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4. Nonlinear Systems of Three or More Ordinary Differential Equations

4.1. Systems of First-Order Ordinary Differential Equations;x = x(t), y = y(t),z = z(t)

  1. _ax_′ = (b − c)yz, _by_′ = (c − a)zx, _cz_′ = (a − b)xy.
  2. _ax_′ = (b − c)yzf(x, y, z, t), _by_′ = (c − a)zxf(x, y, z, t), _cz_′ = (a − b)xyf(x, y, z, t).
  3. _x_′ = a(y − x), _y_′ = bx − y − xz, z_′ = −_cz + xy.Lorenz equations.
  4. _x_′ = _cF_2 − _bF_3, _y_′ = _aF_3 − _cF_1, _z_′ = _bF_1 − _aF_2, where F n = F n(x, y, z, t).
  5. _x_′ = _czF_2 − _byF_3, _y_′ = _axF_3 − _czF_1, _z_′ = _byF_1 − _axF_2, where F n = F n(x, y, z, t).
  6. _x_′ = x(_cF_2 − _bF_3), _y_′ = y(_aF_3 − _cF_1), _z_′ = z(_bF_1 − _aF_2), where F n = F n(x, y, z, t).
  7. _x_′ = h(z)_F_2 − g(y)_F_3, _y_′ = f(x)_F_3 − h(z)_F_1, _z_′ = g(y)_F_1 − f(x)_F_2where F n = F n(x, y, z, t).

7a. Systems of nonlinear ODEs with homogeneous right-hand sides.

4.2. Systems of Second-Order Ordinary Differential Equations; x = x(t), y = y(t), z = z(t)

  1. _x_″ = F x, _y_″ = F y, _z_″ = F z, where F = F(r), r = (_x_2 + _y_2 + _z_2)1/2.
  2. _x_″ = xF, _y_″ = yF, _z_″ = zF, where F = F(x, y, z, t, _x_′, _y_′, _z_′).
  3. _x_″ = _F_1, _y_″ = _F_2, _z_″ = _F_3, where Fn = Fn(t, _tx_′ − x,_ty_′ − y, _tz_′ − z).
  4. _x_″ = _cF_2 − _bF_3, _y_″ = _aF_3 − _cF_1, _z_″ = _bF_1 − _aF_2, where F = F(x, y, z, t, _x_′, _y_′, _z_′).

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