Symmetry Breaking and Iterated-Map Networks

The logistic map is a simple mathematical model commonly used to explore discrete-time systems. L’Her, et al. [1] implemented an electronic circuit that mimics the behavior of coupled logistic maps to great precision. While L'Her’s circuit provides the basis for our experimental exploration of logistic map networks, it lacks practical control of initial conditions. We reproduced L’Her’s circuit and introduced automated manipulation of the initial conditions and of the coupling voltage through an Arduino Uno Microcontroller. This additional control allows for a close exploration of symmetric states and symmetry-broken states that the coupled logistic maps exhibit under specific conditions. Indeed, for fixed values of the growth rate, the manipulation of initial conditions produces an array of symmetric and symmetry-broken states. We will use this data to generate basins of attraction for the corresponding growth parameters. Conversely, if we fix the growth rate and iteratively increment the coupling strength, we experimentally observe interesting bifurcation behavior involving the symmetric, symmetry-broken, as well as phase-shifted solutions.

[1] L’Her, A., et al., Eur. Phys. J. B (2016) 89: 81