Preface
The influence of a periodic forcing term on the Duffing equation is considered in this section.
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Introduction to Linear Algebra with Mathematica
Glossary
Forced Duffing equation
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We plot the stability boundary for the forced Duffing equation
\[
\ddot{x} + x(t) + \frac{1}{6}\, x^3 = \frac{1}{3}\,\cos \left( \frac{3t}{5} \right) , \qquad x(0) =a, \quad \dot{x}(0) = b .
\]
pfun = ParametricNDSolveValue[
{x''[t] == -x[t] + x[t]^3/6 + (1/3)*Cos[3*t/5], x[0] == a,
x'[0] == b},
x, {t, 0, 100}, {a, b}];
fun[a_?NumericQ, b_?NumericQ] := Module[ {res}, res = Quiet[pfun[a, b]]; Boole[res["Domain"] === {{0., 100.}}] ]; plot = ContourPlot[fun[a, b], {a, -0.6, 2}, {b, -1, 1}, PlotPoints -> 50, MaxRecursion -> 3, Axes -> True, AxesOrigin -> {0, 0}] |
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| Stability boundary. | Mathematica codes. |
|
We plot the stability boundary for the forced Duffing equation
\[
\ddot{x} + x(t) + \frac{1}{6}\, x^3 = \frac{1}{3}\,\sin \left( \frac{3t}{5} \right) , \qquad x(0) =a, \quad \dot{x}(0) = b .
\]
pfun = ParametricNDSolveValue[
{x''[t] == -x[t] + x[t]^3/6 + (1/3)*Sin[3*t/5], x[0] == a,
x'[0] == b},
x, {t, 0, 100}, {a, b}];
fun[a_?NumericQ, b_?NumericQ] := Module[ {res}, res = Quiet[pfun[a, b]]; Boole[res["Domain"] === {{0., 100.}}] ]; plot = ContourPlot[fun[a, b], {a, -1.5, 1.5}, {b, -0.6, 1.5}, PlotPoints -> 50, MaxRecursion -> 3, Axes -> True, AxesOrigin -> {0, 0}] |
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| Stability boundary. | Mathematica codes. |
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