  We carry out a compact phase space analysis of a non-canonical scalar field
theory whose Lagrangian is of the form $F(X)-V(\phi)$ within general
relativity. In particular, we focus on a kinetic term of the form $F(X)=\beta
X^m$ with power law potential $V_0 \phi^n$ and exponential potential $V_0
e^{-\lambda\phi/M_{Pl}}$ of the scalar field. The main aim of this work is to
investigate the genericity of nonsingular bounce in these models and to
investigate the cosmic future of the bouncing cosmologies when they are
generic. A global dynamical system formulation that is particularly suitable
for investigating nonsingular bouncing cosmologies is used to carry out the
analysis. We show that when $F(X)=\beta X^m$ ($\beta<0$), nonsingular bounce is
generic for a power law potential $V(\phi) = V_0 \phi^n$ only within the
parameter range $\left\lbrace \frac{1}{2}<m<1,\,n<\frac{2m}{m-1}\right\rbrace$
and for an exponential potential $V(\phi) = V_0 e^{-\lambda\phi/M_{Pl}}$ only
within the parameter range $\left\lbrace\frac{1}{2}<m\leq1\right\rbrace$.
Except in these cases, nonsingular bounce in these models is not generic due to
the non-existence of global past or future attractors. Our analysis serves to
show the importance of a global phase space analysis to address important
questions about nonsingular bouncing solutions, an idea that may and must be
adopted for such solutions even in other theories.
