Systems and Delays (2026)
I’d be interested in learning what, mathematically, makes these oscillations inevitable Say you model your inventory as df/dt = −f(t − τ) / r Where τ is the delay, and r is the response delay divisor, like your post.
- ▪I’d be interested in learning what, mathematically, makes these oscillations inevitable Say you model your inventory as df/dt = −f(t − τ) / r Where τ is the delay, and r is the response delay divisor, like your post.
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I’d be interested in learning what, mathematically, makes these oscillations inevitable Say you model your inventory as df/dt = −f(t − τ) / r Where τ is the delay, and r is the response delay divisor, like your post. A standard result is that the first-order delay differential equation dx/dt = −ax(t − τ) is stable when aτ < π/2. (e.g. see corollary 3.3 in Stépán's Retarded Dynamical Systems, https://www.mm.bme.hu/~stepan/mm/book/retarded_dynamical_systems.pdf) With a = 1/r that gives τ/r < π/2, i.e., r > 2τ/π Applying that to your examples, if τ = 5, you need r > 10/π ~ 3.18 So the system ends up stable when r = 6, but not with r = 2 or r = 1.
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