# Bifurcation Flight Dynamic Analysis of a Strake-Wing Micro Aerial Vehicle
## 2 Bifurcation and Continuation Analysis
### Bifurcation Theory and Bifurcation Types  
In our case bifurcation analysis is applied to autonomous dynamical system of general form:  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{\mu}) \tag{1}\]  
where \(\mathbf{x}\in\Re^{n}\) is vector of \(n\) state variables, \(\mathbf{\mu}\in\Re^{m}\) is vector of \(m\) control parameters, \(\mathbf{f}\) is nonlinear vector field governing system dynamics. The bifurcation for Equation (1) is every qualitative change of the phase portrait occurring upon the passage of parameter \(\mathbf{\mu}\) through a certain point \(\mathbf{\mu}_{0}\). Point \(\mathbf{\mu}_{0}\) is called the bifurcation point for Equation (1). By establishing the set of parameters \(\mathbf{\mu}=\mathbf{\mu}_{0}\) and selecting the initial conditions \(\mathbf{x}(t=0)=\mathbf{x}_{0}\), the system of differential Equation (1) can be integrated for the selected initial condition. In this way, one can study the evolution of vector field \(\mathbf{x}\) over time. In order to thoroughly study the dynamics of the system, this process can be repeated an infinite number of times starting from different initial conditions and for an innumerable fixed combination of parameter values \(\mathbf{\mu}\). This task is exhausting and tedious. The major problem in this exercise is the selection of initial conditions, which is a non-trivial task when dealing with systems that are nonlinear. An alternative and more efficient approach to the analysis of nonlinear dynamical systems described by Equation (1) is based on the asymptotic bifurcation and continuation method. The bifurcation and continuation method begins with the calculation of steady states of the equilibrium type of the Equation (1), which comes down to solving a system of nonlinear algebraic equations:  
\[\dot{\mathbf{x}}=0\ \rightarrow\ \mathbf{f}(\mathbf{x},\mathbf{\mu})=0 \tag{2}\]  
and computing the eigenvalues of the Jacobi matrix:  
\[\mathbf{J}=\frac{\partial\mathbf{f}}{\partial\mathbf{x}} \tag{3}\]  
in each equilibrium state. The numerical scheme for solving both problems is called the continuation algorithm. It is an algorithm of the predictor-corrector type. A continuation algorithm is used to solve the system of Equation (2) as a function of a single parameter of the system \(\mathbf{\mu}\in\mathbf{\mu}\). In other words, the task comes down to determining the zeros of family \(\mathbf{f}:\Re^{n}\rightarrow\Re^{n}\), namely, to determine the solutions to stationary, non-linear algebraic Equation (2). The dimension of the \(\mathbf{\mu}\) parameter vector is called the bifurcation dimension. For a one dimensional case \(\mathbf{\mu}\equiv\mathbf{\mu}\in\Re^{1}\). The bifurcation theory concerning non-linear ordinary differential equations deals with a system of first-order differential Equation (1), which is a mathematical model of a dynamical system in an \(n\)-dimensional Euclidean space \(\Re^{n}\). If the system (1) has asymptotically stable stationary solutions \(\mathbf{x}\)=0, then for all solutions \(\mathbf{x}(0)\) belonging to this neighbourhood:  
* trajectory \(\mathbf{x}(t)\) fulfils the condition: \(|\,\mathbf{x}(t)\,|\,<\,\varepsilon\) for \(t>0\);
* \(|\,\mathbf{x}(t)\,|\,\rightarrow\,0\) for \(t\rightarrow\infty\). Solving the problem involves finding answers to the question of how a parameter change \(\mathbf{\mu}\in\mathbf{\mu}\) locally affects the neighbourhood of point \(\mathbf{x}=\mathbf{x}_{0}\).