i

o

(b)

0 >- o

o

'

o12

oi+

X

I

(d)

(c)

L

J

,

,

,

i

.

.

.

.

i

IHT1

i " " "

::

HT2.

P3T

,

L

,

i

I

.

.

.

.

,

O.5

i

~. . . .

I

,

,

,

i

u'3 d

'

L

0 X

d

IS~NT1

,

-0.5

0.6

SNT21 / - "

-0.74

-0.73

-0.72

-0.71

-0.7

Fig, 12. Asymmetric 3T-orbits: (a) phase portrait for v = - 0 . 7 1 , /3 = 0.5; (b) a pair of triple pulse homoclinic orbits v ~--0.7252, /3 = 0.4; (c) bifurcation diagram; (d) partial bifurcation set. In this enlargement of a region of fig. 3, near intersection of curves HP (dotted here) and D2, the line HT corresponds to three-pulse homoclinic orbits (HT1 and HT2 in bifurcation diagram) and SNT to the saddle-node exhibited by the principal branch P3T of this 3T-orbits (SNT1 and SNT2 in bifurcation diagram).

for the the the

E. Freire et al. / Case study in autonomous electronic circuit

in fig. 13a, go around each non-trivial equilibrium once and then again surrounding the three equilibria. The bifurcation diagram, for/3 = 0.6 (13c), presented in fig. 13, shows that, as in the case of 3T-asymmetric orbits, these limit cycles appear in isolated branches organized around two values (HT1 and HT2) where a pair of saddle connections occurs (13b). In this case, the symmetric orbit undergoes a symmetry-breaking

bifurcation at the points P1 and P2 (between these two points it is non-stable) and also at P3. From this point we have continued the branch A2 of asymmetric orbits. When /3 = 0 . 4 the bifurcation behaviour is richer because in this case the asymmetric orbits, born in a symmetry-breaking bifurcation (on branches A1 and A2) undergo a period-doubling cascade (the first flip bifurcation branches are

"L

(a)

245

. . . . . . .

i

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,

.

.

.

.

.

.

.

.

dl

>- o

i

-0.5

-0.8

0.5

0

--0.4

0

0.4

0.8

X

X

(c)

(d)

iHTI

3~.~

:

.T1

A2

!

:

p

A1

::

A

s

i

s

i

:HTZ o

'

-0.75

_0174

' I/

,

_0J.73

I

|l

,

'

,

,

,

,

-0.8

I

,

,

,

,

-0.75

-0.7

v

(e)

•

,

,

,

,

,

i

,

,

I

,

,

N c5

c~ ,

l

J

-0.5

l

i

,

,

I

0 X

,

i

,

,

I

0.5

,

l

I

-0.5

,

,

,

,

l

0

,

0.5

X

Fig. 13. Symmetric 3T-orbits: (a) phase portrait for 1, = -0.7482, /3 = 0.6; (b) a triple pulse homoclinic orbit for 1, ~ - 0 . 8 1 2 6 , /3 = 0.4; bifurcation diagram for (c)/3 = 0.6 and (d)/3 = 0.4; asymmetric periodic orbits resulting from (e) pitchfork (p = - 0 . 7 5 5 4 , /3 = 0.4) and (f) flip (v = -0.7098, fl = 0.4) bifurcations. P1, P2 and P3 in (c) mark symmetry-breaking bifurcation points. HT1 and HT2 indicate the presence of homoclinic orbits. Points A and B in (d) correspond to saddle-node bifurcations. Branches A1 and A 2 are of asymmetric periodic orbits and branches F1 and F2 of double-period asymmetric orbits. S indicates a branch of symmetric periodic orbits.

E. Freire et al. / Case study in autonomous electronic circuit

246

denoted by F1 and F2). Similar branches will a p p e a r - where the symmetric orbit recovers its stability just before to undergo the saddle-node bifurcation- from the upper branch S near the saddle-node bifurcation points A and B. In fig. 13b we show one of the corresponding homo° clinic orbits. Moreover, these in the branch HT1 are subsidiary [12] of the saddle connections of HA1 organizing the branches of the principal asymmetric periodic orbits (see fig. 9b). In fig. 13e, f one of the asymmetric limit cycles and its first subharmonic also appear. We will complete this subsection drawing, in fig. 14, symmetric 5T-, 7T- and 9T-orbits and one asymmetric 5T-orbit. The shape of these symmetric orbits suggest the possible existence of heteroclinic connections between the nontrivial equilibria, a possibility that will be investigated in future work. This extremely rich variety of periodic oscillations, due to the presence of Sil'nikov homo-

clinic orbits, has to be completed with other limit cycles. For example, we have found the asymmetric 4T-orbit shown in fig. 15 that seems to be related to the homoclinic orbit presented in the same figure. Further work will be carried out on this subject.

