Nonlinear surge response behavior of a multipoint mooring system under harmonic wave excitation is analyzed to investigate various instability phenomena such as bifurcation, period-doubling, and subharmonic and chaotic responses. The nonlinearity of the system arises due to nonlinear restoring force, which is modeled as a cubic polynomial. In order to trace different branches at the bifurcation point on the response curve (amplitude versus frequency of excitation plot), an arc-length continuation technique along with the incremental harmonic balance (IHBC) method is employed. The stability of the solution is investigated by the Floquet theory using Hsu’s scheme. The period-one and subharmonic solutions obtained by the IHBC method are compared with those obtained by the numerical integration of the equation of motion. Characteristics of solutions from stable to unstable zones, chaotic motion, solutions, etc., are identified with the help of phase plots and Poincaré map sections.