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

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, (Omega)1 = 1}
sol1 = NDSolve({x'(t) == (x(t)^4) a1 Sin((Omega)1 t) - a1 Cos((Omega)1 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, (Omega)1 = 3, a2 = 0.2, (Omega)2 = 4}
sol2 = NDSolve({x'(t) == (x(t)^4 + y(t)^4) a1 Sin((Omega)1 t) - a1 Cos((Omega)1 t), y'(t) == (x(t)^4 + y(t)^4) a2 Sin((Omega)2 t) - a2 Cos((Omega)2 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 $omega_2$.*