Compute the linearization of an operator representing the Van der Pol oscillatorLet Gμ:C∞(R)→C∞(R) be an operator defined asGμ[x]=x″+μ(x2−1)x′+xWhere C∞(R) is the space of infinitely-differentiable functions defined on R. The case Gμ=0 describes the Van der Pol oscillator.Define the linearization of Gμ at x to be the operator:Lxμ[v]=limh→0Gμ(x+hv)−Gμ(x)hQuestion: compute Lxμ[v] for any x, v∈C∞(R) and μ∈R.The operator Lxμ[v] looks similar to the limit definition of the derivative, but I don't think I am simply meant to differentiate Gμ. Finding Gμ(x+hv)−Gμ(x) just gives a really long expression that doesn't look like it simplifies to anything useful.

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Compute the linearization of an operator representing the Van der Pol oscillatorLet Gμ:C∞(R)→C∞(R) be an operator defined asGμ[x]=x″+μ(x2−1)x′+xWhere C∞(R) is the space of infinitely-differentiable functions defined on R. The case Gμ=0 describes the Van der Pol oscillator.Define the linearization of Gμ at x to be the operator:Lxμ[v]=limh→0Gμ(x+hv)−Gμ(x)hQuestion: compute Lxμ[v] for any x, v∈C∞(R) and μ∈R.The operator Lxμ[v] looks similar to the limit definition of the derivative, but I don&#039;t think I am simply meant to differentiate Gμ. Finding Gμ(x+hv)−Gμ(x) just gives a really long expression that doesn&#039;t look like it simplifies to anything useful.

Roy Brady · Accepted Answer

Step 1You can think that x depends on a parameter&amp;nbsp;λ&amp;nbsp;and write&amp;nbsp;xλ, with&amp;nbsp;x=x0, and&amp;nbsp;(ddλ)∣λ=0xλ=v. So computeStep 2Lxμ[v]=ddλ∣λ=0Gμ[xλ]=v&amp;nbsp;&#039;&#039;+2μ×&#039;v+μ(x2−1)v&#039;+v,with the product rule. The point is that&amp;nbsp;x↦x′&amp;nbsp;and&amp;nbsp;x↦x&amp;nbsp;″&amp;nbsp;are linear operators in the variable x, so the total derivatives of those, at x, evaluated at v, are simply v&#039; and v&#039;&#039;.

Compute the linearization of an operator representing the

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