New physical insights in dynamical stabilization: introducing Periodically Oscillating-Diverging Systems (PODS)

Dynamical stabilization is the ability of a statically diverging stationary state to gain stability by periodically modulating its physical properties in time. This phenomenon is getting recent interest because it is one of the exploited feature of Floquet engineering that develops new exotic states of matter in the quantum realm. Nowadays, dynamical stabilization is done by applying periodic modulations much faster than the natural diverging time of the Floquet systems, allowing for some effective stationary equations to be used instead of the original dynamical system to rationalize the phenomenon. In this work, by combining theoretical models and precision desktop experiments, we show that it is possible to dynamically stabilize a system, in a “synchronized” fashion, by periodically injecting the right amount of external action in a pulse wave manner. Interestingly, the Initial Value Problem underlying this fundamental stability problem is related to the Boundary Value Problem underlying the determination of bound states and discrete energy levels of a particle in a finite potential well, a well-known problem in quantum mechanics. This analogy offers a universal semi-analytical design tool to dynamically stabilize a mass in a potential energy varying in a square-wave fashion.

All numerical data in this work have been generated from the considered system equations Eqs.(1)–(14), with the approaches described in this work and the cited works, using Python. Therefore, it is possible to completely reproduce the data from the information given in this work.

A. Lazarus and S. Protière are grateful for financial support from Sorbonne University, France (EMERGENCES grants).

Authors and Affiliations

1. Institut Jean Le Rond d’Alembert, CNRS UMR7190, Sorbonne Université Paris, Paris, France

   Alvaro A. Grandi, Suzie Protière & Arnaud Lazarus

Corresponding author

Correspondence to Arnaud Lazarus.

Conflict of interest

The authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest, in the subject matter or materials discussed in this manuscript.

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Appendix 1: Experimental P.O.D.S

In this appendix, we present in detail the experimental P.O.D.S built in the laboratory. In Fig. 12a, the metallic marble has a mass \(m= 28\) g that is attached to a plexiglass rod of length \(L = 6.2\) cm. The rod is then connected to another rod allowing it to rotate only in one plane. The marble is centered below the electromagnet (with typical holding force of 1000 N). Thanks to a Controllino card, we can turn ON and OFF the electromagnet in a very controlled and accurate manner in time. For the recording of the experimental responses, we place the electromagnetic inverted pendulum in front of a white LED to enhance the contrast and record the motion of the metallic marble with a Basler camera CMOS with 150 frames per second. The electromagnet is connected to a generator where we can select the value of the electrical current i. The electrical current is responsible of the intensity of the electromagnetic force near the inverted pendulum. The stronger the value of i, the stronger the electromagnetic field. By turning ON the electromagnet, the electromagnetic force will modify the effective gravitational field near the inverted pendulum, directly affecting the natural time scale of the inverted pendulum. To highlight this, the response for \(i=0\) and \(i=0.48\) A is shown in Fig. 12b, c, respectively. The natural response in Fig. 12b for \(i=0\) is a diverging one with a natural time scale to be \(1/\omega (0)=0.09\) s obtained by fitting an exponential function to the response. For \(i=0.48\) A, the response is damped oscillations about \(\theta (t) \approx 0\) characterized by a natural frequency \(\omega (0.48) = 19.5\) rad/s obtained by doing a Fast Fourier Transformation of the oscillatory response.

Appendix 2: Influence of damping

In this appendix, we look at the influence of viscous damping on the stability diagram of Fig. 3. For practical purposes, we add a reduced damping term \(\xi \) in Eq. (7) so that the linearized equation of motion about the trivial fixed point \((q(t),\dot{q}(t))=(0,0)\) is now

on a given period \(T=T_\mathrm{{D}}+T_\mathrm{{O}}\), with \(I=mL^2=1.076 \times 10^{-4}\) kg.m\(^2\), \(V_\mathrm{{D}}=-\frac{1}{2}I\omega (0)^2=-1.04\) mJ and \(E = V_\mathrm{{D}}+\Delta V=\frac{1}{2}I\omega (0.48)^2=3.24\) mJ. The stability diagram in the modulation parameter space \((T_\mathrm{{O}},T_\mathrm{{D}})\) is given in Fig. 13 where the influence of damping is shown in pink regions (here \(\xi =0.05\)). The influence of viscous damping is a well-known narrowing of the tip of the instability tongues. Interestingly, the tip of the stability tongues does not disappear when viscous damping is added. Although Eq. (16) seems more accurate than the undamped version Eq. (7) that we use in this article because our electromagnetic pendulum is indeed damped during \(T_\mathrm{{O}}\), the undamped stability diagram seems in better agreement with our experimental data.

