### 金融代写|金融工程作业代写Financial Engineering代考|BE953

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|Phase Diagrams for Linear Dynamical Systems

The following autonomous linear system is considered
$$\dot{x}=A x$$
The eigenvalues of matrix $A$ define the system dynamics. Some terminology associated with fixed points is as follows:

A fixed point for the system of Eq. (1.27) is called hyperbolic if none of the eigenvalues of matrix $A$ has zero real part. A hyperbolic fixed point is called a saddle if some of the eigenvalues of matrix $A$ have real parts greater than zero and the rest of the eigenvalues have real parts less than zero. If all of the eigenvalues have negative real parts then the hyperbolic fixed point is called a stable node or sink. If all of the eigenvalues have positive real parts then the hyperbolic fixed point is called an unstable node or source. If the eigenvalues are purely imaginary then one has an elliptic fixed point which is said to be a center.
Case 1: Both eigenvalues of matrix $A$ are real and unequal, that is $\lambda_1 \neq \lambda_1 \neq 0$. For $\lambda_1<0$ and $\lambda_2<0$ the phase diagram for $z_1$ and $z_2$ is shown in Fig. 1.4. In case that $\lambda_2$ is smaller than $\lambda_1$ the term $e^{\lambda_2 t}$ decays faster than $e^{\lambda_1 t}$. For $\lambda_1>0>\lambda_2$ the phase diagram of Fig. $1.5$ is obtained.

In the latter case there are stable trajectories along eigenvector $v_1$ and unstable trajectories along eigenvector $v_2$ of matrix $A$. The stability point $(0,0)$ is said to be a saddle point.
When $\lambda_1>\lambda_2>0$ then one has the phase diagrams of Fig. 1.6.
Case 2: Complex eigenvalues:
Typical phase diagrams in the case of stable complex eigenvalues are given in Fig. 1.7.
Typical phase diagrams in the case of unstable complex eigenvalues are given in Fig. 1.8.

Typical phase diagrams in the case of imaginary eigenvalues are given in Fig. 1.9.
Case 3: Matrix $A$ has nonzero eigenvalues which are equal to each other. The associated phase diagram is given in Fig. 1.10.

## 金融代写|金融工程作业代写Financial Engineering代考|Saddle-Node Bifurcations of Fixed Points

The considered dynamical system is given by $\dot{x}=\mu-x^2$. The fixed points of the system result from the condition $\dot{x}=0$ which for $\mu>0$ gives $x^*=\pm \sqrt{\mu}$. The first fixed point $x=\sqrt{\mu}$ is a stable one whereas the second fixed point $x=-\sqrt{\mu}$ is an unstable one. The phase diagram of the system is given in Fig. 1.14. Since there is one stable and one unstable fixed point the associated bifurcation (locus of the fixed points in the phase plane) will be a saddle-node one.

The bifurcations diagram is given next. The diagram shows how the fixed points of the dynamical system vary with respect to the values of parameter $\mu$. In the above case it represents a parabola in the $\mu-x$ plane as shown in Fig. 1.15.

For $\mu>0$ the dynamical system has two fixed points located at $\pm \sqrt{\mu}$. The one fixed point is stable and is associated with the upper branch of the parabola. The other fixed point is unstable and is associated with the lower branch of the parabola. The value $\mu=0$ is considered to be a bifurcation value and the point $(x, \mu)=(0,0)$ is a bifurcation point. This particular type of bifurcation where the one branch is associated with fixed points and the other branch is not associated to any fixed points is known as saddle-node bifurcation.

In pitchfork bifurcations the number of fixed points varies with respect to the values of the bifurcation parameter. The dynamical system $\dot{x}=x\left(\mu-x^2\right)$ is considered. The associated fixed points are found by the condition $\dot{x}=0$. For $\mu<0$ there is one fixed point at zero which is stable. For $\mu=0$ there is still one fixed point at zero which is still stable. For $\mu>0$ there are three fixed points, one at $x=0$, one at $x=+\sqrt{\mu}$ which is stable and one at $x=-\sqrt{\mu}$ which is also stable. The associated phase diagrams and fixed points are presented in Fig. 1.16.

The bifurcations diagram is given next. The diagram shows how the fixed points of the dynamical system vary with respect to the values of parameter $\mu$. In the above case it represents a parabola in the $\mu-x$ plane as shown in Fig. 1.17.

## 金融代写|金融工程作业代写Financial Engineering代考|Phase Diagrams for Linear Dynamical Systems

$$\dot{x}=A x$$

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