  A dynamical system is characterized as linear or nonlinear depending on the nature of the equations of motion describing the target system. Consider a differential equation system, such as \(d\bx/dt = \bF\bx\) for a set of variables \(\bx = x_1, x_2, \ldots, x_n\). These variables might represent positions, momenta, chemical concentration or other key features of the target system, and the system of equations tells us how these key variables change with time. Suppose that \(\bx_{1}(t)\) and \(\bx_{2}(t)\) are solutions of the equation system \(d\bx/dt = \bF\bx\).  If the system of equations is linear, it can easily be shown that \(\bx_{3}(t) = a\bx_{1}(t)+ b\bx_{2}(t)\) is also a solution, where \(a\) and \(b\) are constants. This is known as the principle of linear superposition. So if the matrix of coefficients \(\bF\) does not contain any of the variables \(\bx\) or functions of them, then the principle of linear superposition holds. If the principle of linear superposition holds, then, roughly, a system behaves linearly: Any multiplicative change in a variable, by a factor \(\alpha\) say, implies a multiplicative or proportional change of its output by \(\alpha\). For example, if you start with your stereo at low volume and turn the volume control one unit, the volume increases by one unit. If you now turn the control two units, the volume increases two units. This is an example of a linear response. In a nonlinear system, such as (Lorenz), linear superposition fails and a system need not change proportionally to the change in a variable. If you turn your volume control too far, the volume may not only increase more than the number of units of the turn, but whistles and various other distortions occur in the sound. These are examples of a nonlinear response. 