Approximate solution of a nonlinear ODE in the form of a Fourier series containing the coefficients of the initial ODE

Question

In this topic we considering nonlinear ODE:

$\frac{dx}{dt}= (x^4) \cdot a_1 \cdot sin(\omega_1 \cdot t)-a_1 \cdot sin(\omega_1 \cdot t + \frac{\pi}{2})$ - Chini ODE

https://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor%2FChini

And system of nonlinears ODE:

$\frac{dx}{dt}= (x^4+y^4) \cdot a_1 \cdot sin(\omega_1 \cdot t)-a_1 \cdot sin(\omega_1 \cdot t + \frac{\pi}{2})$

$\frac{dy}{dt}= (x^4+y^4) \cdot a_2 \cdot sin(\omega_2 \cdot t)-a_2 \cdot sin(\omega_2 \cdot t + \frac{\pi}{2})$

Chini ODE's NDSolve in Mathematica:

pars = {a1 = 0.25, ω1 = 1}
sol1 = NDSolve[{x'[t] == (x[t]^4) a1 Sin[ω1 t] - a1 Cos[ω1 t], x[0] == 1}, {x}, {t, 0, 200}]
Plot[Evaluate[x[t] /. sol1], {t, 0, 200}, PlotRange -> Full]

System of Chini ODE's NDSolve in Mathematica:

pars = {a1 = 0.25, ω1 = 3, a2 = 0.2, ω2 = 4}
sol2 = NDSolve[{x'[t] == (x[t]^4 + y[t]^4) a1 Sin[ω1 t] - a1 Cos[ω1 t], y'[t] == (x[t]^4 + y[t]^4) a2 Sin[ω2 t] - a2 Cos[ω2 t], x[0] == 1, y[0] == -1}, {x, y}, {t, 0, 250}]
Plot[Evaluate[{x[t], y[t]} /. sol2], {t, 0, 250}, PlotRange -> Full]

There is no exact solution to these equations, therefore, the task is to obtain an approximate solution.

Using AsymptoticDSolveValue was ineffective, because the solution is not expanded anywhere except point 0.

The numerical solution contains a strong periodic component; moreover, it is necessary to evaluate the oscillation parameters. Earlier, we solved this problem with some users as numerically: Estimation of parameters of limit cycles for systems of high-order differential equations (n> = 3)

How to approximate the solution of the equation by the Fourier series so that it contains the parameters of the original differential equation in symbolic form, namely $a_1$, $\omega_1$, $a_2$ and $\omega_2$.

I would be grateful for any help!

Answer 1

This solution addresses the first nonlinear ODE. The second system, two coupled nonlinear ODEs, can be treated in a similar way. To begin, rescale t and a1 to eliminate ω1, simply as good practice. Here are solutions for four values of a1, the first three obtained using code of the form:

With[{a1 = 0.1}, NDSolveValue[{x'[t] == (x[t]^4) a1 Sin[t] - a1 Cos[t], 
    x[0] == x[2 Pi]}, x[t], {t, 0, 2 Pi}]];
p1 = Plot[%, {t, 0, 2 Pi}, AxesLabel -> {t, x}, 
    LabelStyle -> {Black, Bold, 15}, PlotLabel -> "a1 = 1/10"]

However, experimentation shows that the maximum value of a1 for which periodic solutions exist is just above a1 = 2.374. For a1 near that value, instead use:

With[{a1 = 2.374}, FixedPoint[NDSolveValue[{x'[t] == (x[t]^4) a1 Sin[t] - a1 Cos[t], 
    x[0] == #}, x[2 Pi], {t, 0, 2 Pi}, MaxStepSize -> 0.00001] &, -20., 
    SameTest -> (Abs[#1 - #2] < 10^-5 &)]]
With[{a1 = 2.374}, NDSolveValue[{x'[t] == (x[t]^4) a1 Sin[t] - a1 Cos[t], x[0] == %}, 
    x[t], {t, 0, 2 Pi}, MaxStepSize -> 0.00001]];
p4 = Plot[%, {t, 0, 2 Pi}, PlotRange -> All, AxesLabel -> {t, x}, 
    LabelStyle -> {Black, Bold, 15}, PlotLabel -> "a1 = 2.374"]

GraphicsGrid[{{p1, p2}, {p3, p4}}, ImageSize -> Large]

Evidently, for values up to about a1 = 1, the solution is approximately sinusoidal, but much less so for larger values.

To obtain symbolic expressions for a Fourier expansion of the solution of the ODE, as requested in the question, begin by assuming that x[t] == 0 in the right side of ODE, solve it to obtain a first approximation to the solution, insert it into the right side of the ODE, solve again, and iterate to obtain power series in a1:

Clear[a1];
Nest[Collect[DSolveValue[{x'[t] == a TrigReduce[Sin[t] #^4] - a Cos[t]}, x[t], t] 
    /. C[1] -> 0, {_Sin, _Cos}, Simplify] &, 0, 4];
sa = Collect[Expand[%] /. (a^n_ /; n > 13) :> 0, {_Sin, _Cos}, Simplify]
(* (-((5 a^5)/8) + (6067 a^13)/7680) Cos[t] + ((5 a^5)/48 - 
   (1070681 a^13)/3225600) Cos[3 t] + (-(a^5/80) + (2118019 a^13)/16128000) Cos[5 t]
   - (62353 a^13 Cos[7 t])/2257920 + (5917 a^13 Cos[9 t])/1612800 - 
   (9277 a^13 Cos[11 t])/35481600 + (67 a^13 Cos[13 t])/5990400 + 
   (-a + (63 a^9)/160) Sin[t] - 17/80 a^9 Sin[3 t] + (67 a^9 Sin[5 t])/1200 - 
   (37 a^9 Sin[7 t])/6720 + (a^9 Sin[9 t])/2880 *)

Applying this expression to a1 = .75 yields a curve indistinguishable to the eye from that obtained numerically. Of course, for larger values of a1 (but still less than 1) more Fourier terms and larger powers of a1 are needed.

Fourier series can be obtained for a1 > 1 but not as symbolic functions of a1, by using the approach illustrated in my answer to 145760. For instance,

a1 = 2; m = 11;
f[t_] := Sum[c[n] Cos[n t] + s[n] Sin[n t], {n, 1, m, 2}]
TrigReduce[f'[t] - (f[t]^4) a1 Sin[t] + a1 Cos[t]];
eq = Join[Thread[Simplify[Coefficient[%, Cos[# t]] & /@ Range[1, m, 2]] == 0], 
    Thread[Coefficient[%, Sin[# t] & /@ Range[1, m, 2]] == 0]];
sf = f[t] /. FindRoot[eq, Join[Array[{s[#], .1} &, (m + 1)/2, {1, m}], 
    Array[{c[#], .1} &, (m + 1)/2, {1, m}]]] 
(* -1.09709 Cos[t] - 0.293228 Cos[3 t] - 0.0992651 Cos[5 t] - 
   0.0448225 Cos[7 t] - 0.0219299 Cos[9 t] - 0.0115066 Cos[11 t] - 
   0.762219 Sin[t] - 0.0697846 Sin[3 t] - 0.0430293 Sin[5 t] - 
   0.0328824 Sin[7 t] - 0.0205152 Sin[9 t] - 0.0134693 Sin[11 t] *)

Plotting this expression yields a curve indistinguishable from that computed numerically above. In principle, it might be possible to solve the equations just above without inserting a numerical value for a1, but doing so would require symbolically solving twelve coupled polynomial equations, each of 12th order. In practice, doing so would be very time-consuming.