卡莱曼（Carleman）线性化 

近似线性化仅可撷取非线性系统的丰富的动力学中的一部分. 计算在范德堡尔振荡器的相空间中不同点的不同阶数的 Carleman 线性化的有效性.

范德堡尔振荡器的模型.

In[1]:=

X vanderpol[\[Mu]_] := AffineStateSpaceModel[ {{Subscript[x, 2], -Subscript[x, 1] - \[Mu] (-1 + \!\(\*SubsuperscriptBox[\(x\), \(1\), \(2\)]\)) Subscript[x, 2]}, {{ }, {}}}, {Subscript[x, 1], Subscript[x, 2]}, {}, {Automatic, Automatic}, Automatic , SamplingPeriod -> None]

从初始点 p 对100个时间单位的响应.

In[2]:=

X vanderpolResponse[\[Mu]_, p_] := OutputResponse[{vanderpol[\[Mu]], p}, {}, {t, 0, 100}]

其相位描述.

In[3]:=

X vanderpolPhasePortrait[\[Mu]_] := With[{v = Normal[vanderpol[\[Mu]]][[1, 1]] /. x_[\[FormalT]] :> x}, StreamPlot[v, {Subscript[x, 1], -3, 3}, {Subscript[x, 2], -3, 3}, StreamStyle -> "PinDart"]]

近似阶数为 n 的卡莱曼线性化系统的响应.

In[4]:=

X approxResponse[\[Mu]_, p_, n_] := With[{\[ScriptCapitalC] = CarlemanLinearize[vanderpol[\[Mu]], n]}, OutputResponse[{\[ScriptCapitalC]["TransformedSystem"], Last /@ \[ScriptCapitalC]["EmbeddingTransformation"] /. Thread[{Subscript[x, 1], Subscript[x, 2]} -> p]}, {}, {t, 0, 100}]]

可视化不同初始点的实际和近似轨迹.

In[5]:=

X Manipulate[ Show[vanderpolPhasePortrait[\[Mu]], ParametricPlot[ Evaluate[{vanderpolResponse[\[Mu], p]}~ Join~(approxResponse[\[Mu], p, #] & /@ n)], {t, 0, 100}, PlotLegends -> Join[{"Exact"}, n]], ImageSize -> Medium], {{p, {0.01, 0}}, {-3, -3}, {3, 3}, Locator}, {{\[Mu], 0.6, "Parameter (\[Mu])"}, 0, 1.2, 0.1}, {{n, {1, 4}, "Approx. system(s)"}, CheckboxBar[Dynamic@n, {1, 2, 4, 6, 8, 10}] &}, SaveDefinitions -> True, AutorunSequencing -> {1, 2}]

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