Self-Excitation Effects in Complex Systems

• Self-excitation effects are defined as internal feedback mechanisms that autonomously amplify fluctuations without relying on external periodic forcing.
• They manifest in diverse domains such as nonlinear dynamics, stochastic processes, and physical devices, with examples including Hawkes processes, plasma oscillations, and nanomechanical systems.
• Mathematical frameworks like bifurcation theory and renewal decompositions rigorously delineate thresholds, attractor properties, and stability conditions for self-excited phenomena.

Self-excitation effects arise when a physical or mathematical system contains feedback mechanisms that allow its internal dynamics to spontaneously amplify fluctuations or oscillate without external periodic forcing. In such systems, the energy for sustained oscillation, enhanced response, or increased event rate is sourced from the system’s reservoir (e.g., steady bias, intrinsic nonlinearity, or embedded feedback), rather than exclusively from a time-periodic external drive. Self-excitation phenomena are central in nonlinear dynamics, stochastic processes, plasmas, electromechanical resonators, spintronics, nanodevices, lasers, and information diffusion models. Rigorous mathematical frameworks establish the conditions for self-excited attractors, bifurcation boundaries, and feedback thresholds in discrete and continuous systems.

1. Mathematical Frameworks for Self-Excitation

Self-excitation is rigorously characterized via the analysis of system attractors and bifurcation theory. In smooth autonomous dynamical systems

$\frac{dx}{dt} = f(x,\mathbf{p}),$

a self-excited attractor A is defined by the property that its basin of attraction intersects any neighborhood of at least one unstable equilibrium $x^*$ of the system; typically, such attractors emerge via local bifurcations (e.g., Hopf, saddle-node of limit cycles) as a system parameter $\mathbf{p}$ is varied and eigenvalues $\lambda_i$ of the Jacobian $J = \partial f/\partial x|_{x^*}$ cross the imaginary axis (Kiseleva et al., 2016).

In stochastic processes, self-excitation corresponds to mechanisms where the instantaneous rate (intensity) or amplitude at time $t$ depends explicitly on the system's history, generating endogenous clusters or bursts. Key examples include Hawkes processes, self-excited multifractal processes, and delayed-feedback pulse trains (Bonnet et al., 2021, Filimonov et al., 2010, Ruschel et al., 2020).

Self-excited phenomena are also formalized in coupled chains (e.g., frictional or Van der Pol oscillators) via the study of spatially localized discrete breathers, nucleation thresholds, and propagating excitation fronts (Shiroky et al., 2019).

2. Self-Excitation in Hawkes-Type Point Processes

Hawkes processes exemplify self-excitation in continuous-time stochastic systems. For a univariate process, the conditional intensity function is

$$\lambda(t) = \left(\mu + \sum_{T_i < t} \alpha e^{-\beta (t-T_i)}\right)^+,$$

where $\mu$ is the baseline rate, $\alpha$ is the excitation (positive) or inhibition (negative) amplitude, and $\beta$ is the decay rate (Bonnet et al., 2021). Each event instantaneously increases $\lambda$ by $\alpha$, with the influence decaying exponentially.

Two mathematical regimes are distinguished:

• Self-exciting ($\alpha > 0$): Each event boosts future intensity, favoring event clustering. The process admits a branching (cluster) representation and the stationary event rate is amplified to $\mu/(1-m)$, where $m = \int h(u) du$ is the branching ratio.
• Self-inhibiting ($\alpha < 0$): Each event depresses future intensity; for sufficiently strong inhibition, the process displays extended intervals where $\lambda(t)=0$ and standard cluster representations fail, requiring exact renewal-theoretic constructions (Costa et al., 2018).

Maximum likelihood estimation for Hawkes processes requires precise handling of the restart times in inhibition regimes, as naive compensator approximations fail when zero-intensity intervals proliferate (Bonnet et al., 2021). Rigorous renewal decompositions and exponential concentration bounds quantify the impact of self-excitation and kernel structure on process statistics (Costa et al., 2018).

