I am doing bifurcation analysis using XPP-AUTO for autonomous system (many examples available in web sources). But in the case of non autonomous system, particularly forced duffing oscillator (x ̈ + b x ̇ + ωx + βx 3=fsin(wt)), how to I do the bifurcation analysis. Is there any calculations in the numeric area which is found in the XPP-AUTO workspace.
I have never used XPPAUTO, but in Auto the system should have following form:
x = u[2];
y = u[3];
ss = x * x + y * y;
f[0] = u[1];
f[1] = -b*y[1]-omega*y[0]-beta*y[0]*y[0]*y[0]+f*x
f[2] = x + w * y - x * ss;
f[3] = -w * x + y - y * ss;
w -frequency of exciation
f - amplitude of excitation
The above system is now autonomous.
as a initial condition:
tpi = atan(1.) * 8;
par[10] = tpi / bet;
u[0] = 0; u[1] = 0;
u[2] = sin(tpi * t); u[3] = cos(tpi * t);
or if you want to follow existing period solution, just start from this solution.
one more addition: f[2] and f[3] is a real form of Stuart-Landau system. This system has stable harmonic solution with frequency w and amplitude equal to unity.
I don't know this software, but as others equivalent with a large spectrum of operations, it is necessary to consider them as a first rough step for bifurcation studies, which imply more profound analyses. Indeed bifurcation phenomena have generally a sensitive dependence with respect to parameters, and to the ODE discretization method, the integration being frequently ill conditioned, with low integration accuracy the results are meaningless, with high integration accuracy the solution evolves too slowly. So numerical studies must be accompanied by a different approach, especially for this ODE generating harmonics of increasing rank, sub-harmonics, and fractional harmonics. For your problem, as a first step, I recommend this excellent paper dealing with periodic solutions for a Duffing's equation:
Hiroshi Kawakami "Bifurcation of periodic responses in forced dynamic nonlinear circuits: computation of bifurcation values of the system parameters". IEEE Transactions on circuits and systems. Vol. CAS-31, n° 3, March 1983.
Moreover, from the choice of two parameters, the parameter plane is foliated (i.e. made up of sheets associated with a well defined periodic solution). When the other parameters vary the bifurcation curves structure of this plane can undergo qualitative changes. This point escapes to too ambitious software operating "automatically", not permitting to detect regions corresponding to several stable solutions. An example of such an approach is given by the two following papers based on the understanding of the parameter plane foliation:
Article: Behavior of Harmonics Generated by a Duffing Type equation with a Nonlinear Damping: Part I. Hédi Khammari, Christian Mira, Jean-Pierre Carcassés . International Journal of Bifurcation and Chaos 01/2005; 15:3181-3221.
Behavior of Harmonics Generated by a Duffing Type equation with a Nonlinear Damping: Part II.
Hédi Khammari, Christian Mira. International Journal of Bifurcation and Chaos 01/2009; 19:1227-1254.
see: http://eng.ikar.udm.ru/sb/sb22e.htm
http://eng.ikar.udm.ru/files/zip/sb22e.zip
3. Resonance in nonlinear systems.
3.1. Simple computational method for nonlinear dynamic systems.
3.2. About Kapica pendulum beyond and within of parametric resonance zone.
3.3. Dynamic stability of saddles point in autonomic systems.
3.4. About stability of unstable states, bifurcation, chaos of nonlinear dynamic systems.
3.5. Discreteness, chaos and evolution in nonlinear dynamic systems.
Another paper dealing with bifurcation analysis of the non autonomous Duffing equation (approach via identification of the foliated parameter plane):
"Bifurcation Structures Generated by the Nonautonomous Duffing Equation" C. Mira, M. Touzani-Qriouet, H. Kawakami. International Journal of Bifurcation and Chaos 01/1999; 9(07):1363-137.
See the attached file
Dear,
I suggest you "Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students"
http://books.google.co.in/books?id=Qg8ubxrA060C&printsec=frontcover#v=onepage&q&f=false
I suggest using Knut instead, there you don't need any tricks as it can handle non-autonomous systems natively.
https://gitorious.org/knut/pages/Home
Standart approach:
dx/dt = y; dy/dt = -by - ωx - βx^3 + f*sin(z); dz/dt =w, then pass to polar coordinates, looking for resonances (nonzero or jumps in time average of rhs), separation upon fast and slow motions, analytical asymptotics near zero, then numerical Poincare maps or Lie series analytics at large amplitudes, if need
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