homework and exercises - Understanding transverse oscillation in 1 mass, 2 spring systems

Lately I have been working through some nice problems on mass-spring systems. There are tons of different configurations - multiple masses, multiple springs, parallel/series, etc.

A few possible configurations are listed here: https://ccrma.stanford.edu/CCRMA/Courses/152/vibrating_systems.html. I think the most interesting one is the 1 mass, 2 spring system with transverse motion:

It is written on that page:

"If the springs are initially stretched a great deal from their relaxed length (but not distorted), the vibration frequency is nearly the same as for longitudinal vibrations.

If the springs are initially stretched very little from their relaxed length, the "natural" frequency is much lower and the vibrations are nonlinear (nonsinusoidal) for all but the smallest of $y$-axis displacements."

If I understand correctly, this means that the system approaches simple harmonic motion when the initial displacement $y_0$ is either very small, or very large.

I tried to see if I could prove this myself by looking at the period of oscillations as a function of the initial displacement $y_0$, but I am having some trouble:

Suppose the mass is a distance $\Delta y$ from the equilibrium position. Then each spring has length $\sqrt {a^2+\Delta y^2}$, so each spring is stretched by $\sqrt {a^2+\Delta y^2}-a$ from its original length. Therefore, each spring exerts a restoring force of $k(\sqrt {a^2+\Delta y^2}-a)$. The magnitude of the net restoring force on the mass is then

$|F_r|=2k(\sqrt {a^2+\Delta y^2}-a)\sin(\theta)=2k(\Delta y-a\sin(\theta))$

Substituting $\sin(\theta)=\Delta y/\sqrt {a^2+\Delta y^2}$, we get:

$|F_r|=2k\Delta y(1-a/\sqrt {a^2+\Delta y^2})$

So we have the differential equation:

$y''=-(2k/m)y(1-a/\sqrt {a^2+y^2})$.

If $y$ is very large, then $y''\approx -(2k/m)y$, which is simple harmonic motion with period $2\pi\sqrt{2k/m}$. This makes sense, because if $y$ is very large, the two springs essentially act in parallel, so we effectively have a 1 spring system in longitudinal motion with spring constant $2k$, which gives the same result.

Now if $y$ is very small, I'm not sure which approximation formulas to use to make things come out nicely. According to the web page, I should get the result that the period will be larger than $2\pi\sqrt{2k/m}$.