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The main objective of the present paper is to study the transition from periodic regular mot ion to chaos in a two degrees of freedom dynamical system by changing control parameters. The nonlinear differential equations governing motion of the system a rc derived from the Lagrange equations. | Vietnam Journa l of l\llecha ni cs, VAST, Vol. 29, No . 3 (2007), pp . 353 - 3711 Special Iss ue Dedicated to the Memory of Prof. Nguyen Van Dao ON THE TRANSITION FROM REGULAR TO CHAOTIC BEHAVIORS IN THE TWO DEGREES OF FREEDOM DYNAMICAL SYSTEM NGUYEN VAN KHANG AND NGUYEN HOANG DUONG Hanoi University of T echnology Abstract. The main object ive of the present pape r is to study the transition from period ic reg ular mo t ion to chaos in a two degrees of fr eedo m dynamical system by changing control parameters. The nonlinear differential equations governin g motion of the system a rc derived from the Lagrange equations. I3y use of the Poincare map , the dynami cal behavior is identified based on nume rical solutions of the ordinary differential equations. The Lyapunov ex ponent and the frequ ency spectrum a re calc ul ated to ide nti fy chaos. From numerical simulations, it is indica ted that the periodic, quasi-periodic and chaotic motions occur in the conside red system. 1. INTRODUCTION Fig. 1. l'viechani cal model In recent yea.rs, the study of chaot ic behaviors a nd strange attractors in deterministic nonlinear systems has undoubtedly developed into one of the main topics i11 the study of nonlinear phenomena in dynamical systems performed by engineers and applied scicutists [1-10]. Many important and interest ing applicat ions are found in nonlinear oscillatio11s. which docs not come as a surprise since nowadays the classical met hods of solution for such nonlinear problems are well known, and also since the new and rather a bstract mathematical tools arise in a rather nat ural way in nonlinear oscillations . .r-v1uch interest has been devoted to the study of chaos in nonlinear osci llators of the Duffing type [l, 7, 8], Mathieu type [l, 12, 13] and Van der Pol type [7, 10, 15, 16]. Let us consider here the two-degree-of-freedom dynamical system (Fig. 1). It is a system with connects t he Duffing and the linear oscillators. 354 Nguyen Van Khang and .
