1. Introduction
Recently it has been discovered that a nonlinear physical system can degenerate into a chaotic, unpredictable time-development, apparently undistinguishable from random noise, although the system is perfectly deterministic in the sense of being described by well-behaved equations of motion with no stochastic, fluctuating ingredients. Mathematicians and theoretical physicists have made great progress in understanding this transition to chaos (see Ref. [1] and Ref. [2]). Here we will describe such phenomena in the dynamical two-well potential system.
2. Description of the system
Consider a horizontal plane (see Fig. 1) on which a bar is situated. One of the ends of the bar is pinned (without friction) to the plane at a point O so that the end A of the bar can move along the circular trajectory on the plane. Another bar is pinned (without friction) to the bar OA at a point A. These bars are assumed to be rigid and weightless. The masses M and m are attached to the points A and B of the bars respectively. The bar AB and mass m can rotate around the pinning point A with a uniform angular speed. The point A of the system is joined to a fixed point C on the plane by a linear spring and a viscous damper. As may be seen from Fig. 1, when the bar AB does not rotate, there are four static equilibrium positions of the system. Two of these positions are stable when the spring is not deformed. Other two are unstable when the bar AB and the point C on the plane lie along the straight line. Two stable equilibrium positions will be situated in two-well potential. We examine the forced vibration due to rotation of the bar AB.
3. Equation of motion
Let O be the origin of the Cartesian system of coordinates xOy
(Fig. 1), L -- the length of the bar OA, l -- the length of the bar
AB, l1 -- the distance between the points O and C,
-- the angular speed of the counterclockwise rotation of the bar
AB, -- the angle formed by the bar OA with respect to the
axis x, -- the initial value of
(t=0), -- the angle formed by bar AB with respect to the line
OA, -- the initial value of the (t=0), k -- the
spring's stiffness, -- the coefficient of viscous
damping. In order to obtain the differential Lagrange equation representing
the motion of the mechanical system under consideration
The kinetic energy T consists of two parts corresponding to the masses
M and m
The velocity V consists of velocities of two motions (each for separate
bar) and is given by a sum of vector's products
For the kinetic energy of whole system one obtains
Let us calculate the potential energy of the system now. If is
the length of non-deformed spring and l2 is the length of the deformed
spring (when the angle formed by the bar OA with respect to the axis x
is ), then the potential energy is the work or energy stored in
the spring during the displacement . The potential energy of
the system is equal to
Therefore as the basic energy expressions of the system are known, it is
possible to obtain the equation of motion in the form
4. Numerical analysis and results
Let us consider a conservative system for which . This system displays chaotic and quasi-periodic motion (periodic motion has not been found during the numerical experiments). Poincaré maps in Figs. 2a-e show the effect of intermittency (chaotic and quasi-periodic motions change into each other). The Poincaré maps were obtained for the following values of the parameters: l/L=0.5, m/M=1.0, , , . Values of the ratio l1/L were 0.2, 0.209, 0.26, 0.3, 0.4 for the cases represented in Figs. 2a-e respectively. The numerical calculation revealed a non-monotonic dependence of system behaviour on the parameter l1/L. The seven distinct circumferences in the Poincaré map, represented in Fig. 2a, were obtained for the parameter l1/L=0.2. This is well known case of quasi-periodic motion (Ref. [2]). Initial increase in the parameter, up to the value of 0.26, is accompanied by distinct circumferences in corresponding Poincaré map merge into diffused circumference, see Figs. 2a-c. The width of the circumference obtained for l1/L=0.26 (Fig. 2c) is at least one order of magnitude greater than the width of the circumference obtained for l1/L=2.2 (Fig. 2a). Further increase in the parameter drives system back to the quasi-periodic motion, see Figs. 2d,e and 3a. The Poincaré map from Fig. 3a presents circumferences characteristic of quasi-periodic motion similar to that presented in Fig. 2a.
Another intermittency is present in Figs. 3a-c. These Poincaré maps were
obtained for the following values of parameters:
Now let us consider the non-conservative system. In the example under
consideration the values of parameters were