<p>The nonlinear dynamics of a viscoelastic fluid saturating a heated porous medium are investigated numerically under the Cattaneo–Christov non-Fourier heat flux formulation. The momentum equation is governed by the Oldroyd-B constitutive model, which accounts for stress relaxation and strain retardation, while the hyperbolic energy equation incorporates a finite thermal time lag through the Cattaneo number. A six-dimensional nonlinear autonomous system is derived via a truncated Galerkin expansion and integrated using a fourth-order Runge–Kutta scheme. The stability of the conductive and convective equilibria is examined analytically, and the onset conditions for stationary and oscillatory convection are established as explicit functions of the elasticity parameters and the Cattaneo number. The nonlinear dynamics are characterized through bifurcation diagrams, Lyapunov exponent spectra, phase portraits and dynamical regime maps in the parameter planes. The results revealed that the chaotic, periodic and multiperiodic behavior exhibited by the system of viscoelastic flow in porous medium can be controlled by the parameters of stress relaxation and strain retardation times as well as the Cattaneo–Christov number for different values of Darcy–Rayleigh number. These findings offer quantitative guidance for the suppression of undesirable chaotic mixing in polymer extrusion processes and viscoelastic polymer flooding in porous oil reservoirs, where the thermal relaxation time and fluid elasticity constitute experimentally accessible modulation parameters.</p>