Details

A form of the equation was first proposed to model an optical bistable resonator system [1]. The route to chaos as increases to is described in [2]. For larger values of the solutions look and behave statistically like Brownian motion.

Snapshot 1: just above the value , where the stable quilibrium changes from a node to a focus

Snapshot 2: just above the value , at which there is a Hopf bifurcation and the appearance of a limit cycle

Snapshot 3: just above the value , at which a pitchfork bifurcation occurs, leading to two coexisting limit cycles

Snapshot 4: the other limit cycle for the same value of ; the initial condition is very close to that of snapshot 3

Snapshot 5: just above the value , at which a period doubling occurs

Snapshot 6: just above the value , where chaos first occurs; note how the solution looks much like Brownian motion

Snapshot 7: just above the value where a periodic window appears

[1] K. Ikeda and K. Matsumoto, "High-Dimensional Chaotic Behavior in Systems with Time-Delayed Feedback," Physica D, 29, 1987 p. 223.

[2] J. C. Sprott, "A Simple Chaotic Delay Differential Equation," Physics Letters A, 366, 2007 pp. 397–402.