# NDSolve Problem with step size and Stiffnes

I need to solve a 5 coupled non-linear differential equation system. There seems to be no apparent singularity but whenever I tried to evaluate the system in Mathematica. It gives the error

NDSolve::ndsz: At t == 0.3396721805460672`, step size is effectively zero; \
singularity or stiff system suspected

Here is the code I am trying to run

A2 := Flatten[{x, y, z, w, j} /. NDSolve[
{Derivative[2][w][t] == -6.73*j[t] - 35.3*w[t] + 20.1*j[t]^2*w[t] + 30.28*w[t]^3 - 34.15*w[t]*x[t] + 89.72*w[t]*x[t]^2 + 20.18*w[t]*y[t] - 71.16*w[t]*x[t]*y[t] + 5.99*w[t]*y[t]^2 + 3.21*w[t]*z[t] -14.51*w[t]*x[t]*z[t] + 4.63*w[t]*y[t]*z[t] + 0.54*w[t]*z[t]^2,

Derivative[2][j][t] == -2.26*j[t] + 2.19*j[t]^3 - 1.34*w[t] + 4.03*j[t]*w[t]^2 - 1.87*j[t]*x[t] + 3.05*j[t]*x[t]^2 - 0.81*j[t]*y[t] +
4.74*j[t]*x[t]*y[t] + 1.38*j[t]*y[t]^2 - 0.65*j[t]*z[t] + 1.76*j[t]*x[t]*z[t] + 1.52*j[t]*y[t]*z[t] + 0.24*j[t]*z[t]^2,

Derivative[2][x][t] == 114.69 - 0.247*j[t]^2 - 0.903*w[t]^2 + 1.863*x[t] + 0.808*j[t]^2*x[t] + 4.74*w[t]^2*x[t] -2.23*x[t]^2 + 3.41*x[t]^3 - 0.66*y[t] + 0.62*j[t]^2*y[t] - 1.88*w[t]^2*y[t] - 0.24*x[t]*y[t] - 1.04*x[t]^2*y[t] - 0.39*y[t]^2 + 0.55*x[t]*y[t]^2 + 0.22*y[t]^3 + 0.35*z[t] + 0.23*j[t]^2*z[t] - 0.38*w[t]^2*z[t] + 0.89*x[t]*z[t] + 0.13*x[t]^2*z[t] - 0.22*y[t]*z[t] - 0.23*x[t]*y[t]*z[t] + 0.07*y[t]^2*z[t] + 0.12*z[t]^2 + 0.33*x[t]*z[t]^2 -
0.087*y[t]*z[t]^2 + 0.047*z[t]^3,

Derivative[2][y][t] == 5.26 - 0.152*j[t]^2 + 0.76*w[t]^2 - 0.954*x[t] + 0.89*j[t]^2*x[t] -2.68*w[t]^2*x[t] - 0.17*x[t]^2 - 0.49*x[t]^3 + 1.83*y[t] + 0.52*j[t]^2*y[t] + 0.45*w[t]^2*y[t] - 1.13*x[t]*y[t] +0.78*x[t]^2*y[t] - 1.2*y[t]^2 + 0.96*x[t]*y[t]^2 + 0.68*y[t]^3 + 0.37*z[t] + 0.28*j[t]^2*z[t] + 0.17*w[t]^2*z[t] -0.32*x[t]*z[t] - 0.16*x[t]^2*z[t] - 0.031*y[t]*z[t] + 0.21*x[t]*y[t]*z[t] + 0.065*y[t]^2*z[t] - 0.054*z[t]^2 - 0.124*x[t]*z[t]^2 + 0.037*y[t]*z[t]^2 - 0.018*z[t]^3,

Derivative[2][z][t] == -13.58 - 0.92*j[t]^2 + 0.909*w[t]^2 +
3.76*x[t] + 2.49*j[t]^2*x[t] - 4.1*w[t]^2*x[t] + 4.76*x[t]^2 + 0.48*x[t]^3 + 2.7*y[t] + 2.1*j[t]^2*y[t] + 1.3*w[t]^2*y[t] -
2.43*x[t]*y[t] - 1.24*x[t]^2*y[t] - 0.11*y[t]^2 + 0.809*x[t]*y[t]^2 + 0.163*y[t]^3 + 1.986*z[t] + 0.684*j[t]^2*z[t] +
0.31*w[t]^2*z[t] + 2.67*x[t]*z[t] + 3.63*x[t]^2*z[t] - 0.81*y[t]*z[t] - 1.86*x[t]*y[t]*z[t] + 0.28*y[t]^2*z[t] +
0.68*z[t]^2 + 1.52*x[t]*z[t]^2 - 0.42*y[t]*z[t]^2 + 0.24*z[t]^3,
x[0] == 0,
y[0] == 0,
z[0] == 0,
w[0] == 0,
j[0] == 0,
Derivative[1][x][0] == 0.5,
Derivative[1][y][0] == 0.6,
Derivative[1][z][0] == 0.7,
Derivative[1][w][0] == 0.8,
Derivative[1][j][0] == 0.9}, {x, y, z, w, j}, {t, 0, 10},
Method -> {"DoubleStep", Method -> {"StiffnessSwitching", Method -> {"ExplicitRungeKutta", Automatic}}}]]

I have tried several different methods like DoubleStep, StiffnessTest and reducing the Stepsize but none of these helped.

• Looks like a singularity, which would be a feature of your system and not an error. Do you have any evidence that the system should not have a singularity? – Michael E2 Dec 18 '17 at 16:32
• My reasoning for not expecting singularities comes from physics. Solutions of these equations will represent a kink solution for a classical field theory which means that at least w and j should go to some fixed value while t goes to +- infinity. Could that singularity be taken care of. – halfmetal Dec 18 '17 at 16:59
• It's not obvious to me from the pde's why the solution should blow up, but if you plot x,y,z,w,j for your solution from t = 0 to your blow up time at about t = 0.33, you can see that all their slopes at the end become nearly vertical. – Bill Watts Dec 23 '17 at 1:53