## LEs of autonomous continuous systems of fractional order

version 2.0.3 (2.99 KB) by
Program to compute LEs of autonomous continuous systems of commensurate Caputo's Fractional Order

Updated 8 Jul 2022

**slight changes**
The program, determines the Lyapunov Exponents (LEs) of a commensurate Fractional Order (FO) autonomous continuous-time system modeled by Capto's derivative  (for non-commensurate order see )
Because there are continuous citations of the paper presenting this program for Lyapunov Exponents (LEs) and due to some (minor) errors existing in the codes, here there are the slightly improved variants of related codes
Thus, the names of the several used codes are synchronized so that the use of programs is easier now
Note that there are the following possibilities:
For commensurate order, all programs uses FDE12.m to integrate the FDEs, https://goo.gl/XScYmi, and the function containing the extended system (see e.g. LE_RF.m or LE_RF_p.m), which have to be in the same folder
1. The program, FO_Lyapunov.m, determine LEs of the (extended) system (see e.g. LE_RF.m) as function of time.
2.1 A modified variant of FO_Lyapunov (FO_Lyapunov_q.m) determines the LEs of a system (see e.g. LE_RF.m) for a given value of q.
2.2 FO_Lyapunov_q.m can be also used to obtain the variation of LEs as function of q. For this purpose it is used by run_FO_LE_q.m
3.1 Another modified variant FO_Lyapunov (FO_Lyapunov_p.m) determines the LEs of a system (see e.g. LE_RF_p.m) for a given value of p.
3.2 FO_Lyapunov_p.m can be used to obtain the variation of LEs as function of p. For this purpose it is used by run_LE_FO_p.m
As mentioned in , 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)
For non-commensurate order see FO_NC_Lyapunov.m which uses another program to integrate non-commensurate FDEs (see ) .
 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
 Marius.-F. Danca, Matlab code for Lyapunov exponents of fractional-order systems, Part II: The non-commensurate case, International Journal of Bifurcation and Chaos, 31(12), 2150187, (2021)

### Cite As

Marius-F. Danca (2022). LEs of autonomous continuous systems of fractional order (https://www.mathworks.com/matlabcentral/fileexchange/114595-les-of-autonomous-continuous-systems-of-fractional-order), MATLAB Central File Exchange. Retrieved .

 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

##### MATLAB Release Compatibility
Created with R2022a
Compatible with any release
##### Platform Compatibility
Windows macOS Linux