Self-oscillation in the Lorenz system determined by its memory term

Determine whether the self-oscillatory behavior of the Lorenz system, when represented by the second-order integro-differential equation ddot(q) + (σ+1)·dot(q) − σ(r−1)·q + (1/2)·q^3 + σ·(1 − b/(2σ))·q·∫_0^t e^{−bτ}·q^2(t−τ) dτ = 0, is determined by the influence of the memory term q·∫_0^t e^{−bτ}·q^2(t−τ) dτ.

Background

The paper derives a second-order integro-differential equation (IDE) for the Lorenz system and compares it to the damped, undriven Duffing oscillator. The authors emphasize that, when the Duffing system’s external forcing term is set to zero, its oscillations decay; in contrast, the Lorenz IDE exhibits sustained self-oscillation.

They note that the only structural difference between the undriven Duffing equation and the Lorenz IDE is the presence of a memory (heredity) term in the latter, and elsewhere argue that memory terms are associated with a positive rate of change of kinetic energy over cycles in the chaotic regime. This leads them to conjecture that the memory term is the decisive cause of self-oscillation in the Lorenz system.