mathematics//dynamical systems//discretization

A continuous rate is not the same as an update by steps. In continuous time \(\dot x=Fx\) specifies a **rate**; in discrete time \(x_{k+1}=Tx_k\) specifies the **next state**. The matrices play different roles and normally carry different units:

A continuous rate is not the same as an update by steps. In continuous time x˙=Fx\dot x=Fxx˙=Fx specifies a rate; in discrete time xk+1=Txkx_{k+1}=Tx_kxk+1​=Txk​ specifies the next state. The matrices play different roles and normally carry different units:

x(t)=eFtx(0),xk=Tkx0.x(t)=e^{Ft}x(0),\qquad x_k=T^k x_0.x(t)=eFtx(0),xk​=Tkx0​.

The matrix exponential is eFt=I+Ft+(Ft)2/2!+⋯e^{Ft}=I+Ft+(Ft)^2/2!+\cdotseFt=I+Ft+(Ft)2/2!+⋯; it is not the exponential applied entry by entry.

Which eigenvalues dominate. In continuous time each mode carries eλite^{\lambda_i t}eλi​t, so the real part of λi\lambda_iλi​ matters. In discrete time it carries μik\mu_i^kμik​, so the modulus of μi\mu_iμi​ matters. Linear asymptotic stability needs Re⁡λi<0\operatorname{Re}\lambda_i<0Reλi​<0 or ∣μi∣<1|\mu_i|<1∣μi​∣<1 respectively. On the boundary the Jordan structure matters, and a mode not excited by the initial condition may not show up in a given response. "The largest eigenvalue dominates" has to specify order, excitation and which output is observed.

Exact sampling and Euler's approximation. For a linear autonomous system sampled every Δt\Delta tΔt, T=eFΔtT=e^{F\Delta t}T=eFΔt reproduces the states exactly at those instants. Euler uses TE=I+ΔtFT_E=I+\Delta t FTE​=I+ΔtF. For x˙=−λx\dot x=-\lambda xx˙=−λx, λ>0\lambda>0λ>0, Euler gives xk+1=(1−λΔt)xkx_{k+1}=(1-\lambda\Delta t)x_kxk+1​=(1−λΔt)xk​: the simulation decays only if 0<λΔt<20<\lambda\Delta t<20<λΔt<2, and between 1 and 2 it alternates sign although the continuous solution does not. To also keep non-negativity from x0≥0x_0\ge0x0​≥0, the condition 0≤λΔt≤10\le\lambda\Delta t\le10≤λΔt≤1 is necessary and sufficient here.

Discretized diffusion. For a symmetric Laplacian, Euler on x˙=−Lx\dot x=-Lxx˙=−Lx contracts every disagreement mode if

0<Δt<2λmax⁡(L).0<\Delta t<\frac{2}{\lambda_{\max}(L)}.0<Δt<λmax​(L)2​.

For I−ΔtLI-\Delta t LI−ΔtL to also be a non-negative averaging matrix, Δt≤1/dmax⁡\Delta t\le1/d_{\max}Δt≤1/dmax​ is required, assuming weights with units of rate. Stability, non-negativity and accuracy are three different checks.

A Markov transition matrix can define a discrete process on its own; it does not have to come from discretizing a differential equation. And P=D−1WP=D^{-1}WP=D−1W is not, in general, the exact sample e−LΔte^{-L\Delta t}e−LΔt of the unnormalized diffusion.

Modal powers and exponentials come from modes; effects between samples belong to delay and lag; the stability studied in equilibrium and stability is the system's, not the algorithm's.