Linear and Nonlinear Oscillators
The linear superposition principle which is valid for linear differential equations is no longer valid for nonlinear ones. A physical consequence is that when the given dynamical system admits oscillatory motion, the associated frequency of oscillation is
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The linear superposition principle which is valid for linear differential equations is no longer valid for nonlinear ones. A physical consequence is that when the given dynamical system admits oscillatory motion, the associated frequency of oscillation is in general amplitude-dependent in the case of nonlinear systems, while it is not so in the case of linear systems. Particularly, this can have dramatic consequences in the case of forced and damped nonlinear oscillators, leading to nonlinear resonances and jump (hysteresis) phenomenon for low strengths of nonlinearity parameters. Such behaviours can be analysed using various approximation methods. However, as the control parameter varies further, the nonlinear systems can enter into more and more complex motions through different routes, where detailed numerical analysis and possible analog simulations using electronic circuits can be of much help to analyse them. In this chapter, we will introduce some basic features associated with nonlinear oscillations and postpone the discussions on more complex motions to later chapters. However, before discussing the nature of such nonlinear oscillations, we shall first discuss briefly the salient features associated with a damped and driven linear oscillator in order to compare its properties with nonlinear oscillators.
2.1 Linear Oscillators and Predictability As we have discussed in Chap. 1, physical systems whose motions are described by linear differential equations are called linear systems. If they are associated with oscillatory behaviour, then they are designated as linear oscillators. The characteristic features of such linear systems are their insensitiveness to infinitesimal changes in initial conditions and at the most constant separation of nearby trajectories in phase space. As a consequence the future behaviour becomes completely predictable. To illustrate these ideas, let us consider the simple example of a linear harmonic oscillator of mass Tn, damped by a viscous drag force, and acted upon by an external periodic force. Such a model represents a very large number of physical systems ranging from forced oscillations in an LCR circuit to electron oscillations in an electromagnetic field. The oscillations are then described by an inhomogeneous, linear, second-order differential equation of the form
M. Lakshmanan et al., Nonlinear Dynamics © Springer-Verlag Berlin Heidelberg 2003
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2. Linear and Nonlinear Oscillators
d2 x
m dt 2
+a
,dx dt
,2
+ Wo
.
x = F sm wt ,
or equivalently
d2x dt 2
dx
2
+ ad-t + WoX = f
.
sm wt ,
a' a=- ,
f= F,
m,
m,
w5 =
W'2
-.J!..., m
(2.1)
where x(t) is the displacement of the system, subjected to suitable initial conditions. We choose the initial conditions for convenience to be x(O) = A, x(O) = O. In (2.1), wo/27r corresponds to the natural frequency, a is the strength of the damping, while f and w stand for the forcing amplitude and angular frequency, respectively, of the external force. We will first consider the special cases of (2.1) before looking
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