The onset of Darcy–Bénard regular and chaotic convection in a porous medium is studied by considering phase-lag effects that naturally arise in the thermal non-equilibrium heat transfer problem between the fluid and solid phases. A new type of heat equation is derived for both the phases. Using a double Fourier series and a novel decomposition, an extended Vadasz–Lorenz model with three phase-lag effects is derived. New parameters arise due to the phase-lag effects between local acceleration, convective acceleration, and thermal diffusion. The principle of exchange of stabilities is found to be valid and the subcritical instability is discounted. The new perspective supports the finding of an analytical expression for the critical Darcy–Rayleigh numbers representing, respectively, the onset of regular and chaotic convection. The understanding of the transition from the local thermal non-equilibrium situation to the local thermal equilibrium one is also best explained through the new perspective. In its present elegant form, the extended Vadasz–Lorenz system with three phase-lag effects is analyzed using the largest Lyapunov exponent and the bifurcation diagram. It is found that the lag effects not only give rise to a quantitative difference in the above two metrics concerning chaos, but also present a qualitative difference as well in the form of the very nature of chaos.

Using Fourier representations, an elaborate study of regular cellular-convective and chaotic motions in a ferrofluid is made. Investigation is made on the adequacy or otherwise of the minimal mode in studying such motions. Higher-order modes are also considered by adding modes (vertical/horizontal/combined extension). For higher modes, the extensions yield a dynamical system of order greater than three. The characteristic features of extended ferromagnetic-Lorenz models are analyzed using the largest Lyapunov exponent(LE), second largest LE, bifurcation diagram, and phase-space plots. The effect of additional modes on critical modal-Rayleigh (infinitesimal and finite-amplitude ones) numbers and the Rayleigh number at which transition to chaos occurs are examined to report features of ferroconvection hitherto unseen in previous studies. As both horizontal and vertical modes are increased, our findings infer that the dynamical system displays advanced onset of regular convection and delayed chaotic motion. Vigorous-chaotic motion is seen on adding vertical modes, whereas on adding horizontal modes, intense chaos appears with decreased intensity for large values of the scaled Rayleigh number. Most important finding from the study is that as modes are increased (vertical/horizontal), the transition from regular to chaotic motion is greatly modified and leads the system to a hyper-chaotic state. Conventionally, the chaotic or hyper-chaotic state is intermittent with a periodic/quasi-periodic state but it can be retained in the chaotic or hyper-chaotic state by considering moderate values of the Prandtl number and/or by bringing in the ferromagnetic effect.