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The stroboscopic map of the plane oscillator

Example

Let $ω > 0$ and let $F : R \times R^{2} \to R^{2}$ , $F (t, z_{1}, z_{2}) : = (z_{2}, - ω^{2} z_{1})$ , the vector field of the undamped plane oscillator. Since $F$ does not depend on $t$ , it is $P$ -periodic for every $P > 0$ ; it is continuous and locally Lipschitz in the state variable, so The stroboscopic (time- $P$ ) map of a $P$ -periodic first-order system applies for every $P > 0$ .

For $z = (z_{1}, z_{2}) \in R^{2}$ the curve

$x_{z} (t) = (z_{1} \cos ω t + \frac{z_{2}}{ω} \sin ω t, - ω z_{1} \sin ω t + z_{2} \cos ω t)$

is defined on all of $R$ and solves $x^{'} = F (t, x)$ with $x (0) = z$ . Hence $D_{P} = R^{2}$ for every $P > 0$ , and the stroboscopic map is

$θ_{P} (z) = (z_{1} \cos ω P + \frac{z_{2}}{ω} \sin ω P, - ω z_{1} \sin ω P + z_{2} \cos ω P) .$

In particular, for $P = 2 π / ω$ the map $θ_{P}$ is the identity of $R^{2}$ , so every initial state is a fixed point; for $P = π / ω$ it is $θ_{P} (z) = - z$ , whose only fixed point is $0$ .

Facts & Assumptions

Given: A number $ω > 0$ and the field $F (t, z_{1}, z_{2}) = (z_{2}, - ω^{2} z_{1})$ .

[L1]

The stroboscopic map of a continuous $P$ -periodic field, locally Lipschitz in the state variable, is defined on $D_{P} = {z : [0, P] \subseteq J_{z}}$ by $θ_{P} (z) = x_{z} (P)$ , where $x_{z}$ is the unique maximal solution through $(0, z)$ ; every solution through $(0, z)$ is a restriction of $x_{z}$ (The stroboscopic (time- $P$ ) map of a $P$ -periodic first-order system, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval).

[L2]

The derivatives of sine and cosine are $(\sin)^{'} = \cos$ , $(\cos)^{'} = - \sin$ , with $\sin 0 = 0$ , $\cos 0 = 1$ . The shift identities $\sin (x + π) = - \sin x$ , $\cos (x + π) = - \cos x$ give $\sin π = \sin 2 π = 0$ , $\cos π = - 1$ , $\cos 2 π = 1$ (Quarter-turn values and shifts by pi/2 and pi). Vector-valued curves are differentiated componentwise (Sine and cosine defined by their real power series, The derivatives of sine and cosine are cosine and minus sine, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L3]

The system is autonomous, so $F (t, z) = F (0, z)$ for all $t$ . For $z, w \in R^{2}$ ,

$∥ F (t, z) - F (t, w) ∥_{2}^{2} = (z_{2} - w_{2})^{2} + ω^{4} (z_{1} - w_{1})^{2} \leq \max {1, ω^{4}} ∥ z - w ∥_{2}^{2},$

so $F$ is globally Lipschitz in the state variable with constant $\max {1, ω^{2}}$ and is continuous; it satisfies the Picard-Lindelof hypotheses (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Verification

technique · direct; verify the explicit solution and evaluate the time-$P$ map

1.1givenL2algebra

Differentiating the two components of the displayed $x_{z}$ with respect to $t$ using [L2] gives $x_{z}^{'} (t) = (- ω z_{1} \sin ω t + z_{2} \cos ω t, - ω^{2} z_{1} \cos ω t - ω z_{2} \sin ω t)$ , whose first component is the second component of $x_{z}$ and whose second component is $- ω^{2}$ times the first component of $x_{z}$ ; hence $x_{z}^{'} = F (t, x_{z})$ on all of $R$ , and $x_{z} (0) = (z_{1}, z_{2}) = z$ because $\sin 0 = 0$ and $\cos 0 = 1$ .

2.1step 1.1L1L3

The curve of step 1.1 is a solution on all of $R$ ; by [L1] and [L3], every solution of the initial value problem with data $(0, z)$ is a restriction of the unique maximal solution $x_{z}$ , whose domain is an open interval. Since a solution with domain $R$ exists, that maximal domain is all of $R$ ; hence $[0, P] \subseteq J_{z}$ for every $P$ and $D_{P} = R^{2}$ . Substituting $t = P$ in step 1.1 gives the displayed formula for $θ_{P} (z)$ for every $P > 0$ .

3.1step 2.1L2algebra∎

If $P = 2 π / ω$ then $\cos ω P = 1$ and $\sin ω P = 0$ , so $θ_{P} (z) = (z_{1}, z_{2}) = z$ for every $z$ ; if $P = π / ω$ then $\cos ω P = - 1$ and $\sin ω P = 0$ , so $θ_{P} (z) = (- z_{1}, - z_{2}) = - z$ , which equals $z$ only for $z = 0$ . This proves the two special cases.

Remarks

The example verifies the iterate identity of Iterates of the stroboscopic map, and equilibria as fixed points implicitly: since $D_{P} = R^{2}$ here, every iterate is defined and $θ_{P}^{k} (z) = x_{z} (k P)$ ; for $P = 2 π / ω$ this makes every iterate the identity.
The map $θ_{P}$ is a linear map of $R^{2}$ and preserves the area element; no symplectic or stability conclusion is drawn here, since those notions belong to the symplectic items and to the stability definitions of this collection.

The stroboscopic map of the plane oscillator

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources