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We have investigated how kinetic instabilities such as the Weibel instability (WI), the mushroom instability (MI), and the kinetic Kelvin-Helmholtz instability (kKHI) are excited in jets without and with a toroidal magnetic field, and how such instabilities contribute to particle acceleration. In this work we use a new jet injection scheme where an electric current is self-consistently generated at the jet orifice by the jet particles which produce the toroidal magnetic field. We perform five different simulations for a sufficiently long time to examine the non-linear effects of the jet evolution. We inject unmagnetized e^{\pm} and e^{-} - p^{+} (m_p/m_e = 1836), as well as magnetized e^{\pm} and e^{-} - i^{+} (m_i/m_e = 4) jets with a top-hat jet density profile into an unmagnetized ambient plasmas of the same species. We show that WI, MI, and kKHI excited at the linear stage, generate a non-oscillatory x-component of the electric field accelerating and decelerating electrons. We find that the two different jet compositions (e^{\pm} and e^{-} - i^{+}) display different instability modes respectively. Moreover, the magnetic field in the non-linear stage generated by different instabilities is dissipated and reorganized into new topologies. A 3D magnetic field topology depiction indicates possible reconnection sites in the non-linear stage where the particles are significantly accelerated by the dissipation of the magnetic field associated to a possible reconnection event.