Bipolar motor: rotation, parametric instabilities and chaos

A bipolar motor is a simple system which exhibit a great number of nonlinear phenomena. It is composed of a compass in an oscillating magnetic field. It is an experiment that can be easily presented to students for Chaos route demonstration and it was its purpose in the years 1980–1990. Several authors developed experiments [1–5], models [6] and simulations [7, 8] of the compass deterministic chaotic behavior. However, despite deep studies of the chaos characteristic, a part of possible regimes has been put aside by the community: the rotation and the oscillation of the compass. The goal of this study is to set measurements, analytical calculations and simulations of those bipolar motor periodic regimes. The framework of this paper is the parameter space made up of the magnetic field amplitude B and its frequency ω_B. We will explicit how periodic regimes surround the chaotic regions and also detail the parametric instabilities thresholds.

The paper is organized as follow: section 2 describes the bipolar motor. Section 3 provides the results on the rotation regime. Section 4 is about the space parameter region that contains the chaotic motion. Section 5 details the parametric instability of the system. Section 7 presents dynamics limitations of the rotation regime.

The experimental device is illustrated in the scheme figure 1. The large coil (30 cm diameter) produces the magnetic field that drives the compass dynamics. The compass (3 cm diameter) is fixed inside the coil with two bearing balls. Two hall effect sensors are set around the compass, one along the coil axis and the other perpendicular to it. The combination of the two signals allows a direct measurement of the compass angle θ(t) at a frequency rate of 1 kHz. The compass and the sensors are described in details in [9]. The magnetic field is controlled by a program that sets its amplitude B and its frequency ω_B. Modifications of B and ω_B can be triggered at a chosen compass angle θ_i and rotation speed ω_i, which will be the initial condition of the compass.

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The compass dynamics is governed by the following equation:

$$\begin{eqnarray}&&J\displaystyle \frac{{{\rm{d}}}^{2}\theta }{{\rm{d}}{t}^{2}}=-M{ \mathcal B }(t)\sin \theta -F\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}t}.\end{eqnarray} \tag{ 1 }$$

θ is the angle between the magnetic field direction and the compass magnetic moment M = 0.77 A m². J is the compass inertial moment equal to 8.35 × 10⁻⁷ kg m². F is the fluid friction coefficient equal to 0.10 × 10⁻⁵ kg m² s⁻¹. Methods used to determine the system parameters are explained in appendix A. The magnetic field has a sinusoidal form: ${ \mathcal B }(t)\,=B\sin ({\omega }_{B}t)$. Depending on the frequency, the magnetic field can reach up to 10 mT. There are several possible regimes presented in figure 2. The compass can oscillate between the two directions of the coil axis at the magnetic field frequency (a). It can have a chaotic behavior (b). It can rotate at the magnetic field frequency (c). Finally, the compass can stop its rotation and align along one direction of the coil axis (d).

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We aim at sorting the conditions of observation of those regimes in the parameter space (B, f_B). We start by considering the rotation regime.

When the compass is rotating with the magnetic field, the model equation (1) can be simplified using the periodicity of the rotation. The average speed of the compass is ω_B and the compass average acceleration is null. The Lorentz force has to compensate the friction of the fluid: this is only possible if the compass has an average phase shift delay with the magnetic field, quoted as ϕ. The solution of the rotating regime has the form: θ = ω_Bt + ϕ. Casting this solution in equation (1) and making the average of this equation over one turn, one obtain:

$$\begin{eqnarray}&&\cos \phi =-\displaystyle \frac{2{\omega }_{B}F}{{MB}}.\end{eqnarray} \tag{ 2 }$$

Assuming a positive rotation, ϕ is limited by two values: −π or −π/2. In the limit of null friction, ϕ = π/2 and the Lorentz force does not need to sustain the rotation. The motion can be described by quarters of T_B, the period of the magnetic field oscillation. During the first quarter of T_B, the compass goes from −π/2 to 0. The magnetic field increases from 0 to B and accelerates the compass. During the second quarter of T_B, the magnetic field reduces from B to zero and the compass decelerates. The phase shift of −π/2 produces a null Lorentz force in average over half a turn. For a large fluid friction, $\phi \to -\pi$. The Lorentz force is always accelerating the compass but all the energy provided by the magnetic field is consumed into the fluid friction. If the frequency f_B increases or the magnetic field amplitude decreases, the energetic balance in half a period may be negative and the rotation condition is lost. The rotation is possible if $| \cos \phi | \lt 1$ which sets our first limitation in the parameter space $[B,{f}_{B}]$. The rotation is sustained if

