An exploration into rings of oscillators using the Ricker Model

In this paper, we explore the response of homogeneous and non-homogeneous rings of oscillators using the Ricker model. We begin in chapter two by introducing the Ricker model and its bifurcation plot. We observe regions of the bifurcation plot for which we obtain periodic solutions and other regions for which we obtain chaotic solutions. Whether a response is periodic or chaotic can be determined by its Lyapunov Characteristic Exponent (LCE), which is discussed in chapter three. In chapter four, we observe the evolution of a single node and see how the system can spontaneously change from an apparent chaotic response to a periodic response. We find that, when acting independently, these oscillators display a chaotic response for given values of the intrinsic growth rate. However, when connecting these oscillators to their nearest neighbors, the response becomes periodic. We then observe the dynamical behavior of connected rings of oscillators by changing the initial conditions. We explore symmetric node responses where we find that the projection of node values can be the same when reflected about an oscillator. We vary the values of system parameters such as the connection strength to each node and the number of connected nodes in a ring, and determine for which of these values result in periodic solutions. In chapter five, we explore a ring of non-identical oscillators. By plotting LCEs for varying values of the connection strength, we find that the global response of non-homogeneous rings is complex. Lastly, in chapter six, we explore the dynamical behavior of coupling rings of non-homogeneous oscillators. The work in this paper parallels the work done by Hubertus von Bremen in [1]; here we focus exclusively on the Ricker model.