Numerical stability diagrams of the upright vertical electromagnetic pendulum governed by Eq. (16) when the current i(t) is modulated with a piecewise constant T-periodic function. During \(T_\mathrm{{D}}\), \(i=0\) and the upright pendulum is diverging with a natural time scale \(1/\omega (0)\) where \(\omega (0)=11.1\) rad/s. During \(T_\mathrm{{O}}\), \(i=0.48\) A and the pendulum is oscillating with a natural frequency \(\omega (0.48)=19.5\) rad/s. Blue regions represent dynamically stable \((T_\mathrm{{O}},T_\mathrm{{D}})\) for the damped (\(\xi =0.05\)) and undamped (\(\xi =0\)) scenario. Pink regions represent dynamically stable and unstable \((T_\mathrm{{O}},T_\mathrm{{D}})\) for the damped and undamped case, respectively. White regions represent unstable couple \((T_\mathrm{{O}},T_\mathrm{{D}})\) for both \(\xi =0\) and \(\xi =0.05\)

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The thing is that there is a paradox when trying to predict the stable motions of the electromagnetic pendulum governed by the damped time-periodic Eq. (16). The upright electromagnetic pendulum is indeed doing damped oscillations when the electromagnets are ON and is diverging when the electromagnets are OFF. But if the electromagnets are turned ON during \(T_\mathrm{{O}}\) and OFF during \(T_\mathrm{{D}}\) in a piecewise constant periodic fashion, Eq. (16) will always predict that \(q(t) \rightarrow 0\) when \(t \rightarrow \infty \) in the case of stable couples \((T_\mathrm{{O}},T_\mathrm{{D}})\). However, in the experiment, the upright pendulum will always oscillate with a finite amplitude even for very long time, because although damped during \(T_\mathrm{{O}}\), the latter is periodically diverging during \(T_\mathrm{{D}}\) so the slightest imperfection that remains after the damped oscillations time \(T_\mathrm{{O}}\) will be amplified. A mass periodically doing damped oscillations and whose initial conditions are periodically “shuffled” by a diverging period is not correctly predicted by a time-periodic system like Eq. (16) and is maybe just difficult to predict at all because of the seemingly random nature of the symmetry breaking associated with the diverging time. This aspect, somehow similar to the so-called micro-chaotic oscillation of (mechanical) systems stabilized by digital control [25], could be interesting to investigate in a near future.

Appendix 3: Particle in a finite potential well

To find the discrete energy levels E for a mass under the effect of a finite square wave potential well of length \(T_\mathrm{{O}}\) and potential depth \(\Delta V\) (Fig. 14) can be written as

Finite square wave potential well of length \(T_\mathrm{{O}}\) and potential depth \(\Delta V\)

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and \(\Psi (-\infty )=\Psi (+\infty )=0\), where \(\Psi (t)\) is a wave function, I is the moment of inertia of the mass and \(\mathcal {U}(t)\) is the fixed square wave potential. Outside of the box \(T_\mathrm{{O}}\), the potential is \(\Delta V\) and zero for t between \(-T_\mathrm{{O}}/2\) and \(T_\mathrm{{O}}/2\). So, the wave function can be considered to be made up of different wave functions at different ranges of t, depending on whether t is inside or outside of the box. Therefore, the wave function can be defined as:

1.1 Wave function inside the box

For the region inside the box, \(\mathcal {U}(t) = 0\), Eq. (17) reduces to

Equation (18) is a linear second-order differential equation with \(E>0\), so it has the general solution

where \(k = \sqrt{2E/I}\) is a real number and A and B can be any complex numbers.

1.2 Wave function outside the box

For the region outside the box, \(\mathcal {U}(t) = \Delta V\), Eq. (17) reduces to

There are two possible families of solutions depending on whether E is greater than \(\Delta V\) (the particle is free) or E is less than \(\Delta V\) (the particle is bound in the potential). In this analysis, we focus on the latter (\(E<\Delta V\)), so the general solution is an exponential of the shape

where \(\alpha = \sqrt{2(\Delta V-E)/I}\) is a real number and F and G can be any complex numbers. Similarly, for the other region outside the box:

where H and I can be any complex numbers.

