https://books.google.com/books?id=shYNuW0B0fsC&pg=PA149&lpg=PA149&dq=find+constants+of+the+motion+for+a+chaotic+flow&source=bl&ots=tOxJknD-do&sig=ACfU3U1omccUJsfTcWGcNA3VpIYc8BL6Ng&hl=en&sa=X&ved=2ahUKEwjs-v2XuKzpAhUI8BQKHR1-DucQ6AEwA3oECAcQAQ#v=onepage&q=find%20constants%20of%20the%20motion%20for%20a%20chaotic%20flow&f=false

Dear Julien Clinton Sprott

I think the following article

Invariants and numerical methods for ODEs

https://doi.org/10.1016/0167-2789(92)90246-J

may include your request to find a numerical approach to detect invariants of (Chaotic) motion.

Best regards

Thanks to you both for those interesting references. the paper by Varvoglis is concerned with Hamiltonian systems and shows properties satisfied by constants of the motion, but not an algorithm for finding them. Likewise, the paper by Gear deals with numerical integration methods that preserve the invariants once they are known, but it does not offer methods for finding them.

Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law, but I don't know how to apply that to a dissipative (non-Hamiltonian) system such as a generalized Lorenz system.

Probably this is a good research problem for an especially gifted student.

Let me give an example that I constructed to illustrate what I am seeking and offer it as a challenge. The following 4-D system has a simple constant of the motion that reduces the dynamics to 3 dimensions. Who can find it?

x' = 4(y - x)

y' = 16x - y - u

z' = xy - z

u' = (x^2 + 4z)y - 5xz

Use initial conditions (1, 2, 6, 6). You will notice that it has two zero Lyapunov exponents, a line of equilibria, and initial conditions that act as bifurcation parameters. The attractor should look familiar to anyone who plots it. Hint: the constant is zero for the given initial conditions.

Dear Julien Clinton Sprott

I think you meant by the simple constant is the curve

X(t)= (t, t, t2 , 15t)

Also, we have the equilibrium state at the origin.

Best regards

Issam -- The constant of motion involves only the state variables f(x, y, z, u) = constant, not the time t. Yes, the origin is an equilibrium, but there are infinitely many others.

Dear Julien Clinton Sprott

I agree, observe that f(X(t)) = f( (t, t, t2, 15t) = 0 for every t in R.

x(t)= t, y(t)=t, z(t)=t2 , u(t)=15t.

It is an invariant curve of your dynamical system.

Best regards

Those solutions do not satisfy the equations:

dx/dt = 4(y - x) = 4(t - t) = 0 but dx/dt = 1. You can solve them parametrically to find the line of equilibria, but not the constant of motion I was challenging you to find. The solution is bounded and chaotic.

I am talking about the invariants in general without any restrictions or initial conditions. Initial conditions are applied to the general solution to find the exact solution, if any.

You said that < dx/dt = 4(y - x) = 4(t - t) = 0 but dx/dt = 1 >

It is not clear! Initial conditions doesn't mean that

dx/dt=1, dy/dt=2, dz/dt=6 and du/dt =6 !

Yes, equilibrium points are trivial invariants of a system, but those are not the ones that are of interest to me since in this case they are unstable. The challenge remains for anyone who can find the hypersurface f(x, y, z, u) = constant on which the chaotic attractor for the system lies.

Dear Professor,

C = u-xz

Or with no difference, C = xz-u

However, obviously, I don't know the answer of the main question :(

I declare Sajad the winner of the competition. The constant C = xz - u is determined by the initial conditions: C = xo*zo - uo which happens to be zero for the given conditions. Now tell us how you found the answer so that someone might generalize your method to more complicated examples.

That is a great honor, Shifu

Indeed, I looked at the patterns you described in "Comment on “How to obtain extreme multistability in coupled dynamical systems”

Trained by them, I just guessed the answer! So, there is no method :(

Sorry for the disappointment I caused :)

Dear's Julien Clinton Sprott ,Sajad Jafari

Congratulations.

Your collective works show a good team.

Indeed the invariant curve

x(t)=t, y(t)=t, z(t)= t2 , and u(t)=15t is unstable and out of you interest.

