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The three-dimensional isotropic oscillator: the stroboscopic map at half period is minus the identity, every state is periodic, the periodic points are dense, and the map is neither mixing nor sensitive

Physical scope

A declared-data check on the physical one-label force model with mass m0=1 kg, force F_phys(T,X,V)=-(m0/t0^2)X and potential U_phys(X)=m0||X||^2/(2t0^2), using ell0=1 m and t0=1 s. The formulas use numerical coordinates q=X/ell0, v=t0 V/ell0 and t=T/t0, normalized mass 1 and potential ||q||^2/2. The stroboscopic map at numerical half-period pi is minus the identity and at 2pi is the identity; the first-return map to q_y=0, v_y>0 is the identity. All states return, including the constant zero state. Periodic points of the map are dense, but the map is neither mixing nor sensitive and is not chaotic in the stated sense. Physical periods multiply by t0. All parameters are declared model data; no measurement is reported.

Example

Declared model data and normalization. Fix one label, $Q : = R^{3}$ , $m_{0} : = 1 k g$ , $ℓ_{0} : = 1 m$ and $t_{0} : = 1 s$ . The physical force map is $F_{p h y s} : R \times Q \times R^{3} \to R^{3}$ , $F_{p h y s} (T, X, V) = - (m_{0} / t_{0}^{2}) X$ , in newtons, with $T$ in seconds, $X$ in metres and $V$ in metres per second. This smooth globally state-Lipschitz map and mass $m_{0}$ specify a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance. Put $t = T / t_{0}$ , $q = X / ℓ_{0}$ and $v = t_{0} V / ℓ_{0}$ . Newton's equation becomes $q^{''} = - q$ since $\ddot{X} = (ℓ_{0} / t_{0}^{2}) q^{''}$ . The normalized force is $F (t, q, v) : = F_{p h y s} (t_{0} t, ℓ_{0} q, (ℓ_{0} / t_{0}) v) / (m_{0} ℓ_{0} / t_{0}^{2}) = - q$ ; $m = 1$ is its normalized mass. With $U_{p h y s} (X) = m_{0} ∥ X ∥_{2}^{2} / (2 t_{0}^{2})$ and $E_{0} = m_{0} ℓ_{0}^{2} / t_{0}^{2}$ , $U = U_{p h y s} / E_{0}$ gives, in the chart $Ω : = R^{3}$ the constant kinetic metric is $M : = I_{3}$ and the potential is $U (q) : = \frac{1}{2} ∥ q ∥_{2}^{2}$ of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart. The field and motion through $(q_{0}, v_{0})$ are $G (q, v) = (v, - q), q (t) = q_{0} \cos t + v_{0} \sin t, v (t) = - q_{0} \sin t + v_{0} \cos t .$ Regard the autonomous force model as $P$ -periodic for any $P > 0$ , and let $Ψ_{P}$ be its stroboscopic map of The stroboscopic map of a periodic model and the Poincare return map of a transversal section: domains, iterates, fixed points and the local C1 diffeomorphism property clause (ii). The complete flow makes each $Ψ_{P}$ defined on all of $R^{6}$ . Numerical period $P$ means physical period $t_{0} P$ ; all state norms below use the dimensionless coordinates $(q, v)$ . Use the section $Σ_{+} : = {(q, v) : q_{y} = 0, v_{y} > 0}$ ; its first-return map sends each section state to the state at its least positive return time, with the local $C^{1}$ construction supplied by The stroboscopic map of a periodic model and the Poincare return map of a transversal section: domains, iterates, fixed points and the local C1 diffeomorphism property clause (iii).

Assertions to check. (i) The displayed curves are the model's motions; (ii) $Ψ_{π} = - i d$ and $Ψ_{2 π} = i d$ , so every state of the model is periodic under the flow with period $2 π$ and every point is a periodic point of $Ψ_{π}$ with period dividing $2$ (the zero state has a constant motion; only nonzero states give nonconstant periodic motions); (iii) the first-return map of $Σ_{+}$ is the identity; (iv) with the Euclidean metric of the phase chart, $Ψ_{π}$ on $X : = R^{6}$ is continuous and has dense periodic points, but is not topologically mixing and has no sensitivity constant, hence is not chaotic in the stated sense of Chaos in the stated sense for maps of the model: topological mixing, dense periodic points and sensitive dependence, as one precise list among several.