$$\begin{eqnarray}&&2{\omega }_{B}F/{MB}\lt 1.\end{eqnarray} \tag{ 3 }$$

The phase shift equation (2) is compared to measurements in figure 3. The two limits of ϕ are observed including the rotation lost at B = 0.2 mT when ϕ ≃ −π. The rotation condition equation (3) is also measured experimentally by slowly reducing the amplitude of the magnetic field during the rotation regime for several fixed frequencies. Figure 4 shows that those measurements and the analytical limit are in a very good agreement. The interpretation of the phase shift ϕ suggests that there is an oscillation of θ(t) around ϕ during the rotation, see the insert in figure 2(c). This can be casted in the solution of the equation as a function δ(t): θ = ω_Bt + ϕ + δ(t). We obtain by gathering this form of the solution and the equilibrium equation (2) into the model equation (1):

$$\begin{eqnarray}\begin{array}{rcl}J\displaystyle \frac{{{\rm{d}}}^{2}\delta }{{\rm{d}}{t}^{2}} & = & -\displaystyle \frac{{MB}}{2}\cos (2{\omega }_{B}t+\phi )\\ & & -\delta \displaystyle \frac{{MB}}{2}(\sin (2{\omega }_{B}t+\phi )-\sin (\phi ))\\ & & -F\displaystyle \frac{{\rm{d}}\delta }{{\rm{d}}t}.\end{array}\end{eqnarray} \tag{ 4 }$$

In the case of small oscillations (δ(t) < 1), the nonlinear second right hand side term can be neglected compared to the first which is the source of the oscillations. The solution for δ(t) must have a null average to satisfy the rotation constraint. Then the solution is:

$$\begin{eqnarray}\begin{array}{rcl}\delta (t) & = & \displaystyle \frac{{\omega }_{0}^{2}}{4{\omega }_{B}({(F/J)}^{2}+4{\omega }_{B}^{2})}\\ & & \times \left(\displaystyle \frac{F}{J}\sin (2{\omega }_{B}t+\phi )-2{\omega }_{B}\cos (2{\omega }_{B}t+\phi )\right),\end{array}\end{eqnarray} \tag{ 5 }$$

with ${\omega }_{0}=\sqrt{{MB}/J}$, the eigen frequency of the compass. As expected, the frequency of oscillations is twice the rotation speed: one per half-turn. For a 10 Hz rotation, we have 2ω_B ≫ F/J and we can deduce from equation (5) the amplitude of the second harmonic in ω_B:

$$\begin{eqnarray}&&{\delta }_{2}\simeq \displaystyle \frac{{\omega }_{0}^{2}}{8{\omega }_{B}^{2}}\ \mathrm{rad}.\end{eqnarray} \tag{ 6 }$$

This value is obtained from experimental and numerical data with a Fourier analysis of δ(t) for different B at 10 Hz. There is a good agreement for δ₂ between experimental, numerical and analytical results, showed on figure 5. For the larger magnetic field presented on figure 5, δ₂ is about 0.17 which still satisfies our hypothesis. However the numerical results show a different behavior for B > 6.2 mT. This behaviour will be described in sections 4 and 5.

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Numerical results in figure 5 exhibit a change in the behavior of the system above 6.2 mT. First, we will explore it with the numerical tool² , performing several simulations of the model equation (1) for B = [0 12] mT and f_B = [0 35] Hz. The initial conditions are set with θ_i = ϕ and ω_i = ω_B (ϕ is set to −π +  if the phase condition is not possible due to limit equation (3)). The different regimes, figure 2, are recognized by the program following criterion detailed in appendix B. Figure 6 presents the numerical results: pink for damped regime when the rotation can not be sustained, colored values for rotation regime where the color is associated to the rotation frequency with the colorbar, white for chaotic regimes and black for oscillations. Panels (b) and (c) are zooms with higher resolution. First, the right limit equation (3) is recovered by numerical simulations, see panel (b) in agreement with figure 4. Second, there is a clear delimitation of the apparition of a complex area for high magnetic field and low frequency. Inside this region, there is an alternation between oscillating and rotating regimes, see panel (c), forming a diagram in layers.