Can you show me how the invariant manifold xz - u = c is stable?

Regards

Good question about the stability of the invariant manifold. I suspect it is neutrally stable as evidenced by the pair of zero Lyapunov exponents, but I don't know how to prove that. Certainly, the numerical evidence is that orbits that start on it remain on it for a very long time. Other initial conditions give a foliation of such manifolds.

Now that you know what I'm looking for, I'll pose a slightly more difficult challenge. Who will be the first to find the invariant manifold for the following 4-D chaotic system:

x' = -u - z

y' = x - 2xu

z' = (y - z)/2

u' = x

With initial conditions (0.1, 0.21, 0, 0.3), it has two zero Lyapunov exponents, a line of equilibria, and initial conditions that act like bifurcation parameters. Tell us how you solved it.

Dear Julien Clinton Sprott

Observe that y + u2 - u = c reduces the given system into a 3D system,

and by forwarding substitutions, we can transform the new system into a linear third-order ordinary differential equation in u with constant coefficients in the LHS, and a quadratic polynomial in the RHS. In conclusion, you can determine the invariant manifold.

Best regards.

I declare Issam the winner of the second challenge. Tell us how you found the solution. The 3-D system is the Rossler (1979) prototype-4 system.

Dear Julien Clinton Sprott

I am working on generalizing the Bendixon- Dulac Criterion into higher-order systems. The work is in progress.

Best wishes

So we must eagerly await the publication of Issam's method. Meanwhile, I'll offer a third challenge, for which I don't know the answer, with properties similar to the previous examples. Find the invariant manifold for the following 4-D system:

x' = -x - z + uz

y' = -z - uz + y/5

z' = y + x

u' = z

which has a chaotic attractor for initial conditions (0, 2, -4, -0.4).

Could you please check the second equation.

The equations are copied and pasted directly from my computer program that gives a chaotic attractor with two zero Lyapunov exponents.

I mean, in the second equation,

Is y/5 coefficient for y only or all terms?

I mean y' = -z - uz + 0.2y. Otherwise I would use parentheses.

Dear Professor Sprott,

x = 3u^2+4u+5y-z

Great job! I declare Fahimeh the winner of the third challenge. Now tell us your method for performing this amazing feat.

Dear Fahimeh Nazarimehr ,

You can add an arbitrary constant.

Well Done.

You guys are so good at this, I'll give you a 4th challenge with the same properties as the previous three. Who will be the first to find the invariant manifold for the following 5-D system:

x' = y

y' = z

z' = u

u' = x^2 - 2u + v + xy

v' = xu - yz

It has a chaotic attractor for initial conditions (-3, -5, -20, 0, 7). Extra points for reducing it to the simplest 4-D system (the one with the fewest terms and the fewest nonlinearities) that satisfy the given initial conditions.

Dear Julien Clinton Sprott

v - xz + y2 = c

Best regards

Congratulations to Issam for winning the fourth challenge. From the initial conditions, the constant is c = -28. Now, who can find the simplest equivalent 4-D system that satisfies that constraint?

Remove the last equation and substitute v = xz - y2 - 28

in the fourth equation.

Thank you for the beautiful puzzles. I am done with this thread. Best regards

Good job Issam.

So these examples show that there are N-dimensional dynamical systems that are actually (N-1)-dimensional with an extraneous variable nonlinearly related to the others, in which case the N initial conditions become bifurcation parameters for the (N-1)-dimensional system. This eliminates the extra zero Lyapunov exponent, gives a sensible Kaplan-Yorke dimension, and removes the line of equilibria. It should be clear that one can take any dynamical system and add such an extraneous variable and obtain a system of this type.

When initial conditions behave as bifurcation parameters, the system is said to exhibit "extreme multistability," and many such examples have recently been published. It is my contention that most if not all these systems are precisely of this sort.

That brings me back to the original question which can be rephrased as follows: When presented with an N-dimensional dynamical system with extreme multistability, is there a systematic way to show that it is actually (N-1)-dimensional with an extraneous variable other than just playing around with the equations and trying to find a constant of the motion?