Facts & Assumptions

Given: the declared one-label force model $Q = R^{3}$ , $m = 1$ , $F (T, q, v) = - q$ , its equivalent chart data $M = I_{3}$ , $U (q) = ∥ q ∥_{2}^{2} / 2$ , the field $G$ and the displayed motion formulas.

[L1]

Unique force-model motions (Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder): for the adopted physical smooth force map, every initial datum has a unique maximal motion, and every other solution through that datum is its restriction. The invertible state and time normalization above transfers existence, uniqueness and maximality to the displayed numerical system.

[L2]

Calculus (The derivatives of sine and cosine are cosine and minus sine, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ): $(\sin)^{'} = \cos$ , $(\cos)^{'} = - \sin$ , $\sin 0 = 0$ and $\cos 0 = 1$ , with the sum, constant-multiple and chain rules.

[L3]

The chaos list (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list): a map is chaotic in the stated sense exactly when it is continuous, topologically mixing, has dense periodic points and has sensitive dependence with a positive constant; the negation of topological mixing or of sensitivity therefore suffices to fail the list.

[L4]

The local first-return proposition (A transversal return of a C¹ flow yields a local C¹ Poincaré map, with inverse given by the reversed crossing): for a $C^{1}$ autonomous field and a $C^{1}$ section, two transversal crossings define a local return map, and exclusion of intervening hits from the section closure makes it a first-return map.

[L5]

Trigonometric values and zeros (Quarter-turn values and shifts by pi/2 and pi, The zero sets of sine and cosine and the least positive common period 2 pi): $\sin π = 0$ , $\cos π = - 1$ , and sine and cosine have period $2 π$ , so $\sin (2 π) = \sin 0 = 0$ , $\cos (2 π) = \cos 0 = 1$ using [L2]. For every real $t$ , $\sin t = 0$ exactly when $t = k π$ for an integer $k$ . In particular, the only zero in $0 < t < 2 π$ is $t = π$ .

Verification

technique · direct

1.1givenL1L2algebra

Field and motion: with $M = I_{3}$ and $U (q) = ∥ q ∥_{2}^{2} / 2$ the mechanical field is $G (q, v) = (v, - q)$ (The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart), which is also the normalized field of the declared physical force map. Differentiating the displayed vector formulas gives $\dot{q} = v$ and $\dot{v} = - q$ , and at $t = 0$ they reproduce the initial datum; by [L1] these are the unique maximal model motions.

2.1step 1.1givenL2L5algebra

Stroboscopic maps: evaluating step 1.1 at $t = π$ uses $\cos π = - 1$ , $\sin π = 0$ and gives $Ψ_{π} (q, v) = (- q, - v)$ ; at $t = 2 π$ it gives the identity. Thus every state returns after $2 π$ , and $Ψ_{π}$ is an involution, so every point of $X = R^{6}$ is a periodic point of $Ψ_{π}$ with period $1$ or $2$ ; in particular the periodic points are all of $X$ and are dense.

2.2step 1.1givenL2L4L5algebra

First return to $Σ_{+}$ : take the open state neighbourhood $W : = {(q, v) : v_{y} > 0}$ and the section function $h : W \to R$ , $h (q, v) : = q_{y}$ , so $h^{- 1} (0) = Σ_{+}$ within $W$ . At every point of $Σ_{+}$ , $D h G = v_{y} > 0$ . A state with $q_{y 0} = 0$ and $v_{y 0} > 0$ has $q_{y} (t) = v_{y 0} \sin t$ and $v_{y} (t) = v_{y 0} \cos t$ , so the next hit of $Σ_{+}$ is at $t = 2 π$ ; at that time the full six-dimensional state returns to its initial value by the formulas of step 1.1. The open segment $0 < t < 2 π$ avoids $\overline{Σ_{+}} = {q_{y} = 0, v_{y} \geq 0}$ : by [L5] its only intermediate zero is at $t = π$ , where $v_{y} = - v_{y 0} < 0$ . Applying [L4] at each reference point gives the local first-return map, and the explicit calculation identifies it with the identity on all of $Σ_{+}$ .

