Nonlinear Dynamics

, Volume 66, Issue 4, pp 681–688 | Cite as

Canard cycles in aircraft ground dynamics

  • J. Rankin
  • M. Desroches
  • B. Krauskopf
  • M. Lowenberg
Original Paper


The loss of lateral stability of an aircraft turning on the ground is associated with a rapid transition from small-amplitude oscillations to large-amplitude relaxation oscillations over a very small parameter range. This phenomenon is shown to correspond to a canard explosion, which is known to be an important feature of systems that exhibit a separation of time scales. While the industry-tested model of aircraft ground dynamics that we consider does not feature an explicit splitting of time scales, we show that, locally near the transition from stable turning to laterally unstable behaviour, the forward speed of the aircraft acts as a slow variable. The associated family of canard cycles is identified, and the canard explosion is shown to be directly related to the successive saturation of tyre forces at the two main landing gears. We present a canonical two-dimensional slow–fast vector field model that captures the key features of this type of canard explosion; it differs from the canard explosion in the archetypical Van der Pol system in terms of the shape of the associated critical manifold.


Aircraft Ground handling Nonlinear dynamics Canard explosion Slow–fast dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brøns, M., Bar-Eli, K.: Canard explosion and excitation in a model of the Belousov–Zhabotinskii reaction. J. Phys. Chem. 95(22), 8706–8713 (1991) CrossRefGoogle Scholar
  2. 2.
    Moehlis, J.: Canards for a reduction of the Hodgkin–Huxley equations. J. Math. Biol. 52(2), 141–153 (2006) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brøns, M.: Bifurcations and instabilities in the Greitzer model for compressor system surge. Eng. Ind. 2(1), 51–63 (1988) Google Scholar
  4. 4.
    Van der Pol, B.: On “relaxation-oscillations”. Philos. Mag. Ser. 7 2(11), 978–992 (1926) Google Scholar
  5. 5.
    Benoît, E., Callot, J.-L., Diener, F., Diener, M.: Chasse au canard. Collect. Math. 32, 37–119 (1981) MathSciNetMATHGoogle Scholar
  6. 6.
    Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. In: Verhulst, F. (ed.) Asymptotic Analysis II. Lecture Notes in Math., vol. 985, pp. 449–494. Springer, New York (1983) CrossRefGoogle Scholar
  7. 7.
    Rankin, J., Krauskopf, B., Lowenberg, M., Coetzee, E.: Operational parameter study of aircraft dynamics on the ground. ASME J. Comput. Nonlinear Dyn. 5, 021007 (2010) CrossRefGoogle Scholar
  8. 8.
    Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: Auto 97: Continuation and bifurcation software for ordinary differential equations, May 2001.
  9. 9.
    Krauskopf, B., Osinga, H.M., Galán-Vioque, J.: Numerical Continuation Methods for Dynamical Systems. Springer, Berlin (2007) MATHCrossRefGoogle Scholar
  10. 10.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, Berlin (1998) MATHGoogle Scholar
  11. 11.
    Lowenberg, M.H.: Bifurcation analysis of multiple-attractor flight dynamics. Philos. Trans. R. Soc., Math. Phys. Eng. Sci. 356(1745), 2297–2319 (1998) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Nguyen, V., Schultz, G., Balachandran, B.: Lateral load transfer effects on bifurcation behavior of four-wheel vehicle system. J. Comput. Nonlinear Dyn. 4(4), 041007 (2009) CrossRefGoogle Scholar
  13. 13.
    Thota, P., Krauskopf, B., Lowenberg, M.: Interaction of torsion and lateral bending in aircraft nose landing gear shimmy. Nonlinear Dyn. 57(3), 455–467 (2009) MATHCrossRefGoogle Scholar
  14. 14.
    Etkin, B.: Dynamics of Atmospheric Flight. Wiley, New York (1972) Google Scholar
  15. 15.
    Pacejka, H.B.: Tyre and Vehicle Dynamics. Elsevier, Amsterdam (2006) Google Scholar
  16. 16.
    Rankin, J., Coetzee, E., Krauskopf, B., Lowenberg, M.: Bifurcation and stability analysis of aircraft turning on the ground. AIAA J. Guid. Control Dyn. 32(2), 499–510 (2009) CrossRefGoogle Scholar
  17. 17.
    Brøns, M.: Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures. Proc. R. Soc., Math. Phys. Eng. Sci. 461(2059), 2289–2302 (2005) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • J. Rankin
    • 1
  • M. Desroches
    • 2
  • B. Krauskopf
    • 2
  • M. Lowenberg
    • 2
  1. 1.NeuroMathComp Project TeamINRIA Sophia AntipolisSophia AntipolisFrance
  2. 2.Faculty of EngineeringUniversity of BristolBristolUK

Personalised recommendations