Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion

This paper focuses on the impact of cross-diffusion for Turing-Hopf instability in a morphochemical model for electrodeposition (DIB) and completes the analysis on the role of cross-diffusion on pattern formation in electrodeposition we recently carried out in [Lacitignola et al. 2018]. We derive and discuss conditions for the destabilization a´ la Turing of the Hopf limit cycle in the neighborhood of the codimension-two Turing-Hopf bifurcation point, resulting in oscillatory Turing patterns. Numerical validation of the above theoretical findings is provided both in the case of Weak and Strong Turing-Hopf instability. Experimental evidence of this kind of phenomenology is then discussed also in relation to the charging and discharging process of a battery.