A noise induced phase transition to chaos in systems with static friction

Rounding can have a significant influence on the limiting dynamics of nonlinear systems. Consider for instance Bernoulli’s map xn+1 = 2 xn mod(1). Its limiting dynamics is expected to be chaotic, because its Lyapunov exponent is positive, the natural logarithm of 2. However when the mapping is iterated on a computer, the times series is initially irregular, but then appears to be attracted to the unstable fixed point at xn=0. Figure 1 shows a typical time series. After each computation, digital computers round the result of the computation, because they can handle only numbers with a limited number of digits. This rounding turns the unstable fixed point at xn=0 into a stable fixed point and suppresses the chaotic motion.