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

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