It is important to emphasize how the homoclinicity in the
Shilnikov scenario for Shilnikov condition: ℜe 3 ℜe > 0
appears. This suggests a rapid injection of flow in the plane of the
stable manifold with a relatively slow winding. The striking fact is
that other authors [60, 61] have developed models, based on
different assumptions, which exhibit similar behavior. In addition,
there is sufficient experimental evidence [62–66] in relation to the
complexity of the dynamics of cancer.
Because the process of metastases presents chaos like Shilni-
kov’s has important biological implications. On one hand, the high
sensitivity of the system to initial conditions determines the inabil-
ity to make long-term predictions in relation to the evolution of the
disease, i.e., the end forecasts are improbable (poor prognosis). In
addition, it gives a high degree of robustness [67, 68], hence the
level of success in the treatment is low, especially at the stage of
metastases [69]. Additionally, the information created during the
evolutionary process of cancer cannot be destroyed [70], which is
manifested in clinical recurrence (relapse) of cancer after a while
that has apparently been “removed.”
It is well known that the tumor progression in prostate cancer
following radical prostatectomy occurs in 20–40% of patients, even
though in pathological stage, approximately 40% of patients who
undergo surgical removal of non-small cell lung cancer without
overt metastases relapse within 24 months after surgery, in breast
cancer, 20% of clinically disease-free patients relapse 7–25 years
after mastectomy [33, 40].
Therefore, the tactic in the treatment of cancer lies not only in
their physical elimination also must change the information created
[71, 72]. Recently, a study [73] has shown that pancreatic cancer
cells can be coaxed to revert back toward normal cells by introdu-
cing a protein called E47. The protein E47 binds to specific DNA
sequences and control genes involved in growth and
differentiation.
In other words, the growth of a primary tumor from a micro-
scopic level, avascular phase, to the macroscopic level, vascular
phase and subsequent onset of metastases, is not simply an accu-
mulation of malignant cells but is a process exhibiting: (1) a
non-linear dynamic, (2) self-organization out of thermodynamic
equilibrium, (3) showing that a high degree of robustness, com-
plexity, and level of hierarchy; which leads to the creation of new
information and learning capacity.
The metastases process is associated with a biological phase
transition type “first-order” through supercritical Andronov-Hopf
bifurcation, coming out of limit cycle and subsequently through a
cascade of bifurcations of the type saddle-foci Shilnikov bifurca-
tions. This could explain the epithelial-to-mesenchymal transition
during metastases [74].142 Sheyla Montero et al.