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I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself.

I know the basics of theory for 2D linear, time-invariant systems, i.e., $$ \dot{x}=a_1x+a_2y \\ \dot{y}=b_1x+b_2y $$ I know that there are explicit exponential solutions, and a ton of theory about fixed points and stability.

I'm wondering if there is any equivalent theory out there for the next higher-degree systems, i.e., $$ \dot{x}=a_1x+a_2y+a_3x^2+a_4y^2+a_5xy \\ \dot{y}=b_1x+b_2y+b_3x^2+b_4y^2+b_5xy $$

I get that this is non-linear, so not nearly as simple. But are there any general solutions or time-scales/eigenvalue-equivalents? Is there a standard approach to looking at this as pieced-together linear approximations? Is there a name for this kind of system or its study that I can look up?

Thanks very much!

dang
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    One thing to try is to first tackle the case of "homogeneous polynomials", i.e. the case without the $a_1x+a_2y$ terms on the right hand side of $\dot x$ and similarly for $\dot y$. But, frankly, I'm not at all sure how to do that case. – Lee Mosher Feb 29 '24 at 14:17
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    If enough parameters are zero so that the equations become uncoupled, you get Bernoulli equations, or more generally Riccati equations. – Lutz Lehmann Feb 29 '24 at 14:56

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A useful search phrase is “quadratic planar vector fields”.

Despite much research, there are still many open problems concerning such systems. For example, Hilbert's sixteenth problem asks for an upper bound for the number of limit cycles of systems of the form $\dot x = p(x,y)$, $\dot y = q(x,y)$, where $p$ and $q$ are polynomials of degree at most $n$. It is known that any given such planar polynomial system can have at most finitely many limit cycles; this was first claimed by Dulac, but it turned out much later that his proof was flawed, and a correct proof was found around 1990. But it's still unknown whether that finite number can be arbitrarily large or not (when you consider all possible systems of that form), even in the simplest nontrivial case $n=2$ (quadratic polynomials, as in your question). See, for example, the Scholarpedia article Limit cycles of planar polynomial vector fields.

There are of course special cases where you can say more, like the famous Lotka–Volterra predator–prey model where you have a constant of motion which causes all nontrivial trajectories to be periodic, but there's no hope of finding a general solution formula which covers all quadratic systems.

Hans Lundmark
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