# Bifurcation Flight Dynamic Analysis of a Strake-Wing Micro Aerial Vehicle
## 5 Bifurcation Flight Dynamic Analysis of a Micro Aerial Vehicle  
Figures 5 and 6 show single-parameter bifurcation diagrams for equilibrium positions: angles of attack \(\alpha\) and angular velocity of banking \(P\) for different values of the bifurcation parameter, namely, elevator deflection angles \(\delta_{e}\) from a range of \(\delta_{e}\in(-35^{\circ},\,30^{\circ})\) for angles of attack equilibrium position, \(\delta_{e}\in(-35^{\circ},\,40^{\circ})\) for banking angular velocity equilibrium positions. Figure 5 shows enlarged bifurcation diagrams for elevator deflections in the range of \(\delta_{e}\in(3^{\circ},\,10^{\circ})\), corresponding to steady states at low and moderate angles of attack for \(\alpha\in(0^{\circ},\,54^{\circ})\), the second area corresponds to a range of high angles of attack \(\alpha\in(65^{\circ},\,83^{\circ})\). Various types of micro aerial vehicle dynamics, corresponding to seven flight regimes, can be classified in these areas. These regimes are marked with the letters A to G. Regime A means steady symmetrical flight equilibrium states, for angles of attack in the range of \(\alpha\in(2^{\circ},\,23^{\circ})\). Area B, for angles of attack in the range of \(\alpha\in(15^{\circ},\,50^{\circ})\), corresponds to various motion states with stable and unstable orbits. Area C, for angles of attack in the range of \(\alpha\in(35^{\circ},\,55^{\circ})\), corresponds to deep steady spirals. Area D corresponds to inverted spirals. Area E corresponds to steady spins. Areas F and G correspond to transient spins. Figure 5: Bifurcation diagram. Quasi-steady states for various elevator deflection angles; \(\alpha(\delta_{e})\). The aircraft equilibrium positions in each of these two areas were analysed depending on the bifurcation parameter, namely elevator deflection angles (symmetrical elevon deflections) \(\delta_{e}\) and asymmetrical elevon deflections \(\delta_{elp}\). During continuation tests, following the methodology described in Section 4, it was assumed that the propeller thrust does not change, and elevon deflections are only symmetrical (or \(\delta_{elp}=0\)). Continuation analyses were conducted for positive and negative changes in the bifurcation parameter (symmetrical elevon deflections \(\delta_{e}\)), starting with \(\delta_{e}=20^{\circ}\). The calculations were conducted for the elevator deflection angle range of change of \(-35^{\circ}\leq\delta_{e}\leq 40^{\circ}\), so that it was possible to detect almost all solutions corresponding to the quasi-steady flight states. At the starting point of the continuation analysis (\(\delta_{e}=20^{\circ}\), \(\alpha=3^{\circ}\)), the aircraft was conducting a steady symmetrical flight. As the elevator deflection angle changed, the elevator angle of attack initially increased, and the deflection angular velocity remained at zero. This dynamic regime was denoted with the letter A. Its corresponding range of angles of attack is \(2^{\circ}<\alpha<25^{\circ}\) (Figures 5 and 7). The range of steady states denoted with the letter A corresponds to a symmetrical flight of the MAV (angular velocity \(P=0\)). When the elevator deflection angle decreases below \(\delta_{e}\)\(\approx\)10 and the angle of attack \(\alpha\approx 28\) degrees, the micro aerial vehicle enters the dynamics regime denoted with the letter B, with both anti-symmetrical, as well as symmetrical motions. They can be associated with unstable Dutch roll, wing-rock oscillations, spiral motion instabilities, as well as unstable phugoids. Ant-symmetrical oscillations result from Hopf bifurcations in B. In this area, when the elevator deflection angle continues to decrease, reaching a value of \(\delta_{e}\approx 4^{\circ}\) and \(\alpha\approx 34^{\circ}\), the spiral mode loses stability, which leads to the bifurcation of the stable and almost symmetrical solution. Two asymmetrical equilibrium position branches appear. At higher angles of attack, the spiral model becomes unstable along the almost symmetrical solution branch.