Posted on Categories:数学代写, 非线性动力系统

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## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|The damped driven pendulum

Another example is that of the damped and driven harmonic oscillator,
$$\frac{d^2 \phi}{d s^2}+\gamma \frac{d \phi}{d s}+\sin \phi=j .$$
This is equivalent to a model of a resistively and capacitively shunted Josephson junction, depicted in fig. 3.3. If $\phi$ is the superconducting phase difference across the junction, the current through the junction is given by $I_J=I_{\mathrm{c}} \sin \phi$, where $I_{\mathrm{c}}$ is the critical current. The current carried by the resistor is $I_R=V / R$ from Ohm’s law, and the current from the capacitor is $I_C=\dot{Q}$. Finally, the Josephson relation relates the voltage $V$ across the junction to the superconducting phase difference $\phi: V=(\hbar / 2 e) \dot{\phi}$. Summing up the parallel currents, we have that the total current $I$ is given by
$$I=\frac{\hbar C}{2 e} \ddot{\phi}+\frac{\hbar}{2 e R} \dot{\phi}+I_{\mathrm{c}} \sin \phi,$$
which, again, is equivalent to a damped, driven pendulum.
This system also has a mechanical analog. Define the ‘potential’
$$U(\phi)=-I_{\mathrm{c}} \cos \phi-I \phi .$$
The equation of motion is then
$$\frac{\hbar C}{2 e} \ddot{\phi}+\frac{\hbar}{2 e R} \dot{\phi}=-\frac{\partial U}{\partial \phi} .$$

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Classification of N = 2 fixed points

Suppose we have solved the fixed point equations $V_x\left(x^, y^\right)=0$ and $V_y\left(x^, y^\right)=0$. Let us now expand about the fixed point, writing
\begin{aligned} & \dot{x}=\left.\frac{\partial V_x}{\partial x}\right|{\left(x^, y^\right)}\left(x-x^\right)+\left.\frac{\partial V_x}{\partial y}\right|{\left(x^, y^\right)}\left(y-y^\right)+\ldots \ & \dot{y}=\left.\frac{\partial V_y}{\partial x}\right|{\left(x^, y^\right)} ^{\left(x-x^\right)}+\left.\frac{\partial V_y}{\partial y}\right|{\left(x^, y^\right)} ^{\left(y-y^\right)}+\ldots . \ & \end{aligned}
We define
$$u_1=x-x^* \quad, \quad u_2=y-y^*,$$
which, to linear order, satisfy
The formal solution to $\dot{\boldsymbol{u}}=\boldsymbol{M} \boldsymbol{u}$ is
$$\boldsymbol{u}(t)=\exp (M t) \boldsymbol{u}(0),$$

where $\exp (M t)=\sum_{n=0}^{\infty} \frac{1}{n !}(M t)^n$ is the exponential of the matrix $M t$.
The behavior of the system is determined by the eigenvalues of $M$, which are roots of the characteristic equation $P(\lambda)=0$, where
\begin{aligned} P(\lambda) & =\operatorname{det}(\lambda \mathbb{I}-M) \ & =\lambda^2-T \lambda+D, \end{aligned}
with $T=a+d=\operatorname{Tr}(M)$ and $D=a d-b c=\operatorname{det}(M)$. The two eigenvalues are therefore
$$\lambda_{\pm}=\frac{1}{2}\left(T \pm \sqrt{T^2-4 D}\right) .$$

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|The damped driven pendulum

$$\frac{d^2 \phi}{d s^2}+\gamma \frac{d \phi}{d s}+\sin \phi=j .$$

$$I=\frac{\hbar C}{2 e} \ddot{\phi}+\frac{\hbar}{2 e R} \dot{\phi}+I_{\mathrm{c}} \sin \phi,$$

$$U(\phi)=-I_{\mathrm{c}} \cos \phi-I \phi .$$

$$\frac{\hbar C}{2 e} \ddot{\phi}+\frac{\hbar}{2 e R} \dot{\phi}=-\frac{\partial U}{\partial \phi} .$$

