International Journal of Bifurcation and Chaos, Vol. 10, No. 7 (2000) 1759–1772 c  World Scientific Publishing Company ANALYSIS OF A NONSYNCHRONIZED SINUSOIDALLY DRIVEN DYNAMICAL SYSTEM O. M ´ ENARD, C. LETELLIER, J. MAQUET, L. LE SCELLER and G. GOUESBET CORIA UMR 6614, Universit´ e et INSA de Rouen, Place Emile Blondel, 76821 Mont Saint-Aignan cedex, France Received June 30, 1999; Revised December 7, 1999 A nonautonomous system, i.e. a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space which is the product of the phase space R m of the undriven system by an extra dimension associated with the external force. The analysis is then performed by taking advantage of the known period of the external force to define a Poincar´ e section relying on a stroboscopic sampling. Nevertheless, it may so happen that the phase synchronization does not occur. It is then more convenient to consider the nonautonomous system as an autonomous system incorporating the subsystem generating the driving force. In the case of a sinusoidal driving force, the phase space is R m+2 instead of the usual extended phase space R m × S 1 . It is also demonstrated that a global model may then be obtained by using m +2 dynamical variables with two variables associated with the driving force. The obtained model characterizes an autonomous system in contrast with a classical input/output model obtained when the driving force is considered as an input. 1. Introduction When one is facing a dynamical system, it is of interest to know whether the system is driven or not by an external force because, if a driving force is identified, the analysis may take advantage of a possible phase synchronization between the driven system and the external force. Examples are pro- vided by the Duffing equations [Gilmore & McCal- lum, 1995] or by experimental driven systems such as a driven laser [Boulant et al., 1997] or a driven thermoionic plasma diode [Mansbach et al., 1999]. When the driving force is identified, the phase space R m of the undriven system is usually extended to R m+1 to describe the driven system. In such a case, the analysis relies on the use of stroboscopic sections to compute Poincar´ e sections and to extract peri- odic orbits [Gilmore & McCallum, 1995; Boulant et al., 1997]. The system is therefore implicitly considered as being phase synchronized with the driving force. Moreover, when the driving force may be recorded simultaneously with a dynamical variable of the system, an input/output model can be obtained [Aguirre & Billings, 1994]. A set of equations is then built to describe the response of the system under the action of the driving force. Nevertheless, it may occur that the system is not well phase synchronized with the driving force. In other cases, the driving force cannot be identified or cannot be measured. Under such circumstances, a dynamical analysis cannot be safely performed in the extended phase space R m+1 and the use of a higher-dimensional phase space becomes more con- venient. Such a phase space is spanned by the dy- namical variables required for a complete descrip- tion of the process governing both the driven sys- tem and the driving force. A driven R¨ ossler system, with a driving force applied to the third equation, is used in this paper to illustrate a situation where a system is not well phase synchronized with the driving force and, a single variable being recorded, where the driving force cannot be measured. 1759