# Bifurcation Theory

Bifurcation theory is a classical analysis method in the study of nonlinear differ­ential equations. Bifurcations are said to occur when nonlinear dynamic systems undergo changes in qualitative behavior. In bifurcation analysis, system steady states, or solutions with all time derivatives set equal to zero, and the stability about those steady states are cal­culated. The steady states are continuous functions of control surface angles. A bifurcation occurs when stability changes from one steady state to the next as a system parameter, such as control surface angle, is varied (Figure 9.12). A particular type of bifurcation known as the Hopf can lead to periodic motions such as wing rock.

A number of investigators, led initially in 1982 by J. V Carroll and R. K. Mehra, have used bifurcation theory in the study of nonlinear airplane motions, including wing rock and spins (Jahnke and Culick, 1994). P Guicheteau in France extended the wing rock application to include unsteady aerodynamic effects, and ONERA ran German-French Alpha Jet flight tests to compare with his theory. Drs. J. B. Planeaux, Jahnke, and Culick have studied bifur­cations in the United States. Also, a nonlinear analogy to linear indicial response methods has been proposed for understanding the response singularities that appear at large angles of attack and sideslip, and large rolling velocities (Tobak, Chapman, and Schiff, 1984).

Bifurcation analysis, in conjunction with piloted simulation, has been recognized as a potential aid in flight test planning (Lowenberg and Patel, 2000). This approach was experimented with using the aerodynamic and mass characteristics of the R. A.E. High Incidence Research [drop] Model, or HIRM, in bifurcation analysis and simulation in the DERA Advanced Flight Simulator, or AFS. The experimenters concluded that simulation validated the nonlinear characteristics predicted by bifurcation analysis. Thus, bifurcation analysis may be used to good effect in planning simulation and flight test programs.