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Classification of $\mathrm{N}=2$ fixed points

$$u_1=x-x^* \quad, \quad u_2=y-y^*,$$

$$\boldsymbol{u}(t)=\exp (M t) \boldsymbol{u}(0),$$

$$P(\lambda)=\operatorname{det}(\lambda \mathbb{I}-M) \quad=\lambda^2-T \lambda+D,$$

$$\lambda_{\pm}=\frac{1}{2}\left(T \pm \sqrt{T^2-4 D}\right) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 非线性动力系统

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## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Pitchfork bifurcation

The pitchfork bifurcation is commonly encountered in systems in which there is an overall parity symmetry $(u \rightarrow-u)$. There are two classes of pitchfork: supercritical and subcritical. The normal form of the supercritical bifurcation is
$$\dot{u}=r u-u^3,$$
which has fixed points at $u^=0$ and $u^=\pm \sqrt{r}$. Thus, the situation is as depicted in fig. $2.4$ (top panel). For $r<0$ there is a single stable fixed point at $u^=0$. For $r>0, u^=0$ is unstable, and flanked by two stable fixed points at $u^*=\pm \sqrt{r}$.

If we send $u \rightarrow-u, r \rightarrow-r$, and $t \rightarrow-t$, we obtain the subcritical pitchfork, depicted in the bottom panel of fig. 2.4. The normal form of the subcritical pitchfork bifurcation is
$$\dot{u}=r u+u^3 .$$
The fixed point structure in both supercritical and subcritical cases is shown in Fig. 2.5.

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Imperfect bifurcation

The imperfect bifurcation occurs when a symmetry-breaking term is added to the pitchfork. The normal form contains two control parameters:
$$\dot{u}=h+r u-u^3 \text {. }$$
Here, the constant $h$ breaks the parity symmetry if $u \rightarrow-u$.
This equation arises from a crude model of magnetization dynamics. Let $M$ be the magnetization of a sample, and $F(M)$ the free energy. Assuming $M$ is small, we can expand $F(M)$ as
$$F(M)=-H M+\frac{1}{2} a M^2+\frac{1}{4} b M^4+\ldots,$$
where $H$ is the external magnetic field, and $a$ and $b$ are temperature-dependent constants. This is called the Landau expansion of the free energy. We assume $b>0$ in order that the minimum of $F(M)$ not lie at infinity. The dynamics of $M(t)$ are modeled by
$$\frac{d M}{d t}=-\Gamma \frac{\partial F}{\partial M},$$
with $\Gamma>0$. Thus, the magnetization evolves toward a local minimum in the free energy. Note that the free energy is a decreasing function of time:
$$\frac{d F}{d t}=\frac{\partial F}{\partial M} \frac{d M}{d t}=-\Gamma\left(\frac{\partial F}{\partial M}\right)^2 .$$

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Pitchfork bifurcation

$$\dot{u}=r u-u^3,$$

$$\dot{u}=r u+u^3$$

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Imperfect bifurcation

$$\dot{u}=h+r u-u^3 .$$

$$F(M)=-H M+\frac{1}{2} a M^2+\frac{1}{4} b M^4+\ldots$$

$$\frac{d M}{d t}=-\Gamma \frac{\partial F}{\partial M},$$

$$\frac{d F}{d t}=\frac{\partial F}{\partial M} \frac{d M}{d t}=-\Gamma\left(\frac{\partial F}{\partial M}\right)^2 .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 非线性动力系统

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Linear differential equations

