FO_Lyapunov_p

Program to compute LEs as function of p of autonomous systems of commensurate Caputo's Fractional Order

August 2023: bugs fixed in n_it and ref for noncommensurate case added

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August 2022: plot modified to overcome the problems with plot function in the last matlab variants

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**slight chnages**

The program, FO_Lyapunov_p, can be used either alone to determined the LEs of a FO system for a fixed fractional order p (see e.g. LE_RF_p.m which contains the extended system), or can be used to obtain the evolution of LEs as function of p, case when the code run_FO_LE_p must be used.

1. To obtain the LEs for a given p, one uses

LE=FO_Lyapunov_p(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q,p);

For example, for the RF system [1]

LE=FO_Lyapunov_p(3,@LE_RF_p,0,0.02,200,[0.1;0.1;0.1],0.02,0.998,1.3)

2. If one intends to obtain the evolution of LEs as function of p one uses

run_LE_FO_p(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q,p_min,p_max,n);

E.g., for the same system RF

run_LE_FO_p(3,@LE_RF_p,0,0.02,200,[0.1;0.1;0.1],0.02,0.998,1.1,1.3,800)

Note that FO_Lyapunov_p.m, LE_RF_p.m, run_FO_LE_p and FDE12.m (used to integrate the system) must be in the same folder.

As mentioned in [1], the relation between h_norm and h is essential. Here both are chosen equal (0.02), but multiple of h for h_norm should be tried (e.g. h=0.002 and h_norm=0.2, but to the detriment of computational time)

[1] Marius-F. Danca and N. Kuznetsov, Matlab code for Lyapunov exponents of fractional order systems, International Journal of Bifurcation and Chaos, 28(05)(2018), 1850067