In this paper, we present a novel approach to characterize the flow and heat transfer characteristics of a tri-hybrid nanofluid (Blood + TiO ${}_2$ + Al ${}_2$O ${}_3$ + Cu) through a curved, thermally heated channel using analytical and numerical frameworks based on a dynamical system. Dynamical system theory enables systematic identification and analysis of fundamental flow phenomena, including equilibrium states, stability, and bifurcations, which are often difficult to capture with traditional methods. Long-wavelength and low-Reynolds reductions yield analytical flow fields, from which the nonlinear dynamical system was constructed. The analytical determination of the general equilibrium points was complicated by the channel geometry and the strong nonlinearity of the model; therefore, a numerical approach based on domain decomposition techniques was adopted to overcome these difficulties. Saddle-node bifurcations, periodic orbits, and heteroclinic orbits provide a framework to characterize recirculating flows and the transitions between different flow states. The impact of physical properties and nanoparticle characteristics, which affect both global and local bifurcation behavior, was also examined. As a novel result, we revealed multiple equilibrium points with distinct bifurcation behaviors, capturing key fluid motion phenomena, such as trapping, governed by the channel geometry and tri-hybrid nanofluid properties. Our results, based on a dynamical systems analysis, refine and extend previous studies by clarifying the stability of equilibrium solutions, identifying key bifurcation regimes, and revealing the underlying nonlinear flow behavior.