Local bifurcation analyses and numerical simulation show that period one single-impact motion, in most cases, undergoes period doubling bifurcation or Hopf bifurcation with change of control parameters.
The numerical simulation shows that the period two cycle of model (3) may be globally asymptotically stable when θ > θ ∗ and θ − θ ∗ is small.
The numerical simulation shows that the period-doubling bifurcation may continue and go to chaos (see Figure 2(d)).
Then, we computed the numerical simulation for two different wave periods of 12.0 and 14.0 s, which were based on the observed wave periods corresponding to the annual maximum wave height at Hualien port.
This indicated that numerical simulation could reproduce the long-period temperature fluctuation at the mixing tee.
Further expansion of simulation period would cause numerical calculation issues.
By means of numerical simulation, in addition to the stable period-1 gait, we found a variety of gait bifurcation phenomena, including the period-doubling bifurcation, the Neimark Sacker bifurcation, the Neimark Sacker-2 bifurcatioNeimark Sacker-2bifurcation, and the Neimark–Sacker-X bifurcation, among which many types have never been reported in previous studies.
Global bifurcations are discussed by using numerical simulation, and the evolution from the period-doubling sequence to chaotic motions is illustrated.
Table 1 Boundary conditions for the numerical simulation Case Sea level (m) Wave height (m) Period (s) 1A 0 10 10 1B 0 15 15 15 2C 5 20 20 5 10 10 2B 5 15 15 2C 5 20 20 Sea level, significant wave height, and period are set as the main conditions on the boundary of the numerical domain.
Finally, numerical simulation is also applied to obtain double-period cascading bifurcations leading to chaos.
Using a numerical simulation, Lachet and Bard (1994) verified that the natural period of the sedimentary site coincides well with the predominant period of H/V.