3.1step 2.1L3algebra

Failure of mixing: let $Z \in X = R^{6}$ be a state with $∥ Z ∥_{2} > 1$ and let $U : = B (Z, 1)$ be the open Euclidean ball (Open ball, closed ball and sphere in a metric space); then $U \cap (- U) = \emptyset$ , and step 2.1 gives $Ψ_{π}^{n} (U) = U$ for even $n$ and $Ψ_{π}^{n} (U) = - U$ for odd $n$ . Taking $V : = U$ in the mixing condition of [L3], the intersection $Ψ_{π}^{n} (U) \cap V$ is empty for every odd $n$ ; consequently no integer $N$ has $Ψ_{π}^{n} (U) \cap V \neq \emptyset$ for all $n \geq N$ , and $Ψ_{π}$ is not topologically mixing on $X$ .

4.1step 2.1step 3.1L3algebra

Failure of sensitivity: for all states $Z, Z^{'} \in R^{6}$ and all $n \geq 1$ , step 2.1 gives $∥ Ψ_{π}^{n} (Z) - Ψ_{π}^{n} (Z^{'}) ∥_{2} = ∥ Z - Z^{'} ∥_{2}$ because each iterate is $\pm i d$ , an isometry; hence for no $δ > 0$ can one find, for every $Z$ and every ball around $Z$ , a nearby state whose iterate is $δ$ -separated. The sensitivity condition of [L3] fails, and with it the chaos list; the periodic-point and continuity clauses alone do not make $Ψ_{π}$ chaotic in the stated sense.

5.1step 2.2step 4.1givenL1L2algebra∎

Reading: the maps use the dimensionless numerical state and time variables defined above. By the chain rule, $X (T) = ℓ_{0} q (T / t_{0})$ obeys $m_{0} \ddot{X} = - (m_{0} / t_{0}^{2}) X$ ; uniqueness transfers through this invertible normalization. The physical first-return time is $2 π t_{0}$ and the physical half-period is $π t_{0}$ . No measurement, sampling or empirical comparison is performed.

Remarks

Why the check matters. It exhibits a model whose stroboscopic map is continuous with dense periodic points yet fails two of the four clauses of the page's chaos list, showing that the list is a genuine restriction rather than a set of automatic consequences of periodicity. It is the discrete-time companion of the finite-time growth checks: rate zero everywhere (Two declared linear models worked through: the harmonic flow has zero finite-time growth in every direction, while the repulsive linear model has exact rates $\pm 1$ and the positive Lyapunov exponent $1$ at its equilibrium) and no chaos.
The return map. On the regular section ${q_{y} = 0, v_{y} \neq 0}$ the first-return time is $π$ , and its return map sends $(q, v)$ to $(- q, - v)$ ; composing this return map twice gives the identity, including on $Σ_{+}$ . The section is transversal because $D h G = v_{y} \neq 0$ on it, in agreement with the hypotheses of The stroboscopic map of a periodic model and the Poincare return map of a transversal section: domains, iterates, fixed points and the local C1 diffeomorphism property clause (iii). For $Σ_{+}$ , its closure is ${q_{y} = 0, v_{y} \geq 0}$ ; the only intermediate zero at time $π$ has $v_{y} = - v_{y 0} < 0$ , so the open reference segment avoids that closure. For the regular two-sided section its closure is ${q_{y} = 0}$ , and no intermediate zero occurs before $π$ . These observations also verify the proposition's extra first-return condition.

The three-dimensional isotropic oscillator: the stroboscopic map at half period is minus the identity, every state is periodic, the periodic points are dense, and the map is neither mixing nor sensitive

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources