E. Chan-López
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Try with the following code: H[px_, pz_, x_, z_] := 1/2 (px^2 + pz^2) + z + 2 (3/4 - Sqrt[x^2 + (z - 1)^2])^2 - 1/8 xpp = D[x[t], t, t] - x[t] (-4 + 3/Sqrt[x[t]^2 + (-1 + z[t])^2]); zpp = D[z[t], t, t]...

Another approach to this problem: (*Code for simple paths*) LineIntegrate[vectorfield_List, varsfield_List, path_List, varpath_, I_List] := Module[{vars, r, s1, s2, g1, g2}, vars := Table[...

Hopf bifurcation analysis The differential system: f1[x_,y_]:=a x (1 - x/k) - b x y; f2[x_,y_]:=-c y + d x y; F[{x_,y_},{a_,b_,c_,d_,k_}]:=Evaluate@{f1[x,y],f2[x,y]}; X={x, y}; μ={a,b,c,d,k}; $$\... View answer Accepted answer 9 votes Try this: PolQuotient[num_, den_, var_] := PolynomialQuotient[num, den, var] + 1/den*PolynomialRemainder[num, den, var] Test: PolQuotient[a - b, a + b, a] (*1 - (2 b)/(a + b)*) Improving the above ... View answer 8 votes Play with the following example:$$ \begin{eqnarray} x(n+1)&=&rx(n)(1-y(n))\\ y(n)&=&x(n), \end{eqnarray} $$with r=\displaystyle\frac{2005}{1000} and three orbits. The phase ... View answer Accepted answer 7 votes Try this: f[x_] = Integrate[Integrate[x^7 (x - 2)^4 (x - 3)^9, x], x] Plot: Plot[f[x], {x, 2, 4.1}, PlotRange -> All, AspectRatio -> 1/GoldenRatio] View answer 7 votes Your approach didn't work because the last condition cannot be solved easily, that is, you need at least one parameter of degree one to solve without complexity. Then, Chris's routine does work ... View answer 6 votes Something like the following: Map[Sort[#] &, list] (*{{{a, 1}, {b, 3}, {c, 5}}, {{a, 5}, {b, 1}, {c, 3}}, {{a, 5}, {b, 3}, {c, 1}}, {{a, 1}, {b, 5}, {c, 3}}}*) A first approximation: MyOrderList[... View answer 5 votes Try this: cont1 = ContourPlot[(Ωx /. {μ -> 1/2, c -> 1, p -> 1/100}) == 0, {x, -2, 2}, {y, -2, 2}, PlotRange -> Automatic, ContourStyle -> {Thickness[0.002], Darker[Red]}, LabelStyle -&... View answer 4 votes With r=3.25, b=2.36, k=0.14, ϵ=1000, and a=0.01 your "limit cycle" appears. r = 3.25; b = 2.36; k = 0.14; ϵ=1000; a = 0.01; sol[t_] = NDSolve[{x'[t] == -2 x[t] + r (k + 1) y[t] + ... View answer Accepted answer 4 votes Version (*12.3.1 for Microsoft Windows (64-bit) (June 19, 2021)*) The following change works for me: Graphics[Circle[], Axes -> True, GridLines -> Automatic, GridLinesStyle -> Directive[... View answer Accepted answer 4 votes With the following code, you can plot the Poincaré sections of the Hénon-Heiles system: With[{icv = {0, 0.36169437164930385, 0.20100851639176504, 0.029106357137938632}}, psection = Reap[NDSolve[{x'... View answer Accepted answer 4 votes The Discrete System:$$ \begin{align} x_{n+1}&=c \left(1-(x_n+y_n)\right)+x_n \left(1-p y_n\right)\\ y_{n+1}&=\left(p x_n+b\right)y_n \end{align}  The Discrete System code: F[{x_, y_}, {b_, ...

Try this: Refine[ToRadicals@FullSimplify[PowerExpand[1/Sqrt[Product[1 + (2 s)/k^3, {k, 1, \[Infinity]}]]], Assumptions -> Element[s, Reals]]]// TraditionalForm

Using ComplexExpand command: A = 5 t^2 + 3 t + 4; ComplexExpand[Conjugate[A]] For expressions within radicals, we must use the Refine command for the ComplexExpand command to work properly. Here is ...

Try this: expr = (-4*dt^2*(c3 + a3*t)^2*t0^4 + dxC^2*(t0^2 + t1^2)^2 + dxM^2*(t0^2 + t1^2)^2 + dxY^2*(t0^2 + t1^2)^2)/(4*t0^4)//FullSimplify Collect[expr, {dxC, dxM, dxY}, FullSimplify] (*-dt^2 (c3 ...

Based on the suggestion of Henrik Schumacher, and the answers of Hausdorff and Daniel Huber, you can do it as follows: Format[braket[{{a_, b_, c_}, {d_, e_, f_}}]] := Row[{"\[LeftAngleBracket]&...

This is what Chris means: a = b = c = d = 1; x0 = 2; y0 = 3; q = 0.2; egyensulypont1 = Graphics[{Blue, Thick, PointSize[0.02], Point[{c/d, a/b}]}]; egyensulypont2 = Graphics[{Red, Thick, ...

Try DeleteCases: list={-1628.12 - 557.34 I, -1628.12 + 557.34 I, -920.324 - 1508.13 I,-920.324 + 1508.13 I, 2107.28, 2145.53, 2207.2, 2289.7,2318.05 - 13.4031 I, 2318.05 + 13.4031 I} DeleteCases[list, ...

First, select all the input cells on the right side, like this: Then, apply Ctrl+8 and save changes.

Try this: IIges[x_, y_] := Sin[x]^2 + Sin[y]^2; xm = 4; {dx[x_, y_], dy[x_, y_]} = D[IIges[x, y], {{x, y}}]; ContourPlot[{dx[x, y], dy[x, y]} == 0, {x, -xm, xm}, {y, -xm, xm}, ContourStyle -> ...

Some modifications for the code in Vitaliy Kaurov's answer: <<MaTeX << c:\CurvesGraphics6\CurvesGraphics6.m Initial condition outside the limit cycle: α = 200/207; a = 4; k = 3; β = ...

The dynamics at the non-trivial equilibrium point: Needs["MaTeX`"] << c:\CurvesGraphics6\CurvesGraphics6.m Non-trivial equilibrium is unstable Non-trivial intersection of zero ...

The vector field defined by your system: f1 = σ (Y - X); f2 = 3 X (ρ - Z) - Y; f3 = X Y - β Z; F = {f1, f2, f3}; V = {X, Y, Z}; The equilibria: P1 = Solve[F == 0, V][[1]] P2 = Solve[F == 0, V][[2]] ...

Try this: T[x_, y_, n_?IntegerQ] /; n >= 0 := If[n === 0, x, Expand[x^2*T[y^2*x, 2 y + 4, n - 1]]] Test: T[x, y, 0] (*x*) T[x, y, 1] (*x^3 y^2*) T[x, y, 2] (*16 x^5 y^6 + 16 x^5 y^7 + 4 x^5 y^8*) ...

Check in the following link: GeometricAlgebra Package.

Try this: P = {};(*empty list*) a = -Pi/2; b = 3 Pi/4; tolorance = 0.0000; f[x_] := Sin[x] && a <= x <= b; For[i = 1, i <= 11, i++, {c = (a + b)/2, If[f[a]*f[c] < tolorance, b = ...

Try Cases: list={94, 35, 47, 29, 86, 40, 46, 72, 87}; twolists=List[Cases[list, Except[_?EvenQ]], Cases[list, Except[_?OddQ]]] (*{{35, 47, 29, 87}, {94, 86, 40, 46, 72}}*) Map[Median, twolists] (*{41,...