HAL Id: jpa-00227611
https://hal.archives-ouvertes.fr/jpa-00227611
Submitted on 1 Jan 1988
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
PERIOD-DOUBLING AND CHAOS IN AN Nth ORDER NONLINEAR DIFFERENTIAL EQUATION
R. Vallée, C. Marriott, P. Dubois, C. Delisle
To cite this version:
R. Vallée, C. Marriott, P. Dubois, C. Delisle. PERIOD-DOUBLING AND CHAOS IN AN Nth
ORDER NONLINEAR DIFFERENTIAL EQUATION. Journal de Physique Colloques, 1988, 49 (C2),
pp.C2-423-C2-426. �10.1051/jphyscol:19882101�. �jpa-00227611�
Colloque C2, Suppl6ment au na6, Tome 49, j u i n
PERIOD-DOUBLING AND CHAOS IN AN N t h ORDER NONLINEAR DIFFERENTIAL EQUATION
R. V A L L ~ E , C. MARRIOTT, P. DUBOIS and C. DELISLE
Laboratoire de Recherches en Optique et Laser, Departement de Physique, Universite Laval, Sainte-Foy. Quebec, GIK 7P4, Canada
Mswne: Un systeme bistable hybride peut &re decrit d'une facon g6nerale par une equation differentielle d'ordre n lorsque n composantes contribuent au temps de r6ponse total du syst6me. I1 s'ensuit une sequence inhabituelle de doublements de &ride menant au chaos.
Le cas asymptotique n - t m est discute.
Abstract: The problem of describing a hybrid bistable device in terms of an nth order non- linear differential equation is considered. The resulting unusual period-doubling route to chaos is analysed as a function of the order of the equation.
1
-
INTRODUCTIONSince the early work of Ikeda /1/ who recast the Maxwell-Bloch equations in terms of a differential-delay equation, the hybrid bistable device (HBD) has been thoroughly studied /2/ as a model for the ring cavity problem. Even though the exactness and relevance of this "model" can be questioned the HBD still in itself remains a very good system for studying unstable phenomena in optical bistability. It has recently been shown /3/ that the HBI) can be more appropriately described by a second order differential-delay equation (DDE) when two different components contribute to the total response time (t). The problem we address here consists of a generaliza- tion to n response times, which can be expressed in terms of the nth order DDE:
where t C ti.The nonlinear function is expressed. by F(X(t-za) ; p ) , p being the control parame- ter and t a the time delay. In the case of the acoustooptic device considered here we have:
We consider the special case where the ti are equal since the coefficient & in front of the mth order derivative term takes the simple form:
n
where Cm is the number of combinations of n quantities taken m at a time.
2
-
ANALYSIS:A) Linear Stability:
The linear stability analysis (LSA) of equation (1) leads to a polynomial of degree n which can be solved numerically in the complex plane. It allows us to establish the onset of
self-oscillation as a function of p and the ratio ta/r. The main result of this analysis is that, contrary to the first and second order equation for which the threshold of self-oscilla- tion is repelled to infinity as ta/z -+0, for n t 3 a self-oscillation occurs for a finite value of u even for a delay ta strictly equal to zero. The threshold of self-oscillation as a funqtion
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882101
C2-424 JOURNAL DE PHYSIQUE
of CI is shown on figLlre la) for the case n = 3. It has an upper bound p3 = 6.02... in the range defined by t a < < < t and a lower bound px = 0.32... corresponding to the discrete model which is approached as ~a becomes larger than t. Furthermore as n is increased the gap between the two limits becomes smaller (for n = 20, y20 = 0.43
...,
see Fig. lb). Another important result from the linear analysis is that the period of the self-oscillation for ~d = 0 approaches 22 for n large. It is interesting to note the parallel with the first order delay differential equation for which the period approaches 2ta when T ~ / T+
m . However the LSA cannot give the form of the solution once it has moved away from the fixed point and numerical simulation is necessary in order to follow the evolution of the solution as p is increased.B ) Numerical Simulation:
Equation ( 1 ) can be solved numerically for an arbitrary order n using a standard Runge-Kutta algorithm. The basic idea behind the simulation is to recast the problem in terms of a system of first order differential equations. Special care must be taken in order to avoid dit-ergence in the computation since the absolute values of the nth derivatives increase rapidly with m. The results obtained from our numerical simulation were compared with the experimental and LSA results in order to ensure their exactness.
PERIOD-2 STABLE
log
( T d / ~ )Fig. 1
-
Threshold of instability of the solutions (a) for n = 3; (b) for n = 20.The basic acoustooptic HBD used in this experiment has been described elsewhere /4/. The main modification to this setup was the introduction in the feedback loop of a series of RC-circuits each separated from their neighbours by an isolation stage of unitary gain. We res- tricted our experiments to the case where all the ti are equal and much larger than the intrin- sic delay of the device, ta. The parameters A and Xb were fixed at 0.35 and zero respectively.
