### 金融代写|金融工程作业代写Financial Engineering代考|Best107

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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
• Statistical Machine Learning 统计机器学习
• 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|Characteristics of the Dynamics of Nonlinear Systems

Main features characterizing the stability of nonlinear dynamical systems are defined as follows [121, 274]:

1. Finite escape time: It is the finite time within which the state-vector of the nonlinear system converges to infinity.
2. Multiple isolated equilibria: A linear system can have only one equilibrium to which converges the state vector of the system in steady-state. A nonlinear system can have more than one isolated equilibria (fixed points). Depending on the initial state of the system, in steady-state the state vector of the system can converge to one of these equilibria.
3. Limit cycles: For a linear system to exhibit oscillations it must have eigenvalues on the imaginary axis. The amplitude of the oscillations depends on initial conditions. In nonlinear systems one may have oscillations of constant amplitude and frequency, which do not depend on initial conditions. This type of oscillations is known as limit cycles.
4. Sub-harmonic, harmonic and almost periodic oscillations: A stable linear system under periodic input produces an output of the same frequency. A nonlinear system,

under periodic excitation can generate oscillations with frequencies which are several times smaller (subharmonic) or multiples of the frequency of the input (harmonic). It may also generate almost periodic oscillations with frequencies which are not necessarily multiples of a basis frequency (almost periodic oscillations).

1. Chaos: A nonlinear system in steady-state can exhibit a behavior which is not characterized as equilibrium, periodic oscillation or almost periodic oscillation. This behavior is characterized as chaos. As time advances the behavior of the system changes in a random-like manner, and this depends on the initial conditions. Although the dynamic system is deterministic it exhibits randomness in the way it evolves in time.
2. Multiple modes of behavior: It is possible the same dynamical system to exhibit simultaneously more than one of the aforementioned characteristics (1)-(5). Thus, a system without external excitation may exhibit simultaneously more than one limit cycles. A system receiving a periodic external input may exhibit harmonic or subharmonic oscillations, or an even more complex behavior in steady state which depends on the amplitude and frequency of the excitation.

## 金融代写|金融工程作业代写Financial Engineering代考|Computation of Isoclines

An autonomous second order system is described by two differential equations of the form
\begin{aligned} & \dot{x}_1=f_1\left(x_1, x_2\right) \ & \dot{x}_2=f_2\left(x_1, x_2\right) \end{aligned}
The method of the isoclines consists of computing the slope (ratio) between $f_2$ and $f_1$ for every point of the trajectory of the state vector $\left(x_1, x_2\right)$.
$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)}$$
The case $s(x)=c$ describes a curve in the $x_1-x_2$ plane along which the trajectories $\dot{x}_1=f_1\left(x_1, x_2\right)$ and $\dot{x}_2=f_2\left(x_1, x_2\right)$ have a constant slope.

The curve $s(x)=c$ is drawn in the $x_1-x_2$ plane and along this curve one also draws small linear segments of length $c$. The curve $s(x)=c$ is known as isocline. The direction of these small linear segments is according to the sign of the ratio $f_2\left(x_1, x_2\right) / f_1\left(x_1, x_2\right)$.
Example 1:
The following simplified nonlinear dynamical system is considered
$$\begin{gathered} \dot{x}_1=x_2 \ \dot{x}_2=-\sin \left(x_1\right) \end{gathered}$$

The slope $s(x)$ is given by the relation
$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)} \Rightarrow s(x)=-\frac{s i n\left(x_2\right)}{x_2}$$
Setting $s(x)=c$ it holds that the isoclines are given by the relation
$$x_2=-\frac{1}{c} \sin \left(x_1\right)$$
For different values of $c$ one has the following isoclines diagram depicted in Fig. 1.1. Example 2:

The following oscillator model is considered, being free of friction and with statespace equations
$$\begin{gathered} \dot{x}_1=x_2 \ \dot{x}_2=-0.5 x_2-\sin \left(x_1\right) \end{gathered}$$

## 金融代写|金融工程作业代写Financial Engineering代考|Characteristics of the Dynamics of Nonlinear Systems

1. 有限逃逸时间：是非线性系统的状态向量收敛到无穷大的有限时间。
2. 多重孤立平衡：一个线性系统只能有一个平衡点，该平衡点收敛于稳态系统的状态向量。一个非线性系统可以有多个孤立的平衡点（不动点）。根据系统的初始状态，在稳态下，系统的状态向量可以收敛到这些平衡点之一。
3. 极限环：对于表现出振荡的线性系统，它必须在虚轴上具有特征值。振荡的幅度取决于初始条件。在非线性系统中，可能存在幅度和频率恒定的振荡，不依赖于初始条件。这种类型的振荡被称为极限环。
4. 次谐波、谐波和几乎周期性振荡：在周期性输入下稳定的线性系统会产生相同频率的输出。一个非线性系统，

1. 混沌：处于稳态的非线性系统可能表现出一种不具有平衡、周期性振荡或几乎周期性振荡特征的行为。这种行为的特点是混乱。随着时间的推移，系统的行为会以类似随机的方式发生变化，这取决于初始条件。尽管动态系统是确定性的，但它随时间演化的方式表现出随机性。
2. 多种行为模式：同一动力系统可能同时表现出上述特征 (1)-(5) 中的一个以上。因此，没有外部激励的系统可能同时表现出多个极限循环。接收周期性外部输入的系统可能会表现出谐波或次谐波振荡，或者在稳态下表现出更复杂的行为，这取决于激励的幅度和频率。

## 金融代写|金融工程作业代写Financial Engineering代考|Computation of Isoclines

$$\dot{x}_1=f_1\left(x_1, x_2\right) \quad \dot{x}_2=f_2\left(x_1, x_2\right)$$

$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)}$$

$$\dot{x}_1=x_2 \dot{x}_2=-\sin \left(x_1\right)$$

$$s(x)=\frac{f_2\left(x_1, x_2\right)}{f_1\left(x_1, x_2\right)} \Rightarrow s(x)=-\frac{\sin \left(x_2\right)}{x_2}$$

$$x_2=-\frac{1}{c} \sin \left(x_1\right)$$

$$\dot{x}_1=x_2 \dot{x}_2=-0.5 x_2-\sin \left(x_1\right)$$

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