4. Sil'nikov chaotic attractors One of the main reasons for the interest provoked in recent years by this electronic circuit is the presence of chaotic oscillations in its dynamics for wide ranges of the parameters, being an autonomous tridimensional system. We will now complete the above description of periodic and homoclinic behaviour, reporting several instances of nonperiodic behaviour obtained in numerical simulation of system (3). We will present some bidimensional flow sections (b)

(a)

'

'

b

i

i

.

I

i

.

.

.

i

.

.

I

L

.

.

I

,

,

L

i

i

cq

c~

d

=

~

I

=

I

=

i

-0.5

I

L

=

,

i

0

I

~

=

i

~

i

-0.5

0.5

h

i

i

0

0.5

X

(d) ~

(c)

>- Q

~-

l -0.5

l

i

L

l

l

l l 0 X

)

l

l

l

l

'

'

'

'

I

i

i

i

i

'

. . . .

I

i

'

0

i

0.5

'

-0.5

0

i

i

L

I

0.5

X

Fig. 14. (a) S y m m e t r i c 5 T - o r b i t ( v = - 0 . 7 2 6 5 , /3 = 0.4). (b) A s y m m e t r i c 5T-orbit ( v = - 0 . 7 1 6 3 7 , 7 T - o r b i t ( v = - 0 . 7 1 1 , /3 = 0.4). (d) S y m m e t r i c 9 T - o r b i t (v = - 0 . 7 2 3 5 , /3 = 0.4).

/3 = 0.4). (c) S y m m e t r i c

247

E. Freire et al. / Case study in autonomous electronic circuit

(a)

i

.

i

i

.

.

.

i

.

I

i

.

.

.

i

d

>-

O

? i

p -0.5

L

i

i

i

f

0

i

i

i

i

I

i

i

0.5

X

i

i

-0.5

i

i

i

0

,

i

i

I

i

0.5

X

Fig. 15. (a) Asymmetric4T-orbit (~, = -0.736, /3 = 0.4). (b) Quadruple-pulse homoclinicorbit (~, ~ -0.7096, fl = 0.4). and unidimensional return maps in order to examine the chaotic attractors. Next, we intend to give a theoretical explanation of the limit sets (and related return maps) appearing in flow sections. A standard analytic approach used in studying the dynamics near a Sil'nikov homoclinic orbit, under strong contraction assumption, will be introduced. The period-doubling cascade undergone by the principal asymmetric periodic orbits originates the presence of a pair of R6ssler chaotic attractors (see fig. 16). The use of a section (the plane y = 0) will give us information about the structure of such attractor. In all the cases shown below, the limit sets of the corresponding Poincar6 map for a specific flow section are seemingly unidimensional islands due to their strong contractivity, which is a consequence of the dissipative character of the system. When v = - 0 . 6 8 two islands appear (see fig. 17a): the upper one, A, corresponds to the cut of the attractor with p < 0; the lower one, B, with )~ > 0. The next return map of the island A onto itself, constructed for two R6ssler attractors, shows the change from spiral type (fig. 17b) to screw type (fig. 17c) [24,10]. If the parameter diminishes, the pair of R6ssler attractors coalesce giving rise to a unique chaotic attractor. We will concentrate on the attractor that coexists with the principal Sil'nikov homoclinic orbit (p--'-0.7377, /3 = 0 . 4 ) , that is on a Sil'nikov attractor shown in fig. 18a. Its flow section basi-

cally presents three islands A, B and C in the 3~ < 0 region - and their symmetric A', B' and C' in )~ > 0 (see fig. 18b). The dotted line indicates ~9---0, SM the intersection with the stable manifold of the origin and the small squares give the position of the three equilibria. The return map of islands A and A' onto themselves is drawn in fig. 18c. In fact, this 1D return map is similar to the several ones presented by King [19] after experiments in his system that have 7/z-symmetry broken. When the homoclinic curve HP reaches the fl = 0 straight line, the Hopf-Sil'nikov point HS appears (see fig. 3). We present a projection of the chaotic attractor existing at this codimension2 point (see fig. 19a). The islands B and C are practically undistinguishable in the flow section (19b). The corresponding 1D return map of A and A ' onto themselves is also shown (19c). The obtained 1D return maps support the supposed chaotic character of the dynamics on the attracting sets. Some of these 1D return maps are single-humped, which are widely studied (cf. [14].). In several instances we find 1D return maps with two humps; this case, of which further study is needed, presents the possibility of coexisting attracting sets. We now try to give some explanations of structures of the attractors which we have observed simultaneously with homoclinic orbits. For that, we consider a 3D-system which is invariant under the Z2-symmetry and has an

248

E . F r e i r e et al.