Fxprimental bifurcation diagrams showing the dynamical evolution of the system as a function of
p for n varying from three to twenty were obtained by detecting electronically the zeros of the first derivative of X(t). Similar results are obtained when we consider the zems of the higher order derivatives.
For n = 3 or 4 the system remains in the Period-2 state until it precipitates to the stable lower branch. For n = 5 to 8 "period-bubbling" /5,6/ is observed as the period-4 oscillation appears for a narrow interval of before returning to the Period-2 waveform (Fig. 2a). A chaotic behavior is observed for n = 9 following what seems to be an infinite period doubling sequence (Fig. 2b). However the system never reaches a fully developed chaotic state since it returns to the Period-2 state before precipitating. Periodic windows begin to be observed within the chaotic domain for n = 11, especially the Period-3 oscillation which appears following an interior crisis (Fig. 2c). A "period-bubbling" of the Period-8 oscillation is shown for n = 20 so that the transition to chaos now occurs through a crisis (Fig. 2d). Note that the Period-3 oscillation for n = 20 looks like a periodic window with four branches on the diagram because of the overshoot appearing approximatively half way through the oscillation period.
For comparison, results of the numerical analysis are shown on figure 3. The agreement with the experiment is almost perfect as far as the sequence (as n is increased) of bifurcation diagrams is concerned. However one can see the discrepancy (growing with n) between the values
.)f n for rchich a qiven bifurcation sequence is obtained experimentally and numerically. For Instance the diagram computed for n = 17 corresponds to what is observed experimentally for n = 20. A possible explanation for this discrepancy could be that the parameters A and X b in the experiment were not precisely those used in the numerical simulation. This is now under
~nvest igation.
Obviousl.y, the question of paramount importance in this problem is whether or not the solution reaches a sort of asymptotic behavior as n is increased. On one hand, the coefficients in from' of the nth derivative decrease very rapidly with m (Especially for large n since the nth derivative term is proportional to l/nn for t S 1 ) . On the other hand, the derivative themselves RrOrJ with n. Therefore each term contributes significantly to the solution. This explains why noticeable change are still visible between bifurcation diagrams corresponding to consecut,ive values of n, even for large n. As a matter of fact the convergence with n is very slow and must be analysed seperately for the periodic and chaotic domains.
For the periodlc domain, it is clear that besides the curious crisis and bubbling phenomena nscurring for relatively small values of n, the period-doubling scenario "ad infinitum" is followed more and more exactly as n is increased. In particular, bifurcation points tend toward those predicted by the discrete model. Incidently, a very good parallel can be established between the bifurcation diagram of the nth order equation without delay as n-+m and that of the first order equatlon with delay as ~d /T -' m.
C2-426 JOURNAL DE PHYSIQUE
The behavior of the chaotic solution of the nth order DE as n is increased is a matter that we have not as yet st~died quantitatively but it is obvious from basic considerations that these solutions grow in complexity with increasing n. Each differential term in the equation corres- ponds to a degree of freedom in the system. An increase in n therefore corresponds to an increase in the dimension of the phase space. One of the major consequences of this complexification of the solutions is the disappearance of the periodic windows corresponding to the "pcycles" of the discrete model. A Period-3 waveform for instance is no longer a solution of the system for n
>
27. Such a phenomenon is interesting since it can be compared to the behavior of the first order DE with delay which shows Period-3 waveforms only for small values of the ratio of td/t. In fact the correspondance between the first order DDE and the nth order DE becomes more and more obvious as n is increased. In the case of the first order DDE it is the ratio ta/t which determines the degree of complexity of the solution even though, strictly speaking, the phase space dimension always remains infinite. In the nth order DE the phase space dimension is finite and equal to n=
t/t, and the asymptotic situation n+m (implying ti + 0 so that t remains finite) corresponds to a system, made of an infinity of infinitely fast components, having an overall "characteristic" time response defined by t./1/ K. IKEDA, Opt. Corn., 30, 257 (1979).
/2/ H.M. GIBES, F.A. HOPF, D.L. WW, and R.L. SHO-, Phys. Rev. Lett., 46, 474 (1981).
/3/ R. VALLEE, P. D'JBOIS, M. COTE, and C. DELISLE, Phys. Rev. A, 36, 1327 (1987).
/4/ J. ~ ~ l . S K I , R . VALLEE, AND C. DELISLE, Can. J. Phys., 6l, 1143 (1983)
/ 5 / M. BIER, and T.C. BOUNTIS, Phys. Lett.
m,
239 (1984)./6/ D.M. HEFFEFNAN, Phys. Lett., =A, 413 (1985).
Fig. 3 - Bifurcation diagrams obtained from the numerical simulation of equation (1) using the coefficient defined in equation (3) and A = 0.35, X b = 0. (a) n = 7; (b) n = 8;
(c) n = 11; (d) n = 17.
