Bifurcations of a limit cycle in nonlinear dynamic systems

Abstract

A skew-symmetry principle that governs the formation of closed and quasiperiodic trajectories is formulated. Bifurcations of a limit cycle in nonlinear dynamic systems are analyzed. The phenomenon of drift is explained. An approximate solution of the limit cycle equations is found through a qualitative analysis Keywords: domain of periodic solutions, bifurcation, drift, skew-symmetry principle Introduction. Many recent publications pay much attention to the bifurcations of the equilibrium states of mechanical systems [7,8]. The study [13] is concerned with modeling the interaction between a rigid body and a medium and with the symmetry and bifurcations of the vector field of dynamic systems.The present paper deals with bifurcation theory and with the symmetry principle in dynamics of systems.A relevant approximate solution was found in [4,10]. Doronitsyn [4] developed a method to plot integral curves on the phase plane when m is great. The method is based on special series in powers of 1/ m.It was pointed out in [5, p. 16] that for any m > 0, the Van der Pol equations has a unique limit cycle, which is stable, on the phase plane ( , )x x 1 2 . This mathematical fact is supported by a physical phenomenon observed experimentally. When m is small, Bogolyubov's theorem can be used to establish whether a limit cycle exists [2]. When m is great, this theorem and the averaging principle are inapplicable and the skew-symmetry principle is used instead.The Van der Pol equation has three domains in the phase space. In the first domain, the trajectory is attracted to the curve on which the angular velocity of oscillations is zero. The periodic solutions evolve in the second domain. The third domain is characterized by aperiodic development of the process.

Complex oscillations in systems subject to periodic perturbation

“…Since the trajectory is attractive as a whole, the LCE signature of the cycle on a torus is ( , , 0 0 0, -) (for the equations of motion (16)). The LCE signature of the synchronized limit cycle of system (16) is (0, -). Figure 9 gives a concentrated presentation of the applied results of the qualitative analysis.…”

“…The principle of symmetry was used in [10,11,15,16] as a criterion for the existence of periodic and quasiperiodic motions. The essence of the principle is that if the axis of symmetry has been found, then any integral curve on its left (below it) is the mirror image of the curve on its right (above it).…”

“…Bifurcations of a Limit Cycle. Introduce a small deviation dx j from the partial solution x j j ( , , , ) = 1 2 3 4 of system (16). The variational equations for system (16) are the following:…”

“…where x 1 = r q cos , x 2 = -r q sin (r is the modulus of a complex variable [9,16]). Equations (20) hold in the domains of periodic and aperiodic solutions of the variational equation.…”

“…If m is small, the roots l 1 2 , ( )x are complex-conjugate for all t. If m is great, there are real roots l 1 2 , ( ) x . Let us plot curves separating the phase-plane domains with aperiodic points on the trajectory of system(12),(16). The boundary of the domain can be found by equating the radicand in the formula for l1 2 , ( ) x to zero.…”