C a s e s t u d y in a u t o n o m o u s

e l e c t r o n i c circuit

¢-,I

(a)

I

0

oq o -~

i

I

I

i

I

I

-0.5

i

i

i

I

I

0

0.5

(b) 0

0

0

L

o

i

i

0.2

1, 0.4

~

L

L

I 0.6

Fig. 16. E v o l u t i o n of the a s y m m e t r i c chaotic attractors (/3 = 0.4): (a) close to the period-doubling cascade (y = - 0 . 6 6 5 6 ) ; (b) in the R r s s l e r region (u = - 0 . 6 8 ) .

equilibrium of saddle-focus type at the origin; such system can be written in the form = -px

- toy + P ( x , y , z ) ,

y~ = oJx - p y + Q ( x , y , z ) , i: = X z + R ( x , y , z ) ,

where p > O , t o > O , h > O and P, Q, R are smooth functions vanishing together with their first derivatives at the origin. We further assume that there are a pair of homoclinic orbits to the

equilibrium point at the origin, denoted F and g'.

We now use a standard construction [10,11,12,16] in order to obtain bidimensional (2D) return maps on flow sections. Let D be a point of the homoclinic orbit F close enough to the origin as to belong to the local stable manifold, that is supposed to be part of the plane z = 0; further, 12 is on the semiplane {y = 0, x > 0), and so ~ = (r, 0 , 0 ) , r > 0 . In an analogous manner we have the point ~ ' = ( - r , 0, 0) of the homoclinic orbit F'. Let 2~ + and ~ - be open parts (adequately chosen) of

E. Freire et al. / Case study in autonomous electronic circuit

(a)

249

to~

d (b)

,:5

N~

+o N=

0

012

014

0.3

0.6

0.35

0,4

0.45

0 5

7n

X (c)

L'3

(5

(5

rr~

,5 I 0.3

,

,

,

i

I

~

~

0,35

J

,

I

i

i

0.4

J

i

I

i

i

i

0.45

,

I

,

,

0.5

Zn

Fig. 17. R6ssler attractors (fl ---0.4): (a) flow section (~, = -0.68); 1D return maps of the island A onto itself: (b) for spiral type attractors (u =-0.67) and (c) for screw type attractors (v =-0.68). Island A corresponds to the intersection between the attractor and the plane y = 0 with p < 0 (island B with )~ > 0).

semiplanes { y = 0 , x > 0 ) and { y = 0 , x < 0 } c o n t a i n i n g O and ~O' respectively (see fig. 20 for the g e o m e t r i c configuration), then, we obtain [11] the following return m a p s on ,~+ and 2 - : T1 : ~ + ---->,~ +

(to z > 0 )

x ~ r + a x z ~ cos( ~ log z + ~bl), f l x z ~ c o s ( s¢ log z + ~b2),

I

Tz: ,~ + ---->,~ -

(to z < 0 )

[x~'-~ - r + a x l z 1' c o s ( ~ log [z[ + ~bl), - - f l x ] z [ ' cos(~: log ]z[ + ~b2),

l-

E. Freire et al.

250

Case study in autonomous electronic circuit

(a)

@4 0

>-

0

0

I

0.5

-0.5

(b)

d

,

,

/..'./. (c) tt3

<:5

.

T'4

T1

~o

N

~3

i

-0.5

0

X

0 5

i

-0.5

i

,

,

i

0

,

,

,

i

I

L

0.5

z,

Fig, 18. Sil'nikov attractor 0" = -0,7377,/3 = 0.4): (a) phase portrait; (b) flow section where several islands (A, B, C) and their symmetric (A', B', C') appear above and below the dotted line that indicates 3~= 0 (also the 3 equilibria are marked with small squares and the intersection of the stable manifold of the origin SM with the plane y = 0 is drawn); (c) 1D return map of islands A and A' onto themselves.

251

E. Freire et al. / Case study in autonomous electronic circuit I

oO

(a)

'

'

'

'

1

'

'

'

'

1

1

i

I

'

'

'

l

. . . .