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To understand these layers, we start the interpretation with a case where T_B = 1/f_B is very high compared to the eigen period of the compass T₀ = 2π/ω₀ and its relaxation time T_F = F/2J. At each T_B/2, the compass has enough time to oscillate and finish the relaxation of the pseudo-periodic motion aligned with B. Then, when B changes its sign, the compass starts each half-turn with a marginally stable position and follows the magnetic field with a randomly positive or negative half-rotation: this creates a chaotic motion, see figure 7. For a second case, T_B is still higher than T₀ but lower than T_F: friction is neglected for this interpretation and ϕ = −π/2. During the magnetic half-period T_B/2, the compass can do a specific number i of oscillations before the magnetic field changes its sign. If i = n + 0.5, the compass oscillates n times and finishes this half-dynamics with an addition π to its phase: this makes a global rotation at ω = ω_B with n oscillations at ω₀ between each half-turns. If i = n, the compass oscillates n times and finishes its motion at ϕ. It is a global oscillation motion at ω = ω_B with oscillations at ω₀ between each half-turns, see insert on figure 7. The other intermediate i does not allow any periodic solutions and drives the compass in a chaotic motion. This interpretation leads to the limit of the layers area when i = 0.5. This explains the alternation of layers. As decreasing f_B increases i, we obtain: rotation (i = 0.5)—chaos—oscillation (i = 1)—chaos—rotation (i = 1.5)—chaos—oscillation (i = 2)—etc.

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We can compare the magnetic field frequency to the number of oscillations at ω₀ to define the regime area. Estimate of ω₀ must account for the large value of θ and include the Borda nonlinear correction. We obtain the following limit:

$$\begin{eqnarray}&&{f}_{B}=\displaystyle \frac{{\omega }_{0}}{4\pi i}\left(1-\displaystyle \frac{{\pi }^{2}}{64}\right).\end{eqnarray} \tag{ 7 }$$

Equation (7) sets the location of the layers in the parameter space $[B,{f}_{B}]$. Figure 8 shows the lines of constant i along with their associated regimes and compared them to the measurements. We find a good agreement between experiment and theory on figure 8 and also recover numerical results of figure 6(c).

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In this interpretation, fluid friction is absent: we are in a region of the parameter space where the magnetic field is strong and its frequency weak which produces a small friction force compared to the Lorentz one. Anyway, it influences several parts of the description. The positions of the layers are modified because the nonlinear correction of equation (7) does not include the oscillation amplitude decrement due to friction: ω₀ tends to its linear value while i increases. The structure of the layers is also modified by F, see figure 9. Increasing the fluid friction blurs the layer's fractal structure and broaden them: they become more visible with less details. The fractal shapes in the layers are probably related to situation where the rotation or the oscillations have a non constant number i: for example we can have a global rotation constituted by tree half-turns in one direction and one turn in the other. For a very large fluid friction coefficient, the relaxation regime of the compass may become sub-critical: the rotation regime would not be available anymore. All those considerations remain out of the scope of our experimental study since the friction coefficient is not tunable.

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There is still a last feature that is not covered by the layers interpretation given in section 4. In figures 8 and 6(c), we can observe an additional oscillation layer, close to i = 0.5, that does not correspond to any value of i. Moreover, we notice in figure 5 that the oscillation δ₂ presents a change of behavior for a magnetic field close to the first transition i = 0.5. We start this investigation by a spectral analysis of the compass oscillation measurements (we removed the rotation part from θ(t) when required). Figure 10 shows the main ω_B harmonics present in those oscillations: the mode 1/2, 1 and 2. Mode 2 has already been explained in section 3. There are two apparitions of mode 1 and the additional oscillation layer has a frequency of ω_B/2. We reproduce those results with simulations in figure 11 with good agreement. In the next paragraphs, we detail successively the first growth of the mode 1, the apparition of this mode 1/2 oscillation regime and the second mode 1 growth.