Master equations to deduce the discrete energy levels \(E_i\) and the corresponding wave function \(\Psi _i(t)\). a Square wave potential fixed at \(u_0^2=5\) gives two intersection points of the master curves which translates into two energy levels \(E_{1,2}\) and the corresponding wave function \(\Psi _{1,2}(t)\) represented in blue and green. b Square wave potential fixed at \(u_0^2=31.25\) gives four intersection points of the master curves which translate into four energy levels \(E_{1,2,3,4}\) and the corresponding wave function \(\Psi _{1,2,3,4}(t)\) represented in blue, green, orange and purple, respectively

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1.3 Wave function for the bound state

For the expression of \(\Psi _1(t)\) in Eq. (22), we see that as t goes to \(-\infty \), the F term goes to infinity. Likewise, in Eq. (23) as t goes to \(+\infty \), the I term goes to infinity. In order for the wave function to be square integrable, we must set \(F = I= 0\).

Next, we know that the overall \(\Psi (t)\) function must be continuous and differentiable. These requirements are translated as boundary conditions on the differential equations previously derived. So, the values of the wave functions and their first derivatives must match up at the dividing points:

giving the system of equations

Finally, the finite square-wave potential well is symmetric (Fig. 14), so symmetry can be exploited to reduce the necessary calculations. This means that the system in Eq. (24) has two sorts of solutions: symmetric and antisymmetric solutions.

1.3.1 Symmetric solutions

To have a symmetric solution, we need to impose \(A=0\) and \(G=~H\). Equation(24) reduces to

and taking the ratio gives

which is the energy equation for the symmetric solutions.

1.3.2 Antisymmetric solutions

For the antisymmetric solutions, we need to have \(B=0\) and \(G=-~H\). Equation(24) reduces to

and taking the ratio gives

which is the energy equation for the antisymmetric solutions.

1.3.3 Master equations

The energy equations (25, 26) cannot be solved analytically. Nevertheless, if we introduce the dimensionless variables \(u=\alpha T_\mathrm{{O}}/2\) and \(v=kT_\mathrm{{O}}/2\), we obtain the following master equations

where \(u_0^2 = \Delta V T_\mathrm{{O}}^2/2I\) and \(v^2 = E T_\mathrm{{O}}^2/2I\). So, for a fixed square-wave potential \((\Delta V,T_\mathrm{{O}})\), the intersections (\(v_i\)) solution of Eq. (27) let us infer the discrete energy levels \(E_i = 2Iv_i^2 /T_\mathrm{{O}}^2\). Then, having the values of \(E_i\) we can deduce the values of \(\alpha _i\) and \(k_i\) and infer the wave function \(\Psi _i(t)\).

Figure 15 shows two examples of application for the master equations (27). In Fig. 15a, the potential barrier \(\Delta V\) and the length of the box \(T_\mathrm{{O}}\) gives \(u_0^2 = 5\). Then, by solving the master equations (27) we obtain two intersections points \((v_1,v_2)\). Then, we deduce the two discrete energy levels \(E_{1,2}\) and the corresponding wave functions \(\Psi _{1,2}(t)\) for this giving square-wave potential (represented in blue and green, respectively, in Fig. 15a). Figure.15b represents another example where \(u_0^2 = 31.25\). The solution of the master equation (27) gives four intersection points. We deduce the discrete energy levels \(E_{1,2,3,4}\) and the corresponding wave functions \(\Psi _{1,2,2,4}(t)\) (represented in blue, green, orange and purple in Fig. 15b).

It is interesting to mention that the resolution previously showed is also the mathematical resolution of the classical problem of a particle trapped in a finite potential well in quantum mechanics [13, 26].

