# Bifurcation Flight Dynamic Analysis of a Strake-Wing Micro Aerial Vehicle
## 2 Bifurcation and Continuation Analysis
### Continuation Software  
Numerical methods for solving bifurcation problems appeared relatively recently and complement the analytical achievements in this field. The following fundamental difficulties encountered when applying these methods can be distinguished:  
Figure 1: Unit circle—diagram showing basic periodic orbit bifurcations—periodic orbit bifurcations. 1. The actual eigenvalue exceeds \(+1\). Periodic limit points appear in this case. 2. The actual eigenvalue exceeds \(-1\). Period doubling bifurcation occurs in this case. In the proximity of this point, a stable periodic orbit with a period of \(T\) becomes unstable and a new stable periodic orbit, with a period of \(2T\) appears. This type of stability loss leads to chaotic motions. 3. Two conjugated eigenvalues leave the unit circle. Stable or unstable trajectories surround an unstable bifurcation orbit [32].  
numerical instabilities associated with calculations in close proximity to bifurcation points,
* issues related to parametrization in close proximity to bifurcation points and limit points,
* structures of bifurcation branches,
* determination whether bifurcation actually takes place,
* problems associated with the convergence of the Newton-Raphson method at singular points.  
The system of Equation (2) can have numerous solutions. These can be isolated solutions and they may not exist at all. There is no theory, which could be used as a base to determine which case you are dealing with. Therefore, this issue is quite challenging, because when using numerical methods, the question whether all solutions have already been found still stands. Among the many methods, the ones applied the most when the \(\mathbf{f}\) functions are smooth is the Newton-Raphson method.  
At the time, there are several software packages available, which are designed to analyse non-linear dynamical systems, e.g., MatCont [33, 34] or KRIT [35]. The most recognizable computer program designed for bifurcation analysis of non-linear dynamical systems is the one developed at the Canadian Concordia University, by a team led by Prof. [PERSON], the AOTO97 package--program developed in the FORTRAN language and AUTO2000--developed in the C language. The latest (as of 2021) version of these packages is the AUTO07P [36]. The description of subsequent versions of the AUTO package can be found in: [37, 38, 39]. AUTO07P was developed in the FORTRAN language, for the UNIX operating system environment. A team at the University of Pittsburgh in America, led by prof. [PERSON], developed XPPAUT [40], which is a version of the AUTO package compatible with the WINDOWS system. A comprehensive description of this package, together with a manual, can be found in the textbook by prof. [PERSON] [41]. A MATLAB system toolbox--_Dynamical System Toolbox_[42] was also created based on the AUTO [43].  
The best-known program designed for the bifurcation analysis of homogeneous ordinary differential equations is that developed by a team lead by Prof. [PERSON] from the Canadian Concordia University called AUTO. Differential equations describing MAV flight dynamics were analysed using an AUTO version implemented in MATLAB (Dynamical System Toolbox [42]).