Scaling Exponents at Boundary Crisis

• The paper demonstrates that scaling exponents capture how abrupt attractor collisions at phase-space boundaries trigger critical transitions in dynamical systems.
• It employs analytical models and numerical simulations to establish universal power-law relations across conservative, dissipative, and stochastic regimes.
• The research highlights the impact of boundary-induced anomalies on scaling laws, offering predictive insights for applications in transport theory and quantum systems.

Scaling exponents near a boundary crisis quantify critical changes in the dynamics of physical, mathematical, and statistical systems as control parameters cross thresholds that destroy or generate attractors at the phase space boundary. This concept is central in nonlinear dynamics, statistical mechanics, transport theory, and critical phenomena, where transitions describe qualitative changes in long-time behavior and scaling exponents relate observable quantities to external parameters in universal fashion. Analysis of these exponents exposes fundamental relationships between geometry, stochasticity, and dynamical properties, with manifestly distinct behaviors in conservative, dissipative, and stochastic regimes. Below, key mechanisms, mathematical formulations, and universal signatures from recent research are systematically outlined.

1. Universal Scaling Relations in Dynamical Boundary Crises

The archetypal signature of a boundary crisis is the abrupt collision of an invariant set (such as a chaotic attractor or saddle) with the basin boundary under variation of a control parameter (e.g., dissipation or external forcing). Scaling exponents emerge in three critical domains:

• Location of Spanning Curves: In area-preserving maps (e.g., wave packet-driven particle acceleration (Oliveira et al., 2011)), the position $|\phi^*|$ of the first invariant curve that bounds chaos scales with the control parameter $\gamma$ as $|\phi^*| \sim \gamma^{2/3}$.
• Transient Lifetime Scaling: Following a crisis, transients exhibit power-law decay in mean lifetime: $n_t \sim \mu^{\zeta}$ with $\zeta \simeq -2$ for dissipation-induced boundary crises, where $\mu$ is the parametric distance from the crisis threshold.
• Supertransient Escape Probability: In non-autonomous or fluctuating environments (e.g., Kuramoto model with inertia (Olmi et al., 17 Mar 2025)), escape probabilities scale as $Q_\infty(\delta) \sim \exp[-\alpha(\ln\delta)^2]$, with $\delta$ denoting distance from the crisis point, revealing stretched exponential behavior rather than pure power laws.

These exponents reflect the geometric and probabilistic structure of the underlying basin boundaries and are universal across wide classes of systems.

2. Mathematical Structures and Power Laws

Precise characterization requires explicit mapping of the dynamics onto minimal models or return maps. Representative formulations include:

• Two-dimensional area-preserving map (wave-packet acceleration):

$$\begin{aligned} \phi_{n+1} &= \phi_n + \gamma \sin(2\pi\beta_n) \ \beta_{n+1} &= \left[ \beta_n - \frac{1}{2\pi|\phi_{n+1}|^{1/2}} \right]\bmod 1 \ \end{aligned}$$

The analytic connection with the standard map allows calculation of critical curves and exponents, with power-law saturation of kinetic energy standard deviation $\omega_\text{sat}\sim\gamma^{2/3}$, and growth $\omega\sim(n\gamma^2)^{1/2}$ for short times.

• Edge and Ghost States in Multistable Climate Models (AMOC (Börner et al., 28 Apr 2025)):

Critical transitions are mediated by the collision of attractors and edge (Melancholia) states, after which the mean lifetime $\tau$ of ghost transients follows $\tau\sim(\lambda_c-\lambda)^{-\gamma}$.

• Stochastic Linear Models for Crisis Escape:

Linearized stochastic propagators within “grey zones” near basin boundaries yield escape probability $G_M(\delta)\approx\exp[-\alpha(\ln\delta)^2]$, capturing the crisis scaling in both low and high-dimensional systems.

These formulations provide both general mechanisms and specific exponents for dynamical regimes near crisis points.

3. Boundary-Induced Anomalous Scaling and Crossover Effects

Boundary effects systematically alter scaling laws and induce crossovers in growth exponents. In non-equilibrium growth processes and diffusion:

• Interface Growth (Edwards–Wilkinson and KPZ Models (Allegra et al., 2013)):

Near a rigid boundary, the interface height profile and local width become functions of the scaled variable $\lambda=x^2/4t$; the local interface width scales as $w\sim t^{\beta_{\text{eff}}}$ with $\beta_{\text{eff}}> \beta_\text{bulk}$ at intermediate times, reverting to bulk scaling at long times.

• Diffusive Infiltration in Fractals (Reis, 2017):

The interface filling scales as $F\sim t^n$ with $n=\nu(D_\text{F}-D_\text{B})$, connecting the single-particle anomalous diffusion exponent $\nu$ and the difference of fractal dimensions for the bulk ($D_\text{F}$) and boundary ($D_\text{B}$). Discrete scale invariance generates log-periodic oscillations superposed on leading power laws.

• First-Passage Times to Fractal Boundaries (Ye et al., 1 Oct 2024):

Survival probability $S(t)$ exhibits apparent power-law decay, $S(t)\sim t^{-\alpha}$, where the local persistence exponent $\alpha(t)$ shows time-dependent plateaus and log-periodic oscillations set by the spatial scaling factor of the boundary.

