Nonlinear Systems: The Three Levels of Complexities

Introduction

Resonators—mechanical, electrical, or optical—are systems that store and exchange energy between conjugate variables and respond strongly near a characteristic frequency \(f_0\) set by geometry and material properties. In the linear regime, superposition holds: the resonance frequency and linewidth \((Q)\) are amplitude-independent, and the response at each frequency is fully determined by \(f_0\), \(Q\) and the drive. Or in other words, while the measured amplitude linearly follows the drive amplitude.

A resonator becomes nonlinear when its restoring force or dissipation depends on amplitude e.g., geometric/material anharmonicity, electrostatic softening, thermoelastic or fluidic effects. The canonical case is stiffness nonlinearity (Duffing) [1], where the effective spring constant acquires an amplitude-dependent term, producing frequency pulling; nonlinear damping and parametric effects are common variants.

As drive increases, behavior spans: (i) linear response with negligible higher-order terms; (ii) weakly nonlinear response with harmonic generation and modest, amplitude-dependent frequency shifts; and (iii) strong nonlinearity with bifurcations, hysteresis, multistability, and routes to chaos. These regimes are ubiquitous in MEMS/NEMS [2,3], piezoelectric [4], and optomechanical devices [5], often first noticed as harmonic distortion or amplitude–frequency coupling. Zurich Instruments lock-ins are well suited to probe this full range: they generate and measure at the drive, demodulate multiple harmonics simultaneously, provide FFT-based broadband views, run bidirectional sweeps to map hysteresis, and use integrated PLL/PIDs to stabilize phase/frequency or trace amplitude-dependent backbones in real time.

1. Status quo: Linear regime

A linear resonator obeys superposition; its response scales with drive and remains at the excitation frequency. The dynamics of such a linear resonator can be described by the following equation:

\(m\ddot{x}+\epsilon \dot{x}+kx = g\cos{\Omega t}\)

The arguments in the equation includes the linear restoring force \(g\), linear damping term , mass m and the sinusoidal driving force at frequency \(\Omega\) with strength \(g\). This equation holds up nicely in the small-signal limit while the resonator responds linearly at the drive frequency with a single stable periodic orbit. In such a limit, the amplitude and phase response follows a simple Lorentzian response.

Figure 1 shows the characterization of a piezo resonator. The piezo resonator is working in self sensing mode with drive and sense mode running on the same system. Using the LabOne Sweeper (\(15-18\) kHz) at \(V_{ac} \approx 100\) mV, the amplitude shows (Figure 1) a standard resonance while the phase is distorted by parasitic capacitances typical of piezo devices. This underscores a practical advantage of lock-ins: phase-sensitive detection reveals parasitics early, enabling compensation if needed, though we do not pursue it here.

2. Weakly nonlinear regime: harmonics and higher-order terms

A typical linear resonator has nonlinear terms which contribute significantly under higher excitations. In such a regime, the system violates superposition and can respond at multiple frequencies, leading to harmonics, bifurcations, and even chaos. Such non-linear systems can be described by adding higher-order terms (e.g., \(\alpha x^2, \beta x^3\)) to the linear restoring force \(kx\) which become prominent as the displacement grows under stronger drive i.e. for \(g>1\).

Therefore, as \(g\) increases modestly, nonlinear terms perturb the motion, producing harmonics of the drive and slight frequency pulling of the fundamental; the response remains single-valued and stable. Using the same piezo resonator, we first look at a broadband view with the LabOne Scope FFT while driving on resonance. \(V_{ac} = 500\) mV yields a clean fundamental near \(17.5\) kHz over a \(230\) kHz span at \(1\) Hz resolution (Figure 2(a)); increasing to \(V_{ac} = 1\) V exposes clear 2nd–7th harmonics, consistent with higher-order nonlinearities as seen in Figure 2(b).

3. Strongly nonlinear regime: Duffing hysteresis and Poincare sections

To better understand complex system behavior, let’s start with the so-called Duffing Resonator. This is a resonator model which is extensively studied especially in the field of Micro or Nano Electro Mechanical Systems (MEMS or NEMS).