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Rotation regime has its mode 2 oscillations described by equation (6). Simulation shows that δ₂ has a completely different behavior for B > 6.2 mT at 10 Hz and for B > 3.2 mT at 7 Hz. This is happening with the growth of a first harmonic oscillation. To explain this change of dynamics, we investigate the nonlinear terms of equation (4) since the oscillations are not negligible anymore compared to the drive. As this first harmonic growth appears at high magnetic field, we simplify equation (4) considering $F\to 0$, $\phi \to -\pi /2$ and a solution sets as $\theta (t)={\omega }_{B}t-\pi /2+\tfrac{{\omega }_{0}^{2}}{8{\omega }_{B}^{2}}\sin (2{\omega }_{B}t)+{\delta }_{1}^{r}(t)$. One gets:

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}}^{2}{\delta }_{1}^{r}}{{\rm{d}}{t}^{2}}+\displaystyle \frac{{\omega }_{0}^{2}}{2}{\delta }_{1}^{r}[1-\cos (2{\omega }_{B}t)]=0.\end{eqnarray} \tag{ 8 }$$

It is a parametric instability equation. Using the Laplace variation method, one gets the following solution:

$$\begin{eqnarray}\begin{array}{rcl}{\delta }_{1}^{r}(t) & = & A(t)\cos ({\omega }_{B}t)+B(t)\cos ({\omega }_{B}t),\\ B(t) & = & {B}_{0}\exp ({kt}),\\ A(t) & = & \displaystyle \frac{2{\omega }_{B}}{{\omega }_{0}^{2}/4-{\omega }_{B}^{2}}{{kB}}_{0}\exp ({kt}),\\ {k}^{2} & = & \displaystyle \frac{1}{4{\omega }_{B}^{2}}\left({\omega }_{B}^{2}-\displaystyle \frac{{\omega }_{0}^{2}}{4}\right)\left(\displaystyle \frac{3{\omega }_{0}^{2}}{4}-{\omega }_{B}^{2}\right).\end{array}\end{eqnarray} \tag{ 9 }$$

The dispersion relation equation (9) indicates that the mode 1 can grow if k² is a positive real. Solving equation (9) indicates a magnetic field threshold of 2.8 mT for f_B = 7 Hz and 5.8 mT for f_B = 10 Hz. This corresponds to the numerical measurements (figures 10 and 5) and to the experimental results (figure 10) despite a slight overestimation of the experimental threshold. This may be explained by the presence of the mode 1, even for the simple rotation regime: the experimental setup has a sub-milimetric excentration.

Regime 1/2 oscillation layer appears suddenly with a very sharp magnetic threshold. As a fist analysis, we define the existence condition of this regime by setting $\theta (t)={\delta }_{1/2}\cos ({\omega }_{B}t/2+{\phi }_{1/2})$ in equation (1), using the Jacobi–Anger development and neglecting the friction. The solution for this regime is then:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\omega }_{B}}{4{\omega }_{0}}{\delta }_{1/2} & = & {J}_{3}({\delta }_{1/2})-{J}_{1}({\delta }_{1/2}),\\ {\phi }_{1/2} & = & \pm \pi /4,\end{array}\end{eqnarray} \tag{ 10 }$$

where J_ν(z) is the Bessel function of the first kind. This amplitude is reported in figure 11 with a very good agreement. The comparison to the experimental measurements is not so good, we recover the magnitude order and the tendency. Equation (10) does not have a solution for every value of the magnetic field for a given frequency f_B: it requires a magnetic field greater than 3.7 mT at 7 Hz. Then, both rotation regime and mode 1/2 oscillation regime are possible in this parameter space $[B,{f}_{B}]$ region. It is not mathematically proved, but experimentally, the oscillating regime is always dominating the rotation one.

As for the rotation regime, the 1/2 oscillation mode has a change of behavior of its amplitude δ_1/2 that appears with the increase of a mode 1 oscillation. Here again, we set the first mode as a small fluctuation: $\theta (t)={\delta }_{1/2}\cos ({\omega }_{B}t/2+{\phi }_{1/2})+{\delta }_{1}^{o}(t)$. We still neglect friction and recover a parametric equation:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{\rm{d}}}^{2}{\delta }_{1}^{o}}{{\rm{d}}{t}^{2}} & = & -{J}_{2}({\delta }_{1/2}){\omega }_{0}^{2}{\delta }_{1}^{o}[1-\cos (2{\omega }_{B}t)]\\ & & -{\omega }_{0}^{2}{\delta }_{1}^{o}\displaystyle \frac{{J}_{0}({\delta }_{1/2})-{J}_{4}({\delta }_{1/2})}{{J}_{2}({\delta }_{1/2})}\sin ({\omega }_{B}t).\end{array}\end{eqnarray} \tag{ 11 }$$

