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Hybrid Dynamical Systems

    
Each time the solution crosses the negative axis, reflect it across the axis:
Wolfram Language code: de[δ_] := {Subscript[x, 1]'[t] == Subscript[x, 2][t], Subscript[x, 2]'[t] == -Subscript[x, 1][t] + 2δ Subscript[x, 2][t] + 1}; ic = {Subscript[x, 1][0] == .5, Subscript[x, 2][0] == .5}; sol = NDSolve[{de[0.1], ic, WhenEvent[And[Subscript[x, 2][t] == 0, Subscript[x, 1][t] < 0], {Subscript[x, 1][t] -> -Subscript[x, 1][t]}]}, {Subscript[x, 1], Subscript[x, 2]}, {t, 0, 500}];
Wolfram Language code: Row[Map[ParametricPlot[{Subscript[x, 1][t], Subscript[x, 2][t]} /. sol, {t, 0, #}, PlotPoints -> 500, PlotRange -> {{-1, 2.5}, {-2, 1.5}}]&, {10, 50, 150}]]
Compare the solutions for several different values of :
Wolfram Language code: resetOsc[δ_] := NDSolve[{de[δ], ic, WhenEvent[And[Subscript[x, 2][t] == 0, Subscript[x, 1][t] < 0], {Subscript[x, 1][t] -> -Subscript[x, 1][t]}]}, {Subscript[x, 1], Subscript[x, 2]}, {t, 0, 500}]; {sol1, sol2, sol3} = Map[resetOsc, {.1, .15, .2}];
Wolfram Language code: Grid[{Map[ParametricPlot[{Subscript[x, 1][t], Subscript[x, 2][t]} /. sol1, {t, 0, #}, PlotPoints -> 500, PlotRange -> {{-3, 3}, {-3, 3}}]&, {10, 50, 150}], Map[ParametricPlot[{Subscript[x, 1][t], Subscript[x, 2][t]} /. sol2, {t, 0, #}, PlotPoints -> 500, PlotRange -> {{-3, 3}, {-3, 3}}, PlotStyle -> Green]&, {10, 50, 150}], Map[ParametricPlot[{Subscript[x, 1][t], Subscript[x, 2][t]} /. sol3, {t, 0, #}, PlotPoints -> 500, PlotRange -> {{-3, 3}, {-3, 3}}, PlotStyle -> Red]&, {10, 50, 150}]}]