A homogeneous linear $N^{\text {th }}$ order $\mathrm{ODE}$,
$$\frac{d^N x}{d t^N}+c_{N-1} \frac{d^{N-1} x}{d t^{N-1}}+\ldots+c_1 \frac{d x}{d t}+c_0 x=0$$
may be written in matrix form, as
$$\frac{d}{d t}\left(\begin{array}{c} \varphi_1 \ \varphi_2 \ \vdots \ \varphi_N \end{array}\right)=\overbrace{\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & \vdots & \vdots & & \vdots \ -c_0 & -c_1 & -c_2 & \cdots & -c_{N-1} \end{array}\right)}^M\left(\begin{array}{c} \varphi_1 \ \varphi_2 \ \vdots \ \varphi_N \end{array}\right) .$$
Thus,
$$\dot{\varphi}=M \boldsymbol{\varphi},$$
and if the coefficients $c_k$ are time-independent, i.e. the ODE is autonomous, the solution is obtained by exponentiating the constant matrix $Q$ :
$$\boldsymbol{\varphi}(t)=\exp (M t) \boldsymbol{\varphi}(0)$$
the exponential of a matrix may be given meaning by its Taylor series expansion. If the ODE is not autonomous, then $M=M(t)$ is time-dependent, and the solution is given by the path-ordered exponential,
$$\varphi(t)=\mathcal{P} \exp \left{\int_0^t d t^{\prime} M\left(t^{\prime}\right)\right} \varphi(0),$$
As defined, the equation $\dot{\varphi}=\boldsymbol{V}(\boldsymbol{\varphi})$ is autonomous, since $g_t$ depends only on $t$ and on no other time variable. However, by extending the phase space from $\mathcal{M}$ to $\mathbb{R} \times \mathcal{M}$, which is of dimension $(N+1)$, one can describe arbitrary time-dependent ODEs.

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Lyapunov functions

For a general dynamical system $\dot{\varphi}=\boldsymbol{V}(\boldsymbol{\varphi})$, a Lyapunov function $L(\boldsymbol{\varphi})$ is a function which satisfies
$$\boldsymbol{\nabla} L(\boldsymbol{\varphi}) \cdot \boldsymbol{V}(\boldsymbol{\varphi}) \leq 0 .$$
There is no simple way to determine whether a Lyapunov function exists for a given dynamical system, or, if it does exist, what the Lyapunov function is. However, if a Lyapunov function can be found, then this severely limits the possible behavior of the system. This is because $L(\varphi(t))$ must be a monotonic function of time:
$$\frac{d}{d t} L(\boldsymbol{\varphi}(t))=\boldsymbol{\nabla} L \cdot \frac{d \boldsymbol{\varphi}}{d t}=\boldsymbol{\nabla} L(\boldsymbol{\varphi}) \cdot \boldsymbol{V}(\boldsymbol{\varphi}) \leq 0 .$$

Thus, the system evolves toward a local minimum of the Lyapunov function. In general this means that oscillations are impossible in systems for which a Lyapunov function exists. For example, the relaxational dynamics of the magnetization $M$ of a system are sometimes modeled by the equation
$$\frac{d M}{d t}=-\Gamma \frac{\partial F}{\partial M},$$
where $F(M, T)$ is the free energy of the system. In this model, assuming constant temperature $T, \dot{F}=F^{\prime}(M) \dot{M}=-\Gamma\left[F^{\prime}(M)\right]^2 \leq 0$. So the free energy $F(M)$ itself is a Lyapunov function, and it monotonically decreases during the evolution of the system. We shall meet up with this example again in the next chapter when we discuss imperfect bifurcations.

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Linear differential equations

$$\frac{d^N x}{d t^N}+c_{N-1} \frac{d^{N-1} x}{d t^{N-1}}+\ldots+c_1 \frac{d x}{d t}+c_0 x=0$$

$$\dot{\varphi}=M \varphi,$$

$$\varphi(t)=\exp (M t) \varphi(0)$$

\left 缺少或无法识别的分隔符

## 数学代写|非线性动力系统代写Nonlinear Dynamics代考|Lyapunov functions

$$\boldsymbol{\nabla} L(\varphi) \cdot \boldsymbol{V}(\boldsymbol{\varphi}) \leq 0$$

$$\frac{d}{d t} L(\varphi(t))=\nabla L \cdot \frac{d \varphi}{d t}=\boldsymbol{\nabla} L(\varphi) \cdot \boldsymbol{V}(\varphi) \leq 0 .$$

$$\frac{d M}{d t}=-\Gamma \frac{\partial F}{\partial M},$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。