数学代写|常微分方程代写ordinary differential equation代考|MATH289

数学代写|常微分方程代写ordinary differential equation代考|Periodic Orbits of Linear Systems

In this section we will consider the existence and stability of periodic solutions of the time-periodic system
$$\dot{x}=A(t) x+b(t), \quad x \in \mathbb{R}^{\bar{n}}$$
where $t \mapsto A(t)$ is a $T$-periodic matrix function and $t \mapsto b(t)$ is a $T$-periodic vector function.

Theorem 2.76. If the number one is not a characteristic multiplier of the $T$-periodic homogeneous system $\dot{x}=A(t) x$, then (2.33) has at least one T-periodic solution.

Proof. Let us show first that if $t \mapsto x(t)$ is a solution of system (2.33) and $x(0)=x(T)$, then this solution is $T$-periodic. Define $y(t):=x(t+T)$. Note that $t \mapsto y(t)$ is a solution of $(2.33)$ and $y(0)=x(0)$. Thus, by the uniqueness theorem $x(t+T)=x(t)$ for all $t \in \mathbb{R}$.

If $\Phi(t)$ is the principal fundamental matrix solution of the homogeneous system at $t=0$, then, by the variation of constants formula,
$$x(T)=\Phi(T) x(0)+\Phi(T) \int_0^T \Phi^{-1}(s) b(s) d s .$$

Therefore, $x(T)=x(0)$ if and only if
$$(I-\Phi(T)) x(0)=\Phi(T) \int_0^T \Phi^{-1}(s) b(s) d s .$$
This equation for $x(0)$ has a solution whenever the number one is not an eigenvalue of $\Phi(T)$. (Note that the $\operatorname{map} x(0) \mapsto x(T)$ is the Poincaré map. Thus, our periodic solution corresponds to a fixed point of the Poincaré map).

By Floquet’s theorem, there is a matrix $B$ such that the monodromy matrix is given by
$$\Phi(T)=e^{T B} .$$
In other words, by the hypothesis, the number one is not an eigenvalue of $\Phi(T)$

数学代写|常微分方程代写ordinary differential equation代考|Stability of Periodic Orbits

Consider a (nonlinear) autonomous system of differential equations on $\mathbb{R}^n$ given by $\dot{u}=f(u)$ with a periodic orbit $\Gamma$. Also, for each $\xi \in \mathbb{R}^n$, define the vector function $t \mapsto u(t, \xi)$ to be the solution of this system with the initial condition $u(0, \xi)=\xi$.

If $p \in \Gamma$ and $\Sigma^{\prime} \subset \mathbb{R}^n$ is a section transverse to $f(p)$ at $p$, then, as a corollary of the implicit function theorem, there is an open set $\Sigma \subseteq \Sigma^{\prime}$ and a function $T: \Sigma \rightarrow \mathbb{R}$, the time of first return to $\Sigma^{\prime}$, such that for each $\sigma \in \Sigma$, we have $u(T(\sigma), \sigma) \in \Sigma^{\prime}$. The map $\mathcal{P}$, given by $\sigma \mapsto u(T(\sigma), \sigma)$, is the Poincaré map corresponding to the Poincaré section $\Sigma$.

The Poincaré map is defined only on $\Sigma$, a manifold contained in $\mathbb{R}^n$. It is convenient to avoid choosing local coordinates on $\Sigma$. Thus, we will view the elements in $\Sigma$ also as points in the ambient space $\mathbb{R}^n$. In particular, if $v \in \mathbb{R}^n$ is tangent to $\Sigma$ at $p$, then the derivative of $\mathcal{P}$ in the direction $v$ is given by
$$D \mathcal{P}(p) v=(d T(p) v) f(p)+u_{\xi}(T(p), p) v .$$
The next proposition relates the spectrum of $D \mathcal{P}(p)$ to the Floquet multipliers of the first variational equation
$$\dot{W}=D f(u(t, p)) W .$$

常微分方程代考

数学代写|常微分方程代写ordinary differential equation代考|Periodic Orbits of Linear Systems

$$\dot{x}=A(t) x+b(t), \quad x \in \mathbb{R}^{\bar{n}}$$

$$x(T)=\Phi(T) x(0)+\Phi(T) \int_0^T \Phi^{-1}(s) b(s) d s .$$

$$(I-\Phi(T)) x(0)=\Phi(T) \int_0^T \Phi^{-1}(s) b(s) d s .$$

$$\Phi(T)=e^{T B} .$$

数学代写|常微分方程代写ordinary differential equation代考|Stability of Periodic Orbits

$$D \mathcal{P}(p) v=(d T(p) v) f(p)+u_{\xi}(T(p), p) v .$$

$$\dot{W}=D f(u(t, p)) W$$

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