We solve it again with the Laplace variation method. The second right hand side terms do not provide any contribution to the mode 1. The first right hand side term leads to the dispersion relation:

$$\begin{eqnarray}&&{k}^{2}=\displaystyle \frac{1}{4{\omega }_{B}^{2}}\left({\omega }_{B}^{2}-\displaystyle \frac{{J}_{2}({\delta }_{1/2}){\omega }_{0}^{2}}{2}\right)\left(\displaystyle \frac{3{J}_{2}({\delta }_{1/2}){\omega }_{0}^{2}}{2}-{\omega }_{B}^{2}\right).\end{eqnarray} \tag{ 12 }$$

Mode 1 is unstable if k² is a positive and real. The magnetic field threshold is equal to 8.5 mT at f_B = 10 Hz, which is out of the range of our experimental setup. At f_B = 7 Hz, it is 4.2 mT: this value is recovered numerically, figure 11. There is a larger error of 0.5 mT with the experiment. If the calculus of the instability threshold is done with the experimental value of δ_1/2 ≃ 3 rad, we obtain 3.8 mT which provides a much better agreement with experimental data. This validates the parametric instability mechanisms.

Two elements remain unexplained here. The first is the prevalence of the oscillating regime over the rotation regime for B > 3.7 mT at f_B = 7 Hz as we said previously. The second is the chaotic dynamics that develops abruptly after the regime 1/2 oscillation pumped a mode 1 oscillation. This may be relative to the three period theorem [10]: the system can possibly exhibit the mode 1/2, 1 and 2 and then becomes chaotic, but this hypothesis is to handle with care.

There is an other limitation to the rotation regime which is not reported in figure 6. Experimentally, the compass lose its rotation at high speed long before the friction limit. This rotation loss is delayed if the frequency enhancement is done slowly. There is a link between the initial condition of the compass and the possibility of getting a rotation regime. To prove this acceleration limitation, we measure numerically and experimentally the possible frequency acceleration of the compass. The experiment consists in several simulations of equation (1) with f_B = 25 Hz, B = 0.75 mT and various initial condition θ_i, ω_i. Those simulations are confirmed by measurements: the same magnetic field is triggered for various initial conditions. Figure 12 gathers the cases where the rotation fails (white background or red crosses) or holds (gray background or blue circle). All the initial conditions do not reach a rotation regime. The compass needs enough kinetic energy and a constructive phase shift with the magnetic field, centered around ϕ = −π/2, in gray in figure 12. Each magnetic field defined by f_B and B has an acceptance area around ϕ. Then the compass moves in the (θ_i, f_i) graph and reaches its stationary state at (ϕ, f_B).

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Figure 12 depends on f_B and B. Then, to accelerate the compass and make a jump from ${f}_{B1}$ to ${f}_{B2}$, we need $({\phi }_{1},{f}_{B1})$ to be in the acceptance area (gray zone) of the acceptable initial conditions graph for $({\phi }_{2},{f}_{B2})$. This condition limits the value of the frequency jump. Moreover, the acceptance area shrinks while 1/B and f_B increase: it is more and more difficult to reach the maximum speed set by equation (3). To illustrate this feature, we perform simulations estimating the maximal frequency jump Δf possible versus f_B at the fixed phase ϕ and fixed B. Result are reported in figure 13 in colored lines for different magnetic field amplitudes. Δf tends to zero while the rotation frequency reaches its maximal value (equation (3), the stars in the figure at the bottom of each lines). This figure also shows the experimental results. We perform successive constant jumps from 10 Hz to the maximum speed reachable. For each jump, phase was kept constant, as the amplitude of the magnetic field which was fixed to 1 mT. This was repeated for Δf = 5, 4, 3 and 1 Hz. As expected, the rotation fails when the experimental Δf gets close or exceeds the maximum jump possible (the black bold line).

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In practice, to reach a point in $[B,{f}_{B}]$, it is more time efficient to make large jumps at high magnetic field and then decrease the magnetic field to the wanted point: indeed, the variations of ϕ with the magnetic field are weaker. But this path is limited, by the magnetic field production which decreases with its frequency. Jumps to accelerate the compass must be smaller as the frequency increases and they must be done slowly to wait for the equilibrium between each jump.