For sufficiently strong drive, the Duffing model admits three steady-state solutions at certain detuning—two stable limit cycles separated by an unstable one—producing a fold (saddle-node) bifurcation of cycles. The frequency response becomes multivalued, leading to jumps and hysteresis; in phase space, coexisting attractors appear as distinct fixed points in a stroboscopic (once-per-period) Poincaré section.

The Duffing resonator is described only by the cubic nonlinearity [1] and for an external drive \(g\gg1\) with a frequency \(\Omega\), it can be written as:

\(\ddot{x} + \epsilon \dot{x} + x + x^{3} + \eta x^{2}\dot{x} = g \cos(\Omega t)\)

Here, we assume normalized responses to assume \(\alpha = 1\) while \(\eta\) represents the nonlinear damping. For the most part of the blog, we will assume this to be \(0\). While the resonance is also normalized and we consider the resonance frequency, \(\Omega_0 = 0\).

Following the discussion in Lifshitz et al. [1] We can derive the amplitude \((a)\) and phase \((\phi)\) response of such a resonator and write it down as:

\(|a|^{2} = \frac{g^{2}}{\left(2\Omega - \frac{3}{4}|a|^{2}\right)^{2} + \left(1 + \frac{1}{4}\eta |a|^{2}\right)^{2}}\)

\(\tan\phi = \frac{1 + \frac{1}{4}\eta |a|^{2}}{2\Omega - \frac{3}{4}|a|^{2}}\)

Numerically solving these equations and plotting them reveal the behavior of amplitude and phase as a function of frequency. Note that for small drives i.e. \(g < 1\) (blue line in Figure 3), the resonator response is linear since the contribution from the nonlinear terms are negligibly small. However, stronger excitation causes the response to deviate strongly from this linear response. The orange and green responses are plotted for excitation \(g=0.5\) and \(g=1.4\) respectively.

Experimentally, we use Zurich Instruments LabOne bidirectional frequency sweeps to probe this behavior. At low drive ( \(V_{ac} = 500\)mV), upward and downward sweeps overlap in both amplitude and phase, consistent with a linear response. Increasing the drive to \(V_{ac} = 3\)V yields a clear hysteresis loop in amplitude and phase (Figure 4), indicating the onset of Duffing-type bistability. Further increasing to \(V_{ac}=6\)V broadens the hysteresis region, as expected when nonlinear coefficients dominate.

Harmonics in strong nonlinear regime

At high drive \(g\gg1\), harmonic contents also becomes significant; using the Zurich Instruments MFLI with the MF-MD Multi-Demodulator option, we simultaneously track up to four harmonics. Bidirectional sweeps of the 2nd–4th harmonics show the bifurcation onset at the same frequency and identical hysteresis widths across harmonics, indicating a coherent Duffing-type nonlinearity imprinting uniformly on all orders.

Poincaré Maps

The Poincaré map (stroboscopic section) for a driven Duffing oscillator samples the state once per drive period T, providing a compact phase-space representation. Fixed points of this map correspond to T-periodic limit cycles; their stability, the appearance of unstable cycles, chaotic attractors, and bifurcation points are readily visualized. Varying parameters reveals transitions between monostable periodic motion, multistability, and chaos.

Rather than solving in the frequency domain (see Eq. 2), we now analyze the time-domain response. Figure 6 shows the transient dynamics after a sudden application of the drive. In the linear regime (\(g<1\)), all initial conditions decay to a unique steady state, so repeated measurements converge to the same relaxed response [Fig. 6(a)]. With Duffing nonlinearity, the system becomes bistable: trajectories settle into one of two coexisting attractors [Fig. 6(b)] basins set by initial conditions and noise. Increasing the nonlinearity or drive amplitude induces successive bifurcations (e.g., period-doubling), culminating in aperiodic, chaotic motion. The transition from a single solution to multiple coexisting solutions is a bifurcation [cite].

Experiments on Poincare trajectory

Experiments on Poincare trajectoryExperimental Poincaré sections were obtained by stroboscopically sampling the in-phase (X) and quadrature (Y) responses of a driven nonlinear resonator with a lock-in amplifier. The lock-in provided both the drive and synchronous demodulation at the drive frequency; downsampling once per drive period T yielded the discrete-time map consistent with the theoretical Poincaré framework.

