Analytic solutions throughout a period doubling route to chaos

We show examples of dynamical systems that can be solved analytically at any point along a period doubling route to chaos. Each system consists of a linear part oscillating about a set point and a nonlinear rule for regularly updating that set point. Previously it has been shown that such systems can be solved analytically even when the oscillations are chaotic. However, these solvable systems show few bifurcations, transitioning directly from a steady state to chaos. Here we show that a simple change to the rule for updating the set point allows for a greater variety of nonlinear dynamical phenomena, such as period doubling, while maintaining solvability. Two specific examples are given. The first is an oscillator whose set points are determined by a logistic map. We present analytic solutions describing an entire period doubling route to chaos. The second example is an electronic circuit. We show experimental data confirming both solvability and a period doubling route to chaos. These results suggest that analytic solutions may be a more useful tool in studying nonlinear dynamics than was previously recognized.