Investigation of the forced Bonhoeffer van der Pol equations via continuation and other methods

Abstract

Previous work used bifurcation diagrams and Lyapunov exponents to explore the dynamics of the forced Bonhoeffer van der Pol equation. Here we use continuation methods to clarify the bifurcations of the periodic orbits which had previously been ambiguous. We discuss the use of numerical methods including Galerkin and the recent 0-1 test for chaos. References

• B. Barnes and R. Grimshaw, Numerical studies of the periodically forced Bonhoeffer van der Pol system, International Journal of Bifurcation and Chaos, 7(12):2653--2689, 1997.
• A. Ben-Tal, D. Shein, S. Zissu, Studying ferroresonance in actual power systems by bifurcation diagram,Electric Power Systems Research, 49, 175--183, 1999. doi:10.1016/S0378-7796(98)00117-5.
• N. Britton, Essential Mathematical Biology, Springer Undergraduate Mathematics Series, Springer-Verlag, 2003.
• H. Croisier, P. Dauby, Continuation and bifurcation analysis of a periodically forced excitable system, Journal of Theoretical Biology, 246:430--448, 2007. doi:10.1016/j.jtbi.2007.01.017.
• E. Doedel and R. Heinemann, Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A$\rightarrow $B$\rightarrow $C reactions, Chemical Engineering Science, 38(9):1493-1499, 1983. doi:10.1016/0009-2509(83)80084-0.
• R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1:445--466, 1961. http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1366333
• G. A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems,Proc. Royal Soc. Lon. Series A, 460:603--611, 2004. doi:10.1098/rspa.2003.1183.
• G. A. Gottwald, I. Melbourne, Testing for chaos in deterministic systems with noise, Physica D, 212:100--110, 2005. doi:10.1016/j.physd.2005.09.011.
• J. Guckenheimer, K. Hoffman and W. Weckesser, The forced van der Pol equation 1: The slow flow and its bifurcations, SIAM J. Applied Dynamical Systems, 2(1):1--35, 2003. doi:10.1137/S1111111102404738.
• H. Hayashi, S. Ishizuka, M. Ohta and K. Hirakawa, Chaotic behavior in the onchidium giant neuron under sinusoidal stimulation, Physics Letters, 88A:435--438, 1982. doi:10.1016/0375-9601(82)90674-0.
• A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to to conduction and excitation in nerve,J. Physiol., 117:500--544, 1952. http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1392413
• D. Li, J. Xu, A method to determine the periodic solution of the nonlinear dynamics system, Journal of Sound and Vibration, 275:1--16, 2004. doi:10.1016/S0022-460X(03)00656-4.
• A. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics---Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science, John Wiley and Sons, 1995.
• M. Nicol, I. Melbourne and P. Ashwin, Euclidean extensions of dynamical systems, Nonlinearity, 14:275--300, 2001. doi:10.1088/0951-7715/14/2/306.
• R. Seydel, Practical Bifurcation and Stability Analysis From Equilibrium to Chaos, Interdisciplinary Applied Mathematics, Springer--Verlag, 1994.
• F. Takens, Detecting strange attractors in turbulence, Lecture Notes in Mathematics - Springer, 898:366--381, 1981.
• M. Urabe and A. Reiter, Numerical computation of nonlinear forced oscillations by Galerkin's procedure, Journal of Mathematical Analysis and Applications, 14:107--140, 1966.