A unified fixed point approach for enriched bivariate mappings with applications to volterra systems and fractional-order chaos.

We develop a unified fixed point theory for enriched bivariate contractions in ordered Banach spaces and in ordered convex metric spaces. Two classes of mappings are introduced and analysed: enriched bivariate Ciric-Reich-Rus contractions (𝒞BCRRCe) and enriched bivariate interpolative Ciric-Reich-Rus contractions (𝒞BICRRCe). For each class we prove the existence and uniqueness of coupled fixed points and establish the geometric convergence of the coupled Krasnoselskij iteration. The methodological contribution is a transparent product space reduction: the coupled averaged operator associated with an enriched bivariate map is shown to be an ordinary (single variable) Ciric-Reich-Rus operator on the product space, so that the classical theory applies directly and every algebraic step is explicit. This reduction yields a sharp sufficient condition for the existence of a coupled fixed point, namely (2a+k2)/(k1+1)+2b<1, where a,b are the contraction constants and k1,k2 are the enrichment constants. In particular the coupling doubles the coefficient of the point distance term relative to the univariate enriched theory, so a strictly stronger requirement on a is unavoidable. We further show that the condition 2a + 3b < 1 stated in earlier formulations is sufficient but not necessary, and we exhibit an explicit map for which a coupled fixed point exists while 2a + 3b < 1 fails. The abstract results are applied first to a Volterra type coupled integral system, and then to the fractional order Thomas cyclically symmetric attractor formulated with the Caputo-Fabrizio derivative, where an averaged mapping yields local in time existence and uniqueness even though the long time dynamics are chaotic. All numerical results are produced by direct computation. A bifurcation diagram and a largest Lyapunov exponent spectrum, computed independently, agree window for window and confirm a period doubling route to chaos with intermediate periodic windows, while the measured convergence factor of the coupled iteration is geometric.