  The nonlinear dynamics of circularly polarized dispersive Alfv{\'e}n wave
(AW) envelopes coupled to the driven ion sound waves of plasma slow response is
studied in a uniform magnetoplasma. By restricting the wave dynamics to a few
number of harmonic modes, a low-dimensional dynamical model is proposed to
describe the nonlinear wave-wave interactions. It is found that two
subintervals of the wave number of modulation $k$ of AW envelope exists, namely
$(3/4)k_c<k<k_c$ and $0<k<(3/4)k_c$, where $k_c$ is the critical value of $k$
below which the modulational instability (MI) occurs. In the former, where the
MI growth rate is low, the periodic and/or quasi-periodic states are shown to
occur, whereas the latter, where the MI growth is high, brings about the
chaotic states. The existence of these states is established by the analyses of
Lyapunov exponent spectra together with the bifurcation diagram and phase-space
portraits of dynamical variables. Furthermore, the complexities of chaotic
phase spaces in the nonlinear motion are measured by the estimations of the
correlation dimension (CD) as well as the approximate entropy (ApEn), and
compared with those for the known H{\'e}non map and the Lorenz system in which
a good qualitative agreement is noted. The chaotic motion thus predicted in a
low-dimensional model can be a prerequisite for the onset of Alfv{\'e}nic wave
turbulence to be observed in a higher dimensional model that and is relevant in
the Earth's ionosphere and magnetosphere.
