I would like to solve the following differential equation system,

$$ \frac{dx}{dt} = C_{1}\cdot x(t) + C_{2}\cdot x(t)^{2} +C_{3}\cdot x(t)\cdot y(t)\\ \frac{dy}{dt} = C_{4}\cdot y(t) + C_{5}\cdot y(t)^{2} +C_{6}\cdot x(t)\cdot y(t)$$

where $$C_{1},C_{2},C_{3},C_{4},C_{5},C_{6} $$ are constants and the initial conditions are given, $$ x(t=0) = xt_{0}\qquad y(t=0) = yt_{0}$$ and the values of the variables in equilibrium when $$ \frac{dx}{dt} = 0 \qquad \frac{dy}{dt} = 0 $$ are also given: $$ x_{0} = 0 \qquad y_{0} = 0$$

Could someone give me any advise or method to help me find the analytic solution of x(t), y(t).

Thanks in advance.

  • 1
    $\begingroup$ That is a quite interesting problem. I'm afraid that there is no analytical solution but for some special cases (perhaps for $C_2=C_3=C_5=C_6=0$). Since the RHS of your equations is not Lipschitz continuous for all $t \in R$, the Picard-Lindelöf theorem can not ensure uniqueness (nor existence) for solutions for infinite time. Considering non uniqueness, but existence, maybe it means that there is some stationary solution for infinite $t$ in the form of an attractor to which more then one solution converges. $\endgroup$ – rafa11111 Apr 17 '18 at 13:55
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    $\begingroup$ I'm wondering if there's a way to represent this in a vectorized format, with some matrix on the RHS that you could diagonalize. That is, the RHS looks like you might be able to do a change of variable and uncouple it by finding principle components or something like that. $\endgroup$ – Adrian Keister Apr 17 '18 at 13:59
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    $\begingroup$ @rafa11111 The RHS is locally Lipschitz in the variables $x,y$. Existence and uniqueness of a local solution is guaranteed. $\endgroup$ – Julián Aguirre Apr 17 '18 at 14:06
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    $\begingroup$ $(0,0)$ is an equilibrium point. Its character (sink, source, saddle,...) will depend on the constants $C_i$, and can be found by linearization. Also, depending on the values of the constants, there may be more equilibria. $\endgroup$ – Julián Aguirre Apr 17 '18 at 14:22
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    $\begingroup$ Such systems are called generalized Lotka-Volterra equations, and there is vast literature on them. For starters, see the monograph by Hofbauer and Sigmund. $\endgroup$ – user539887 Apr 17 '18 at 20:38

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