  In this study, based on the $\varphi^4$ model, a new model (called the
$B\varphi^4$ model) is introduced in which the potential form for the values of
the field whose magnitudes are greater than $1$ is multiplied by the positive
number $B$. All features related to a single kink (antikink) solution remain
unchanged and are independent of parameter $B$. However, when a kink interacts
with an antikink in a collision, the results will significantly depend on
parameter $B$. Hence, for kink-antikink collisions, many features such as the
critical speed, output velocities for a fixed initial speed, two-bounce escape
windows, extreme values, and fractal structure in terms of parameter $B$ are
considered in detail numerically. The role of parameter $B$ in the emergence of
a nearly soliton behavior in kink-antikink collisions at some initial speed
intervals is clearly confirmed. The fractal structure in the diagrams of escape
windows is seen for the regime $B\leq 1$. However, for the regime $B >1$, this
behavior gradually becomes fuzzing and chaotic as it approaches $B = 3.3$. The
case $B = 3.3$ is obtained again as the minimum of the critical speed curve as
a function of $B$. For the regime $3.3< B \leq 10$, the chaotic behavior
gradually decreases. However, a fractal structure is never observed.
Nevertheless, it is shown that despite the fuzzing and shuffling of the escape
windows, they follow the rules of the resonant energy exchange theory.
