  In this work, we develop a comprehensive numerical analysis of the dynamic
response of a Josephson Traveling Wave Parametric Amplifier (JTWPA) by varying
the driving parameters, with a focus on the pathways leading to chaotic
behavior. By tuning the working conditions, we capture the broad spectrum of
dynamical regimes accessible to JTWPAs, delineating the settings under which
transition to chaos occurs. Furthermore, we extend our investigation to device
formed by junctions characterized by a non--sinusoidal current phase relation
(CPR) and exploring the impact of its shape on the amplifier's performance.
Through the study of gain characteristics, Poincar\'e sections, and Fourier
spectra, we provide an in-depth understanding of how non-linearity and CPR
nonsinusoidality influence the JTWPAs' operational effectiveness and stability.
This investigation offers insights into optimizing the device designs for
enhanced performance and robustness against chaotic disruptions, in order to
establish a framework for predicting and controlling JTWPA behavior in
practical applications. This effort will pave the way for the development of
devices with tailored dynamic responses and for advancements in quantum
computing and precision measurement technologies, where stability and high
fidelity are of paramount importance.
