  We present a model of two-kinks resulting from an explicit composition of two
standards kinks of the $\phi^4$ model based on the procedure of Ref.
\cite{uchiyama}. The two-kinks have an additional parameter accounting for the
separation of the standard kinks of $\phi^4$ model. We have shown that the
two-kinks have two discrete internal modes besides the zeroth mode and the
continuous spectrum. This new feature signalizes that the head-on collision a
two-kinks/two-antikinks pair exhibits a rich and complex behavior due to the
additional channel from which the energy of the system can be stored. We have
exhibited the fractal structure associated with the main configurations after
the collision. We have inferred the fractality as the imprint of the nonlinear
exchange of energy into the two discrete internal modes.
