Dynamics of a System in a Coupled Duffing and a Ratchet like Potential Functions

Abstract

This paper considers the dynamics of a system in a coupled
Duffing and Ratchet potentials in a range of the position
coordinates that captures a complete behaviour for the system so
that anything outside this range will only be a repetition because
of the periodicity of the system. Due to computational difficulty of
the ratchet potential a power series approximation method is
applied to transform this part of the potential function to a very
closely fitting polynomial so that the entire potential gradient can
be uniformly expressed by a polynomial. The positions of all the
equilibrium points are evaluated in the range under consideration
and the stabilities of all the equilibrium points are obtained from
the computation of their eigenvalues. Analytic solutions for the
forced and unforced system is found by using two time variable
expansion method. The frequency response curve for the system is
evaluated for the frequency in the range of [0.0-5.0] the result
show stable and unstable response amplitudes obtained by using
simple graphical analysis. Regions where only a single response
amplitude are found and region where multi-valued response
amplitude’s are also seen along with hysteresis and jump
phenomenon region. The nature of the solutions for the forced
system is seen from the simple graphical analysis of the frequency
response equation showing how the system will move around the
equilibrium points. The work also show how the equilibrium
points move as the forcing amplitude is varied for the undamped
case , showing how a pair of equilibrium points move toward each
other collide and disappear through a reverse saddle-node
bifurcation leaving only a single equilibrium point which remain
for all the forcing amplitudes considered. Leading to the
conclusion that for the Duffing–ratchet system considered the
dynamical behaviour found is very similar to that of the purely
Duffing only that for this system there are more equilibrium
points in particular two more saddle equilibrium points come up
as a result of more potential wells of the system. As a result a
more complicated dynamical behaviour is seen. Consequently the
advantages obtained from complex dynamics of the Duffing
system can be better obtained in this Duffing–ratchet system.