# Bifurcation Flight Dynamic Analysis of a Strake-Wing Micro Aerial Vehicle
## 2 Bifurcation and Continuation Analysis
### Bifurcation Theory and Bifurcation Types
Due to the fact that for all \(\mathbf{\mu}\) the Equation (2) is satisfied, and Equation (1) can be expressed as:  
\[\dot{\mathbf{x}}=\mathbf{R}\mathbf{x}+\sigma(\mathbf{x},\mathbf{\mu}) \tag{4}\]where in **R** is a square characteristic matrix (Jacobian matrix) with elements provided by the equation:  
\[\left[\textbf{R}\right]_{i,j}=\frac{\partial f_{i}(\textbf{x}_{0},\textbf{\mu})} {\partial x_{j}}, \tag{5}\]  
and non-linear vector function \(\boldsymbol{\sigma}\) fulfils the conditions:  
\[\boldsymbol{\sigma}(\textbf{x}_{0},\textbf{\mu})=0,\frac{\partial\sigma_{i}( \textbf{x}_{0},\textbf{\mu})}{\partial x_{j}}=0. \tag{6}\]  
and finally:  
\[\dot{\textbf{x}}=\textbf{R}\textbf{x} \tag{7}\]  
The Hartman-Grobman theorem applies within the process of studying the stability of a stationary solutions to equation [15; 16; 19; 20]. It states that if all eigenvalues of a Jacobian matrix **R** of a linearized system (1) lie in the left complex half-plane, i.e.,  
\[\text{Re}\big{(}\lambda_{j}\big{)}<0,i=1,2,\ldots,n \tag{8}\]  
then there is a certain continuous, homomorphic transformation of variables reducing a locally non-linear system of Equation (1) to a linear system. This means that if a stationary solution to a linearized system of equations is asymptotically stable, then also the solution to a non-linear system is stable. The Hartman-Grobman theorem also indicates that every qualitative change in the nature of solutions to a system of non-linear equations describing a dynamical system is indicated by the appearance of zero real parts of the eigenvalues of a linearized system characteristic matrix **R**. In mathematical terms, bifurcation of equilibrium positions takes place, when a eigenvalue of Jacobi matrix (3), (5) of system (1), estimated in the state of equilibrium, intersects the imaginary axis. Similarly, in the case of an oscillating solution, bifurcation occurs when the Floquet multiplier intersects the unit circle. The results shown in this article discuss five bifurcation types; they all have a codification of one, which means that they are encountered when a single continuation parameter changes. Basic bifurcation types are [32]:  
A. A saddle-node bifurcation, also called a saddle limit point, occurs when the real eigenvalue of a Jacobian matrix (5) estimated in a state of equilibrium, intersects the imaginary axis. There is no equilibrium on one side of the bifurcation point (locally), whereas there are two equilibrium branches on the other (e.g., one stable and one unstable) (Figure 1). Hopf bifurcation occurs, when a complex pair of Jacobian (5) eigenvalues, assessed at equilibrium, intersect the imaginary axis. In this case, the equilibrium changes stability and a periodic orbit is formed, which can be stable or unstable (Figure 1). The limit point or periodic orbit fold bifurcation occur when the real Floquet multiplier intersects the unit circle at \(1\); as for the states of equilibrium, then there are no periodic orbits on one side of the bifurcation (locally), whereas there are two periodic orbits on the other (Figure 1). A. A period-doubling bifurcation occurs when the real Floquet multiplier intersects the unit circle at \(-1\). The periodic orbit loses stability when a new period orbit appears with a period (approximately) twice as long (Figure 1). The Neimark-Sacker bifurcation or torus bifurcation appears, when the periodic orbit becomes unstable, namely, when a pair of complex Floquet multipliers intersects the unit circle and an additional oscillation frequency is introduced. The outcome is a torus dynamic, which can be periodic (blocked) or quasi-periodic (Figure 1). To sum up, it can be concluded that:  
The bifurcation theory is a set of mathematical results aimed at analysing and explaining unexpected modifications in the asymptotic behaviour of non-linear systems of differential equations, when their parameters slowly change.