I

'

'

'

'

l

'

'

'

'

l

d

c5

>-

0

c5 I

O0 0

I

i

I

i

-1.5

i

I

-1

i

I

i

-0.5

i

I

. . . .

i

0

. . . .

I

0.5

,

,

,

,

1

I

1.5

X

. . . .

(b)

r

. . . .

i

. . . .

i

. . . .

/

i

d

N

(c)

'~'

I

. . . .

I

. . . .

I

. . . .

I

'

'

'

'

I ~ '

#"

..... ..........

?4

71

~'3

~'2

0

0

N

?1

+=o

./

d

/

I

=

-0.5

0

0.5

. . ,

-1

i

. . . .

-0.5

i

0

. . . .

t

0.5

....

i ,

1

Zn Fig. 19. Hopf-Sil'nikov attractor (compare with figure 18) (v = -1.0264787, fl = 0): (a) phase portrait; (b) flow section; (c) 1D return map.

252

E. Freire et al, / Case s t u d y in a u t o n o m o u s electronic circuit

T3: 2~ - --->2~ -

(to z < 0 )

p

/t

%

¢

-/31xllzl ~ cos(~ log Izl + ,/,0

, -

T 4:~---->2~ +

\

\

l

~

SM/

(toz>0)

{:~--~ r - alxlz~ cos( , log z + ¢l) , -/31xl? cos(V log z + 6 0 , where 6 = p/A, ~ = - to/h and a, /3, ~b,, ~bz are constants. Hereafter, the Sil'nikov condition 6 < 1 is assumed. In the limit of strong contraction [1], x---> r on 2~ + and x - - > - r on ,~-, and the above maps of the plane reduce to the unidimensional maps

/i\

///

/SM

z Y

Fig. 20. Geometrical construction for the Sil'nikov.

~->r ,

TI:

{X

z~-~/3rz ~ cos(~ l o g z + ~b2),

(to z > 0 , x = r )

T2:

( X~-->--r

,

z ~ / 3 r [ z l ~ cos(~

log [zl + 4'2),

(to z < 0, x = r)

T3: { X ~-'> --r z ~ - / 3 r l z l ~ c o s ( ~ log Izl

+

~),

(to z < 0 , x = - r )

~4..{X~'->r~ z~--~ - / 3 r z ~ cos(~ log z + 4 2 ) ,

(to z > 0 , x = - r ) In these conditions we have an attractor included in the union of islands Z + N {x = r} and 2 ~ - N {x = - r } , and the dynamics on it is determined by the 1D-maps Ti, i = 1, 2, 3, 4. Now, this configuration is transported along the homoclinic orbits F and F', producing on the flow section y = 0 a denumerable set of attracting islands formed in neighbourhoods of the points where the homoclinic orbits meet this plane y = 0. T h e above heuristic approach would justify the flow section shown in fig. 18b, at least in those parts of islands A, B, C, A', B', C' nearest the stable manifold SM of the origin (to interchange

the x and z coordinates in fig. 20 helps to see its similarity to fig. 18b). Also the four 1D-maps presented in fig. 18c are quite similar to the graphs of Ti maps, i = 1, 2, 3, 4 and so they are labelled in accordance. The above heuristic analysis, as applied to the H o p f - S i l ' n i k o v case, also works to justify the observed flow sections. As 6 decreases to its limit value 6 = 0 corresponding to the H o p f bifurcation ( p = 0), the focus on the stable manifold of the origin becomes a weak focus and accordingly the islands in the flow section y = 0 tend to be closer. This fact is evident in fig. 19b, where we can see the islands B, C (B', C') are nearer than in fig. 18b.

5. Conclusions

Local and global bifurcations for the autonomous electronic system have been analysed in a p a r a m e t e r plane. A double-zero point, a curve of homoclinic connections and a Hopf-Sil'nikov point stand out as dynamics organizers in the bifurcation set. It is conjectured that the described bifurcation behaviour, much of it numerically observed, would be obtained from analytical study of the triple-zero degeneracy

E. Freire et al. / Case study in autonomous electronic circuit

announced in section 2 (case 5). Its unfolding would need, at least, a three parameter space; until now only biparameter slices of which have been reported.

Acknowledgements The authors wish to thank the referees for their helpful comments and suggestions. This work was partially supported by the Ministerio Espafiol de Educaci6n y Ciencia in the frame of D G I C Y T project PS90-0139 and the Consejeria de Educaci6n y Ciencia de la Junta de Andaluc/a.

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