  The low-mass X-ray binary 4U1705-44 exhibits dramatic long-term X-ray time
variability with a timescale of several hundred days. The All-Sky Monitor (ASM)
aboard the Rossi X-ray Timing Explorer (RXTE) and the Japanese Monitor of
All-sky X-ray Image (MAXI) aboard the International Space Station together have
continuously observed the source from December 1995 through May 2014. The
combined ASM-MAXI data provide a continuous time series over fifty times the
length of the timescale of interest. Topological analysis can help us identify
'fingerprints' in the phase-space of a system unique to its equations of
motion. The Birman-Williams theorem postulates that if such fingerprints are
the same between two systems, then their equations of motion must be closely
related. The phase-space embedding of the source light curve shows a strong
resemblance to the double-welled nonlinear Duffing oscillator. We explore a
range of parameters for which the Duffing oscillator closely mirrors the time
evolution of 4U1705-44. We extract low period, unstable periodic orbits from
the 4U1705-44 and Duffing time series and compare their topological
information. The Duffing and 4U1705-44 topological properties are identical,
providing strong evidence that they share the same underlying template. This
suggests that we can look to the Duffing equation to help guide the development
of a physical model to describe the long-term X-ray variability of this and
other similarly behaved X-ray binary systems.
