MAT244-2013S > Easter and Semester End Challenge

Easter challenge

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Victor Ivrii:
Draw phase portraits :

\begin{gather}
\left\{\begin{aligned}
&x'=-\sin(y),\\
&y'=   \sin (x);
\end{aligned}\right. \tag{a}\\
\left\{\begin{aligned}
&x'=-\sin(y),\\
&y'= \,2\sin (x);
\end{aligned}\right. \tag{b}
\end{gather}

Explain the difference between portraits and its reason

Hareem Naveed:
Attached are the two phase portraits.

In terms of difference between the two; they are level curves of the following functions:
$$H_{a}(x,y) = \cos(y)+\cos(x) \\ H_{b}(x,y) = \cos(y) + 2\cos(x)$$
Level curves are also attached.

From the level curves, in a, the centres are hemmed in by 2 defined separatrices, not so in b where there is only one.

How could I formalize these statements? Intuitively, I can "see" the answer.

Victor Ivrii:

--- Quote from: Hareem Naveed on March 29, 2013, 11:38:46 AM ---Attached are the two phase portraits.

In terms of difference between the two; they are level curves of the following functions:
$$H_{a}(x,y) = \cos(y)+\cos(x) \\ H_{b}(x,y) = \cos(y) + 2\cos(x)$$
Level curves are also attached.

From the level curves, in a, the centres are hemmed in by 2 defined separatrices, not so in b where there is only one.

How could I formalize these statements? Intuitively, I can "see" the answer.

--- End quote ---

Your solution is definitely correct but I want a bit more observation about dynamics, not only about centers but about all trajectories. Start from the very informal description as I am interested more in an understanding than the formal expression

Alexander Jankowski:
This is a nice problem. I can give a little bit of input: in system (b), the rate of change $y'(x)$ is twice that of the same rate in system (a). Perhaps we can treat (b) as a vertical expansion of (a), which is what the contour maps suggest. We can also note that each system has the same critical points $(2 \pi n,2 \pi n)$, where $n$ is an integer.

By inspection, we see that the separatrices in system (a) are lines with slopes $\pi$ and $-\pi$. In fact, by looking more closely at the contour maps, the separatrices appear to have the equations $$y = Â± \pi x + n \pi,$$ where $n$ is an integer. In system (b), the separatrices are sinusoidal functions that oscillate about an equilibrium line that is parallel to the $y$-axis. Finally, because there are infinitely many critical points in each system, there are also infinitely many separatrices.

Victor Ivrii:

--- Quote from: Alexander Jankowski on March 29, 2013, 06:11:46 PM ---This is a nice problem. I can give a little bit of input: in system (b), the rate of change $y'(x)$ is twice that of the same rate in system (a). Perhaps we can treat (b) as a vertical expansion of (a), which is what the contour maps suggest. We can also note that each system has the same critical points $(2 \pi n,2 \pi n)$, where $n$ is an integer.

By inspection, we see that the separatrices in system (a) are lines with slopes $\pi$ and $-\pi$. In fact, by looking more closely at the contour maps, the separatrices appear to have the equations $$y = Â± \pi x + n \pi,$$ where $n$ is an integer. In system (b), the separatrices are sinusoidal functions that oscillate about an equilibrium line that is parallel to the $y$-axis. Finally, because there are infinitely many critical points in each system, there are also infinitely many separatrices.

--- End quote ---

I know that this is a nice problem, I just invented it :D You observe interesting things (correctly) albeit separatrices are only sinusoid-looking but not sinusoid lines.

But what about other trajectories which are not separatrices?