Can a chaotic trajectory solve an algebraic equation?

Question

Given a polynomial ODE in $n$-dimensions of maximal degree $d$ $$\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n$$ we define an algebraic solution as an any curve $\gamma: [t_0,t_1] \rightarrow \mathbb{R}^n$ solving the ODE and fulfilling $$G(\gamma(t))=\sum_{i_{1},\dots,i_{n}=1}^{d'}b_{i_{1},\dots,i_{n}}{\gamma_1(t)}^{i_{1}}\dots {\gamma_n(t)}^{i_{n}}=0 \quad \forall t \in [t_0,t_1] $$ for some real coefficients $b_{i_1,...,i_n}$.

Now we assume the solution $\gamma$ is a solution on a chaotic attractor. This especially means that there is some closed subset (the attractor) $A \subset \mathbb{R}^n$ such that $\Gamma_{x}$ is dense in $A$.

My question is the following: Can $\gamma$ still solve an algebraic equation even though it is a chaotic trajectory?

Answer 1

If I understand your question correctly, then I believe the answer is yes, and an abundance of examples are given by Hamiltonian dynamics. Somewhat by definition, the energy of solutions to Hamiltonian systems is conserved; $H(\gamma(t)) = H(\gamma(0))$ for all $t\in \mathbb{R}$. With this we may define the function $G(\vec{x}) = H(\vec{x}) - H(\gamma(0))$. There also may be additional conserved integrals of motion. But for your question, we just need to look for a polynomial Hamiltonian with chaotic solutions.

One such example is given by the Henon-Heiles system. This is a 2-degree of freedom Hamiltonian ODE with position variables $(x,y)$ and momentum variables $(p_x,p_y)$, and having Hamiltonian

$ H = \frac{1}{2} (p_x^2 + p_y^2) + \frac{1}{2} (x^2 + y^2 ) + \lambda (x^2 y - y^3/3)$

for parameter $\lambda$. As a first-order ODE this is given by:

$ \dot{x} = p_x$

$ \dot{p}_x = -x - 2 \lambda xy $

$ \dot{y} = p_y $

$ \dot{y}_y = -y -\lambda (x^2 -y^2)$

While not every solution is chaotic, the Henon-Heiles system is a canonical example of a chaotic Hamiltonian system, and any solution to the system will preserve the Hamiltonian.