This article studies the phase portraits, chaotic patterns, and traveling wave solutions of the fractional order generalized Pochhammer–Chree equation. First, the fractional order generalized Pochhammer–Chree equation is transformed into an ordinary differential equation. Second, the dynamic behavior is analyzed using the planar dynamical system, and some three-dimensional and two-dimensional phase portraits are drawn using Maple software to reflect its chaotic behaviors. Finally, many solutions were constructed using the polynomial complete discriminant system method, including rational, trigonometric, hyperbolic, Jacobian elliptic function, and implicit function solutions. Two-dimensional graphics, three-dimensional graphics, and contour plots of some solutions are drawn.