In high-dimensional settings, the self-excitation in mutually-interacting point processes is captured by nonlocal drift terms in the McKean-Vlasov equation, yielding contraction phenomena in the steady-state distribution (Wang et al., 2018).

3. Nonlinear Resonance and Self-Excitation in Physical Systems

Self-excitation is intrinsic to many nonlinear physical systems, particularly in plasma physics, nanomechanics, spintronics, and optomechanics.

Plasma Series Resonance (PSR): In RF capacitively coupled plasmas, the bulk plasma inductance and nonlinear sheath capacitance couple to enable spontaneous high-frequency oscillations (PSR) above the driving frequency. Equivalent circuit analysis shows PSR self-excitation occurs if the sheath charge–voltage relation is non-quadratic or the bulk electron plasma frequency is time-dependent; such mechanisms break symmetry cancellation in the circuit and facilitate positive feedback (Schuengel et al., 2016, Schuengel et al., 2016). PIC/MCC simulations confirm spatially localized bursts of electron heating and the generation of energetic beam electrons due to PSR (Schuengel et al., 2016).

Nanomechanical Self-Excitation: Nano-electromechanical systems (NEMS) such as pillars or shuttles can attain self-excited motion when electronic feedback (e.g., DC bias-induced tunneling force) overcomes viscous dissipation, producing oscillations at mechanical resonances. The onset is set by net negative damping, with amplitude scaling $\sim(V_{dc} - V_{th})^{1/2}$, and regimes such as hysteresis and bistability are observed under asymmetrical electrode conditions (0708.1646).

Optomechanical Resonance: Cantilevers in AFM can be driven to self-excitation by photothermal forces when laser power exceeds a threshold $P_{th}$ where damping becomes negative. The amplitude saturates via fourth-order nonlinear damping, producing limit cycles and substantially increased effective quality factor $Q_{eff}\gg Q_0$ (out-of-equilibrium dynamics) (Milde et al., 2022).

Spintronic Oscillators: In perpendicularly magnetized nano-contacts, spin-transfer torque (STT) can drive self-excited droplet solitons. Parametric instabilities arise when the STT excites perimeter modes whose frequency matches half the droplet precession, subject to critical current density and tilt of spin polarization (Xiao et al., 2016). Similarly, indirect excitation by the spin Hall effect requires reflection torque from a tilted pinned layer to produce net positive energy over one precession, enabling self-oscillation (Taniguchi, 2017).

4. Feedback Mechanisms and Thresholds in Self-Excited Devices

Self-excitation universally requires a feedback loop whose net effect is to overcome system losses, crossing a mathematical or physical threshold.

Electronic Circuits: In circuits with negative differential conductance (e.g., quantum dot junctions with R-L-C embedding), feedback via an inductor in the presence of Coulomb and spin blockade allows self-excited oscillations via a Hopf bifurcation at critical inductance $L_c$, with GHz-frequency current and spin oscillations (Radic et al., 2010).

Magnetomotive Instability: Suspended nanoribbons or nanotubes in RLC circuits, under magnetomotive Lorentz forces, can exhibit selective self-excitation of higher-order vibrational modes by tuning circuit resonance frequencies and magnetic field. Instability analysis provides mode-selective threshold criteria $|\beta_n S_n| > 1/Q$ for excitation (Nordenfelt, 2011).

Surface Plasmon Polaritons (SPP): In plasmonic nanocavities, positive feedback from SPP field reflection drives self-excitation in ensembles of polarizable inclusions, yielding coherent SPP modes without stimulated emission (contrast: classical SESP vs. quantum SPASER) (Bordo, 2015).