This work relates complementary results on the bipolar motor used for chaos determination. It provides information on compass locking, rotation or oscillation regimes, such as their location in the parameter space $[B,{f}_{B}]$. All the presented results are supported by measurements, simulations and computations all in good agreement. The surprising presence of parametric instabilities in the periodic regimes was for the first time explored and is probably involved into the chaos generation. We have observed that the chaotic region of the parameter space is in fact an alternation of periodic regimes with chaotic ones, forming a diagram in layers. The complete structure of the parameter space may contain other layers, especially with a system with less friction and with a better experimental resolution which are unaccessible on our system because of a too high friction coefficient of the compact axis.

In this section, we will discuss the different methods used to measure each parameter. This is particularly important as it allows us to link experimental data with numeric simulations, and the different regimes rely deeply on the value of these parameters.

A.1. Magnetic moment

The magnetic field ${\overrightarrow{{ \mathcal B }}}_{{ \mathcal M }}$ generated by the compass is a function of ${ \mathcal M }$. We can deduce its value by measuring ${{ \mathcal B }}_{{ \mathcal M }}$ for different distances z along the axis of the magnetic moment:

$$\begin{eqnarray*}&&{\overrightarrow{{ \mathcal B }}}_{{ \mathcal M }}=\displaystyle \frac{2{\mu }_{0}{ \mathcal M }}{4\pi {z}^{3}}\overrightarrow{{u}_{r}}.\end{eqnarray*}$$

This law is only verified for large distances compared to the compass' size.

We obtain (figure A1):

$$\begin{eqnarray*}&&{ \mathcal M }=(0,77\pm 0,02)\,{\rm{A}}\,{{\rm{m}}}^{2}.\end{eqnarray*}$$

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A.2. Moment of inertia

For a constant magnetic field and small oscillations ($\theta \ll 1$) of $\overrightarrow{{ \mathcal M }}$, the differential equation can be rewritten as:

$$\begin{eqnarray*}&&\ddot{\theta }(t)={ \mathcal B }\theta (t)-F\dot{\theta }(t).\end{eqnarray*}$$

And the oscillation period is:

$$\begin{eqnarray*}&&{\omega }_{0}^{2}=\displaystyle \frac{{ \mathcal M }{ \mathcal B }}{J}-{\left(\displaystyle \frac{F}{2J}\right)}^{2}.\end{eqnarray*}$$

We measure the frequency for several values of ${ \mathcal B }$ and the slope gives us the moment of inertia (figure A2):

$$\begin{eqnarray*}&&J=(8.33\pm 0.01)\times {10}^{-7}\,\mathrm{kg}\,{{\rm{m}}}^{2}.\end{eqnarray*}$$

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A.3. Coefficient of friction

Shuting down the magnetic field while the compass is oscillating at a constant speed (50 Hz) allows us to get the value of the friction coefficient. The differential equation verified by $\omega =\dot{\theta }$ becomes:

$$\begin{eqnarray*}&&\displaystyle \frac{{\rm{d}}\omega (t)}{{\rm{d}}t}=-F\omega (t).\end{eqnarray*}$$

Therefore, speed decreases exponentially (figure A3) and we get:

$$\begin{eqnarray*}&&F=(1.00\pm 0.01)\times {10}^{-6}\,\mathrm{kg}\,{{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}.\end{eqnarray*}$$

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B.1. Code algorithm

B.2. Dynamics sorting algorithm

In order to properly map every regime, it is important to choose a precise enough step size so that a regime is not mistaken for another. The relevant time scales of the equation are T_B (the period of the magnetic field) and T₀ (the fundamental period for the oscillation regimes, ${T}_{0}\propto {{ \mathcal B }}^{-1/2}$). To properly resolve the trajectory of the compass, we must choose :

$$\begin{eqnarray*}&&{\rm{d}}{t}\lt \displaystyle \frac{{T}_{0}}{10},\displaystyle \frac{{T}_{B}}{10}.\end{eqnarray*}$$

Therefore the sampling frequency must depend on these parameters. Even when these conditions are respected, there is still some indetermination on the resolution of the trajectories, as the system can develop harmonics at higher frequencies (as with the small oscillations during rotating regimes).

Furthermore, as some regimes are chaotic and thus deeply dependent on initial conditions, it is impossible to get the real analytic solutions with a discrete time step.

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Indeed, as seen on figure C2, a small error due to resolution issues will increase and the numerical solution will diverge from the analytic one: the longer you wait, the more precise you have to be. Therefore, we used a step size of 0.0001 s to ensure that our regime map was properly resolved during the characteristic time of the simulation.

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