These results highlight that while bulk (translationally invariant) systems display universal scaling, the presence of boundaries—especially with fractal or self-similar structure—creates regime-dependent exponents, oscillatory corrections, and anomalous enhancements.

4. Quantum, Statistical, and Transport Systems: Scaling Near Boundary Crises

In open quantum systems, statistical models, and hydrodynamic transport theory, scaling exponents encode the transition from rapid to slow relaxation as boundaries shift system dynamics:

• Liouvillian Gap Scaling in Boundary-Dissipated Systems (Zhou et al., 2022):

The Liouvillian gap, dictating relaxation rates, scales as $\Delta_g\sim L^{-3}$ for extended states, $\Delta_g\sim e^{-\kappa L}$ for localized states, with $\kappa$ the Lyapunov exponent of eigenstate localization. The transition between power-law and exponential scaling precisely demarcates a dissipative boundary crisis.

• High-Rayleigh-Number Convection Scaling (Tong, 26 Mar 2024, Kazemi et al., 2021):

Nusselt and Reynolds numbers depend nontrivially on Rayleigh number. Conventionally, $Nu\sim Ra^{1/3}$ signifies boundary-limited scaling, whereas $Nu\sim Ra^{1/2}$ signals mixing-length (“ultimate”) scaling. In net-zero heating/cooling scenarios, a continuous spectrum of exponents emerges, corresponding to the fraction of flux crossing the boundary.

• Boundary Finite-Time Scaling in Driven Critical Dynamics (Shu et al., 12 Sep 2025):

Near boundaries, classic Kibble–Zurek scaling generalizes to boundary finite-time scaling (BFTS). For heating, surface exponents $\beta_1$ control order-parameter scaling; for cooling in the ordinary transition, logarithmic corrections predominate: $M_{s1}^2\sim L^{-(d-1)}\log(bv)$, violating naive power-law scaling.

These varied behaviors underscore the essential role of boundary conditions, boundary geometry, and dynamical localization in controlling critical scaling exponents.

5. Exotic and Logarithmic Scaling Laws in Deep Boundary Criticality

Novel scaling behaviors arise in quantum critical points when boundary perturbations decay algebraically into the bulk (Liu, 19 Nov 2024):

• Deep Boundary Criticality: Perturbations $\sim x^{-\alpha}$ induce either irrelevant (power-law scaling) or relevant (logarithmic/fractional power) corrections to expectation values. For the marginal regime $\alpha=1$, leading terms are $m(x)\sim -(\lambda\log x)/(\pi x)$ plus higher-order fractional powers determined by $\beta=|2\lambda-1|$ or $|2\lambda|$.
• Transition of Exponents: As $\alpha$ is tuned through 1, the scaling exponent shifts from pure power law ($1/x$) to logarithmic and nonuniversal fractional behavior. This exemplifies a “crisis” in scaling relations induced by long-range boundary effects.

This scenario illustrates that near a boundary crisis, effective exponents need not be universal and can vary continuously, depending on decay profiles or coupling strengths.

6. Implications, Universality, and Predictive Utility

Scaling exponents near boundary crises offer a robust diagnostic across disciplines:

• Critical Dynamics: Exponents (especially those connected to transient lifetimes and escape probabilities) provide predictive tools for phase transitions, tipping points in climate systems, and loss of stability in coupled oscillators.
• Universality and Regime Structure: Recurrence of similar exponents (e.g., $-2$ in transient scaling, $2/3$ in invariant curve location, $1/2$ in velocity near turning points) demonstrates cross-model universality, though boundary complexity introduces anomalous and regime-specific corrections.
• Stochastic and Nonautonomous Systems: Quadratic logarithmic scaling in escape probabilities (Olmi et al., 17 Mar 2025) reveals a new universal crisis scaling in fluctuating environments, critical for quantifying resilience to perturbations.
• Boundary-Induced Multiscaling: The presence of boundaries, especially fractal or self-similar, produces effective local scaling exponents and oscillatory corrections that must be included for a comprehensive description of real-world phenomena.

7. Table: Representative Scaling Exponents at Boundary Crisis Transitions

Physical/Model Context	Observable or Quantity	Scaling Exponent(s)
Area-preserving map (Oliveira et al., 2011)	Invariant curve position	$2/3$
Same	Kinetic energy Std.Dev. growth	$1/2$; crossover $z\sim -0.65$
Dissipative map (Oliveira et al., 2011)	Transient time after crisis	$-2$
Fractal diffusion (Reis, 2017)	Filling exponent	$n=\nu(D_\mathrm{F}-D_\mathrm{B})$
Open quantum systems (Zhou et al., 2022)	Liouvillian gap (localized phase)	$\Delta_g\sim e^{-\kappa L}$
Nonautonomous crisis (Olmi et al., 17 Mar 2025)	Escape probability	$\sim\exp[-\alpha(\ln\delta)^2]$
Boundary-driven criticality (Shu et al., 12 Sep 2025)	Order parameter (cooling, ordinary)	Logarithmic scaling

All exponents and forms above are directly traceable to the referenced research and represent the critical scaling structure near boundary crises.

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In total, scaling exponents near boundary crises encapsulate both universal and system-specific features, governing transient lifetimes, amplitude growth, critical escape probabilities, and more. Their calculation and interpretation require explicit account of phase-space geometry, boundary structure, stochasticity, and parameter regime, with broad consequences across dynamical systems, statistical models, quantum theory, and geophysical transport.