Integration on $q$-Cosymplectic Manifolds

This paper presents a unified framework for studying dynamics and integration on $q$-cosymplectic manifolds. After outlining the geometric foundations of $q$-cosymplectic structures, we derive new results concerning integrable systems and the characterization of Liouville coordinates, and further investigate the Lie integrability of $q$-evolution systems in this setting. We then develop a Hamilton--Jacobi theory tailored to multi-time Hamiltonian systems, both from an intrinsic geometric perspective and via symplectification techniques. To illustrate the applicability of the framework, we construct a $q$-cosymplectic Hamiltonian model for an extended FitzHugh-Nagumo system, providing a biologically relevant example involving three distinct temporal scales.