We drove the resonator in the \(19-20\)kHz range i.e. within the hysteresis region. Each run started from \(V_{ac}=0\)V; the drive was then stepped to a target amplitude within the hysteretic (Duffing) regime (\(V_{ac}=3\)V), and X,Y were recorded with identical timestamps. This procedure captures the transient trajectory in phase space and its asymptotic state, enabling reconstruction of the Poincaré section.

Within the hysteresis region, the dynamics are bistable: for the same drive frequency and amplitude, trajectories relax to one of two coexisting T-periodic limit cycles. On the Poincaré section these appear as two stable fixed points (Fig. 7), each with its own basin of attraction. An unstable periodic orbit lies between them; its stable manifold forms the separatrix. The final state is set by the initial condition and noise relative to this boundary, leading to probabilistic switching between attractors across repeated trials. Notably, the transient path in phase space is reproducible for a given attractor, even though the selected attractor varies between runs.

Backbone measurement

The backbone curve of a Duffing resonator captures how the resonant frequency shifts with amplitude due to nonlinear stiffness, revealing hardening or softening behavior. Measuring it provides direct identification of key parameters—natural frequency, damping, and the cubic stiffness coefficient—that are essential for accurate modeling and prediction of dynamics. This enables to quantify the nonlinearity of the nonlinear resonator and predict the hysteresis behavior of the system.

Conventional backbone measurements use repeated frequency sweeps at increasing drive to track the resonance peak shift. This approach is time-consuming, susceptible to drift over long runs, and requires substantial post-processing. By contrast, PLL tracking locks at the resonance phase and continuously follows the resonance frequency while the drive amplitude is ramped, allowing tracking of resonant amplitude as a function of frequency in a single pass. This greatly shortens acquisition, minimizes drift, and eliminates peak-finding post-processing; the recorded frequency detuning versus resonant amplitude is the backbone used to extract the nonlinear spring constant. Set the PLL bandwidth to follow slow amplitude-induced shifts (below the resonator linewidth) and run up/down amplitude ramps to capture both Duffing branches.

We begin by phase-locking the piezo resonator near its resonant phase (about \(88.45\) degrees) within the linear regime with \(g\ll1\). While locked, we slowly increase the drive and record the amplitude and phase. As the drive increases, the resonance frequency shifts; using the sweeper’s XY plot, we reconstruct the backbone curve by plotting amplitude and phase versus resonance frequency (Figure 8). Throughout the sweep, we continuously monitor the locked phase shown by the yellow graph.

Conclusion

This work delineates resonator dynamics across linear Lorentzian response, weak nonlinearity with harmonic generation and slight frequency pulling, and strong Duffing behavior with fold bifurcations, hysteresis, and multistability, validated on a piezo by harmonics in FFTs, bidirectional sweeps, Poincaré sections, and a fast PLL‑based backbone. Beyond this bistable regime, chaos can emerge when drive or effective anharmonicity is increased and damping reduced, or via added parametric/intermode coupling. Achieving and diagnosing it demands controlled amplitude/parameter modulation, low‑noise time‑series for phase‑space reconstruction, and quantitative tests (Poincaré maps, broadband spectra, Lyapunov exponents).

All the measurement showed here were performed by the MFLI lock-in amplifier. However, all of our lock-ins are particularly well suited for such measurements.

References

[1] Lifshitz & Cross, "Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators", Review of Nonlinear Dynamics and Complexity, March 5 (2008).

[2] Karabalin, Cross, and Roukes, “Nonlinear dynamics and chaos in two coupled nanomechanical resonators,” Nano Letters (2009).

[3] Nony, Boisgard and Aimé, "Nonlinear dynamical properties of an oscillating tip–cantilever system in the tapping mode" , J. Chem. Phys. 111, 1615–1627 (1999).

[4] Erturk & Inman, "Piezoelectric Energy Harvesting", Wiley (2011). 

[5] Chowdhury et al. "Weak signal enhancement by nonlinear resonance control in a forced nano-electromechanical resonator", Nature Communications 11, 2400 (2020).