### 3.3 Chaotic scalar field dynamics

Many studies of cosmological models generally reduce complex physical systems to simplified or even
analytically integrable systems. In sufficiently simple models the dynamical behavior (or fate) of the
Universe can be predicted from a given set of initial conditions. However, the Universe is composed of many
different nonlinear interacting fields including the inflaton field which can exhibit complex chaotic behavior.
For example, Cornish and Levin [63] consider a homogenous isotropic closed FLRW model with various
conformal and minimally coupled scalar fields (see Section 6.2.2). They find that even these
relatively simple models exhibit chaotic transients in their early pre-inflationary evolution.
This behavior in exiting the Planck era is characterized by fractal basins of attraction, with
attractor states being to (i) inflate forever, (ii) inflate over a short period of time then collapse, or
(iii) collapse without inflating. Monerat et al. [122] investigated the dynamics of the pre-inflationary
phase of the Universe and its exit to inflation in a closed FLRW model with radiation and a
minimally coupled scalar field. They observe complex behavior associated with saddle-type
critical points in phase space that give rise to deSitter attractors with multiple chaotic exits to
inflation that depend on the structure of the scalar field potential. These results suggest that
distinctions between exits to inflation may be manifested in the process of reheating and as a selected
spectrum of inhomogeneous perturbations influenced by resonance mechanisms in curvature
oscillations. This could possibly lead to fractal patterns in the density spectrum, gravitational waves,
cosmic microwave background radiation (CMBR) field, or galaxy distribution that depend on
specific details including the number of fields, the nature of the fields, and their interaction
potentials.
Chaotic behavior can also be found in more general applications of scalar field dynamics. Anninos et
al. [20] investigated the nonlinear behavior of colliding kink-antikink solitons or domain walls described by
a single real scalar field with self-interaction potential . Domain walls can form as topological
defects during the spontaneous symmetry breaking period associated with phase transitions, and can seed
density fluctuations in the large scale structure. For collisional time scales much smaller than the
cosmological expansion, they find that whether a kink-antikink collision results in a bound state or a
two-soliton solution depends on a fractal structure observed in the impact velocity parameter
space. The fractal structure arises from a resonance condition associated with energy exchanges
between translational modes and internal shape-mode oscillations. The impact parameter space
is a complex self-similar fractal composed of sequences of different -bounce (the number
of bounces or oscillations in the transient semi-coherent state) reflection windows separated
by regions of oscillating bion states (see Figure 4). They compute the Lyapunov exponents
for parameters in which a bound state forms to demonstrate the chaotic nature of the bion
oscillations.