Neutrally stable response for \(I=mL^2=\) 0.1076 g.m\(^2\), \(\Delta V = 4.28\) mJ, \(E=0.895\) mJ, \(T_\mathrm{{O}} = 0.21365\) s and \(T_\mathrm{{D}} = 0.8\) s visualized in the elementary time cell \(-T/2< t < T/2\) with \(T=T_\mathrm{{O}}+T_\mathrm{{D}}\). a Evolution of the generalized coordinate q(t) and Floquet eigenfunction \(\Psi (t)\). b Collapse of the trajectories q(t) of a on the Floquet eigenfunction \(\Psi (t)\) and evolution of the associated modulation function V(t) (equivalent to \(\mathcal {H}(q,p,t)\)). The eigenfunction \(\Psi _0(t)\) and eigenvalue \(E_0\) of Eq. (9) are reported on the figure

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Stability diagram of the trivial state \((q(t),\dot{q}(t))=(0,0)\) of the square-wave P.O.D.S. governed by Eqs. (5)–(7) in the dimensionless \(({\Delta \tilde{V}},\tilde{E})\) space. Blue regions indicate that a basin of attraction exist for which the mass is neutrally stable about (0, 0) when white regions show an unstable trivial state. Black and orange lines are the symmetric and antisymmetric master curves from the Liouville eigenproblem in Eq. (9) where we assumed \(T_\mathrm{{D}} \rightarrow T\) with \(T=T_\mathrm{{O}}+T_\mathrm{{D}}\) and \(E<\Delta V\) (or \(V_\mathrm{{D}} < 0\)). The red line is the limit \(E=\Delta V\), i.e., \(V_\mathrm{{D}} =0\). Below this red line, we have \(V_\mathrm{{D}}>0\), the case of a particle in a potential energy with a time-varying local curvature that always remain positive. a \(T_\mathrm{{D}}/T=0.7\). b \(T_\mathrm{{D}}/T = 0.25\)

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Appendix 4: Fundamental bound state for \(T_\mathrm{{O}}=0.21365\) s

The resolution of the Liouville eigenvalue problem Eq. (9) suggested that for \(I=0.1076\,\text {g}\,\text {m}^2\), \(\Delta V = 4.28\) mJ and \(T_\mathrm{{O}} = 0.21365\) s, a modulation function with \(E=0.895\) mJ would stabilize the mass even when the diverging time \(T_\mathrm{{D}}\) is large. This result is summarized in Fig. 9 that showed the “bound states” and “energy levels” of the particle confined in a finite potential well for \(T_\mathrm{{O}} = 0.21365\) s and \(\Delta V = 4.28\) mJ. In Fig. 16, we show the response of the mass governed by the linear Initial Value Problem Eq. (7) when using the modulation function V(t) suggested by the eigenvalue problem Eq. (9). In Fig. 16a, the 100th first periods of the dynamical response q(t) are superposed in the elementary time cell \([-T/2,T/2]\) alongside with its Floquet eigenfunction \(\Psi (t)\) shown in black thin line. As predicted by the Boundary Value Problem, the response is neutrally stable even if \(T_\mathrm{{D}}\) is large. Moreover, upon the correct scaling, one can collapse all the trajectories on a single curve in \([-T/2,T/2]\) that is the Floquet eigenfunction \(\Psi (t)\) of the response as shown in Fig. 16b where we also plot the piecewise constant modulation function V(t) (that is very close to the total energy of the mass) in green line. The eigenvalue and eigenfunction of Eq. (9) are also reported in this figure. As expected, they match with the outcome of our Initial Value Problem. The Boundary Value Problem Eq. (9) is therefore a good design tool to predict what modulation function will dynamically stabilize the mass even for a long diverging time \(T_\mathrm{{D}}\) and what will be the qualitative shape of the oscillatory response over each period.

Appendix 5: Extended stability diagram in the \((\sqrt{\bar{E}},\sqrt{\bar{\Delta V}})\) space

In Fig. 10, we showed the linear stability diagram of our square-wave Periodically Oscillating Diverging System (P.O.D.S.) governed by the dimensionless equation Eq. (13) in the \((\sqrt{\bar{E}},\sqrt{\bar{\Delta V}})\) space for \(0< E < \Delta V\), i.e., \(V_\mathrm{{D}} < 0\) that is the P.O.D.S. formalism. In Fig. 17, we show this stability diagram in the general case that allow \(E > \Delta V\), i.e., \(V_\mathrm{{D}} > 0\) that is the case when the particle is in a potential whose local curvature varies between only positive values, in a square-wave fashion in our case. What we see in Fig. 17 is then the classic instability tongues (white regions) of the Meissner equation Eq. (13) that has been extensively studied in the literature [9, 17, 18, 21].

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