  A non-${\cal{PT}}$-symmetric Hamiltonian system of a Duffing oscillator
coupled to an anti-damped oscillator with a variable angular frequency is shown
to admit periodic solutions. The result implies that ${\cal{PT}}$-symmetry of a
Hamiltonian system with balanced loss and gain is not necessary in order to
admit periodic solutions. The Hamiltonian describes a multistable dynamical
system - three out of five equilibrium points are stable. The dynamics of the
model is investigated in detail by using perturbative as well as numerical
methods and shown to admit periodic solutions in some regions in the space of
parameters. The phase transition from periodic to unbounded solution is to be
understood without any reference to ${\cal{PT}}$-symmetry. The numerical
analysis reveals chaotic behaviour in the system beyond a critical value of the
parameter that couples the Duffing oscillator to the anti-damped harmonic
oscillator, thereby providing the first example of Hamiltonian chaos in a
system with balanced loss and gain. The method of multiple time-scales is used
for investigating the system perturbatively. The dynamics of the amplitude in
the leading order of the perturbation is governed by an effective dimer model
with balanced loss and gain that is non-${\cal{PT}}$-symmetric Hamiltonian
system. The dimer model is solved exactly by using the Stokes variables and
shown to admit periodic solutions in some regions of the parameter space.