Delay Feedback in Lasers: Delayed optical feedback in excitable lasers (Yamada model with saturable absorber) allows periodic pulse trains to be sustained by self-excitation. Intricate bifurcation landscapes involving fold periodic orbits and codimension-two points delimit regions of parameter space supporting stable trains, with conditions for dissipative soliton stability linked to pseudo-continuous spectra (Ruschel et al., 2020).

Digital Self-Excited Loops (SELs): Positive feedback SEL controllers regulate SRF cavities by establishing a self-excited oscillation that locks to cavity resonance even under detuning or perturbations. Digital implementations introduce latency and quantization effects, requiring careful control law design (e.g., PI gains, CORDIC latency, stateful phase resolvers) to maintain the desired amplitude and recover from large excursions (Doolittle et al., 16 Oct 2025).

5. Self-Excitation in Coupled Arrays and Front Propagation

Self-excitation phenomena in spatially extended arrays are marked by localized breathers, nucleation, and propagating excitation fronts.

Coupled Oscillator Chains: In friction-excited or Van der Pol chains, each unit cell can support large-amplitude limit cycles via internal feedback (e.g., velocity-weakening friction or nonlinear damping). Under weak coupling, discrete breathers (localized periodic solutions) are stable; above critical coupling $c_{crit}$, breathers nucleate propagating excitation fronts with speed scaling linearly with coupling coefficient ($v_{front} \sim c$) (Shiroky et al., 2019).

Bifurcation Topologies: These systems universally exhibit bi-stable unit cell dynamics (trivial and large limit-cycle attractors), with front velocity and nucleation thresholds robust to model variation.

6. Statistical, Multifractal, and Network Dynamics

Statistical models with self-excitation capture multifractal and heavy-tailed phenomena.

Self-Excited Multifractal Dynamics: The SEMF process modulates the scale of future increments exponentially by a long-memory functional of past increments:

$$d_n = \sigma \xi_n \exp\left\{-\frac{1}{\sigma}\sum_{i<n} d_i h_{n-i-1}\right\},$$

yielding distributions with power-law tails, long-range volatility correlations, leverage effects, and time-reversal asymmetry. Despite parameter flexibility and memory kernel shape, multifractality and endogenous cluster formation are robust emergent properties (Filimonov et al., 2010).

Information Diffusion: In mutually-exciting opinion models, the macroscopic evolution of interacting agents (opinions) is governed by mean-field limits with nonlocal drift, encoding the reinforcing effect of self-excitation and yielding contraction of distribution support in steady state, preserving diversity (Wang et al., 2018).

7. Applications and Implications Across Disciplines

Self-excitation effects have wide-reaching applications and implications:

• Physical device engineering: Tunable oscillators, frequency combs, ultrafast detectors, quantum mixers, plasmonic generators, sensors with high responsivity to environment perturbations (Bordo, 2022, Macnaughtan et al., 12 Jun 2025, 0708.1646, Bordo, 2015).
• Plasma and surface processing: Optimized electron heating profiles and separate control of ion flux/energy in semiconductor fabrication (Schuengel et al., 2016).
• SRF cavity control: Robust locking and detuning compensation in high-Q RF systems (Doolittle et al., 16 Oct 2025).
• Spintronic and nanomagnetic systems: Controlled auto-oscillation in magnetic droplets and perpendicular ferromagnets (Xiao et al., 2016, Taniguchi, 2017).
• Stochastic modeling: Accurate modeling and inference in event clustering, financial volatility, seismology, epidemic cascades (Bonnet et al., 2021, Costa et al., 2018, Filimonov et al., 2010).
• Laser physics: Deterministic soliton microcomb generation and periodic pulse trains via controlled feedback (Ruschel et al., 2020, Macnaughtan et al., 12 Jun 2025).

The universality of self-excitation mechanisms—positive feedback, nonlinear damping, time-dependent coupling, delayed response, and endogenous reinforcement—underpins their relevance to emergent complexity in both physical and information systems. However, system stability, threshold tuning, and control law design remain critical for exploiting or mitigating self-excitation effects in application domains.