Comparison of nonlinear scales using atmospheric CO2 data

Nonlinear monotonically increasing bounded functions help to visualize and analyze data on various scales. However, many monotonic functions such as logarithm or power laws have either function values or derivatives that become unbounded at some regions of the $x-$ axis. On the other hand, sigmoid or hyperbolic functions become saturated and functionally inactive over a wide region of the $x-$ axis. Therefore, a well-behaved function over $x\in(-\infty,\infty)$ is still a fundamental requirement for data visualization and analysis. Recently introduced algebraic form $ay^3+y=mx$ with parameters $a>0$ and $m$ as slope is one such function that is both well-behaved and smooth i.e., infinitely differentiable on an $xy-$ plane. The slope of this form vanishes as $x\rightarrow\pm\infty$ similar to $y=\log x$. This form can also be interpreted as a continued fraction of $y=mx$ when expressed as $y=mx/(1+ay^2)$. The analytical real solution of the algebraic form can be interpreted as a recursion up to infinite depth. In this work, in addition to the cubic form we introduce higher order polynomials in $y$ that can be expressed as continued fraction of straight lines with tractable analytical solutions. We plot atmospheric CO$_2$ data on the continued fraction of axes for various values of $a$ and $m$.