We study several aspects of FitzHugh–Nagumo's (FH–N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurrences of bifurcation events is much richer. We present a numerical study of an example exhibiting a series of various bifurcations, including subcritical Hopf bifurcations, homoclinic bifurcations and saddle-node bifurcations of equilibria and of periodic solutions. Finally, we study periodically forced FH–N equations. We prove that phase-locking occurs independently of the magnitude of the periodic forcing.

PurposeThe purpose of this paper is to study the steady mixed convection flow over a vertical cone in the presence of surface mass transfer when the axis of the cone is inline with the flow.Design/methodology/approachIn this case, the numerical difficulties to obtain the non‐similar solution are overcome by applying an implicit finite difference scheme in combination with the quasilinearization technique.FindingsNumerical results are reported here to display the effects of Prandtl number, buoyancy and mass transfer (injection and suction) parameters at different stream‐wise locations on velocity and temperature profiles, and on skin friction and heat transfer coefficients.Research limitations/implicationsThermo‐physical properties of the fluid in the flow model are assumed to be constant except the density variations causing a body force term in the momentum equation. The Boussinesq approximation is invoked for the fluid properties to relate the density changes to temperature changes and to couple in this way the temperature field to the flow field.Practical implicationsConvective heat transfer over a stationary cone is important for the thermal design of various types of industrial equipments such as heat exchangers, conisters for nuclear waste disposal, nuclear reactor cooling systems and geothermal reservoirs, etc.Originality/valueThe combined effects of thermal diffusion and surface mass transfer on a vertical cone has been studied.

The nonlinear propagation characteristics of surface acoustic waves on an isotropic elastic solid have been studied in this paper. The solution of the harmonic boundary value problem for Rayleigh waves is obtained as a generalized Fourier series whose coefficients are proportional to the slowly varying amplitudes of the various harmonics. The infinite set of coupled equations for the amplitudes when solved exhibit an oscillatory slow variation signifying a continuous transfer of energy back and forth among the various harmonics. A conservation relation is derived among all the harmonic amplitudes.

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