# Found a cool Non-autonomous Coupled System of ODEs

• TylerH
In summary, the conversation discusses a system with chaotic behavior and spiral-like shapes for different initial values. The vector fields are locally linear, indicating that there may be no serious numerical errors. The system is not solvable and can only be explored through numerical methods. A video is shared to demonstrate the behavior of the system.
TylerH
By a fortuitous mistake in copy/pasting, I happened across this system. It exhibits very chaotic behavior for some initial values and spiral-like shapes for others. (About a 70%/30% split, respectively.)

$x'=cosy+sint$
$y'=sinx+cost$

Here's an album of it plotted from t=0 to 1000. The titles of the plot are x0_y0.
http://imgur.com/a/lbhrX
A lot of those have serious errors in their long term trends, but the overall type of pattern they take is, I believe, realistic. I believe so, because, as you can see in the video below, their vector fields are locally linear. From that, I intuitively draw the conclusion that there could be no serious numerical error, like jumping a phase line, but only small compounding error that only affects long term trend. That is to say, the only kind of numerical error present is the kind inherent to numerical methods, not anything specific to this ODE. (Correct me if my intuitive reasoning is incorrect.)

And, here's a video showing (x0_y0)=(1,1) with surrounding vector field from t=0 to 250:
https://vimeo.com/88323596

If anyone would like, I can post a video that is zoomed closer for the entire video, so as to see the local linearity better in the faster moving parts. I have only uploaded the one below so far because, IMO, it is more fun to watch that the one above.

Last edited:
Also, I've reduced it to a system of very complex 2nd order noncoupled ODEs, but that doesn't seem to help in solving it. Mathematica can't solve it either. I'm almost convinced it isn't solvable. So it seems the only way to explore it is with numerical methods.

## 1. What is a non-autonomous coupled system of ODEs?

A non-autonomous coupled system of ODEs is a set of differential equations that describe the behavior of two or more variables over time, where the equations are dependent on external factors or variables.

## 2. How is a non-autonomous coupled system of ODEs different from an autonomous system?

In an autonomous system, the equations are only dependent on the variables within the system itself, while in a non-autonomous system, the equations are also influenced by external factors or variables.

## 3. What makes non-autonomous coupled systems of ODEs interesting to study?

Non-autonomous coupled systems of ODEs can exhibit complex behaviors and interactions between variables, making them a rich area of study for scientists. They also have many real-world applications, such as in physics, biology, and engineering.

## 4. How are non-autonomous coupled systems of ODEs solved?

Non-autonomous coupled systems of ODEs are typically solved using numerical methods, such as Euler's method or the Runge-Kutta method. These methods use iterative calculations to approximate the solution to the system of equations.

## 5. Are there any limitations or challenges when working with non-autonomous coupled systems of ODEs?

One limitation is that the solution to a non-autonomous system may not always be unique, as it can be affected by the initial conditions and external factors. Additionally, the complexity of the system can make it challenging to find an analytical solution, and numerical methods may not always provide an accurate or complete solution.

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