The butterfly effect, namely the amplification of an initially localised infinitesimal perturbation, is a fundamental hallmark of chaotic systems. While this phenomenon is well understood in low-dimensional chaotic systems, its implications become more complex in spatially extended systems characterised by multiple temporal and spatial scales, such as geophysical flows. In this letter, we investigate the butterfly effect by examining the evolution of infinitesimal localised perturbations in a minimal model of mesoscale geophysical turbulence. We find that localised perturbations exhibit a highly variable dynamics, including an initial transient regime during which perturbation energy decreases while the error spectrum shifts from the injection scale toward the system’s most unstable configuration. The duration of this transient regime spans a broad range and may persist for several characteristic small-scale times, depending on the perturbation’s size, its initial location and the Reynolds number.

We investigate the exponential growth of uncertainty energy in three-dimensional (3-D) Navier–Stokes turbulence, emphasising the intermittent and highly localised amplification/production of uncertainty, a critical factor in understanding the predictability of turbulent systems. From the Navier–Stokes equations, one can identify some key fields contributing to the growth/decay of uncertainty-production term $P_{\varDelta }$: strain rate, vorticity and vortex deformation. The dynamics of these fields is examined in the $Q{-}R$ plane, where $Q$ and $R$ are the second and third invariants of the velocity gradient tensor, to understand their role in the evolution of uncertainty-production term $P_{\varDelta }$. We proceed by estimating committor functions across the entire spatiotemporal domain of the direct numerical simulation (DNS) of turbulence in a periodic domain at different Reynolds numbers. Our estimates of the probability of rare extreme events of the local uncertainty-production term as a function of uncertainty energy, strain rate, vorticity and vortex deformation confirm the role of strain rate in driving uncertainty. Where strain rate and vorticity are too close to their space-average values, stable probabilistic forecasts appear impossible solely based on the fields considered here.

We perform a long-time Lyapunov analysis of plane Couette turbulence up to friction Reynolds number $h^+=269$. The energy of the leading Lyapunov vector is predominantly concentrated in the buffer layer, with characteristic streamwise and spanwise wavelengths $\lambda _1^+\simeq 100$ and $\lambda _3^+\simeq 50$, respectively. These wavelengths are smaller than the corresponding streamwise and spanwise periods of a minimal flow unit. Through Schmidt–Lyapunov analysis, we find that the first and second Lyapunov exponents are close in value, and that the energy spectra of the first Lyapunov vector and the second Schmidt vector have peaks at similar characteristic wavelengths in the buffer layer. These imply that the fastest orbital instability is not organised by a single minimal-flow-unit-scale perturbation, but by a nearly degenerate set of small-scale buffer-layer perturbation directions, which may be interpreted as a superposition of such perturbations. Coarse graining with a Gaussian low-pass filter further shows that the vortical structures of the local Lyapunov vector tend to appear around near-wall large-scale quasi-streamwise vortices. These results suggest that the orbital instability in wall-bounded turbulence is associated with small-scale buffer-layer perturbations influenced by the strain field around near-wall large-scale structures.

The evolution of small perturbations applied to turbulent Couette flow is examined over a long time horizon, until the perturbed flow field becomes completely decorrelated from the original state. To elucidate the fundamental physical processes involved, we focus on the minimal flow unit, where the dynamics of coherent structures is well understood in terms of the self-sustaining process (Hamilton et al. J. Fluid Mech. vol. 287, 1995, pp. 317–348). As expected, in the short term, perturbations exhibit exponential growth governed by the leading Lyapunov exponent. This mechanism is driven by the streamwise-dependent flow, which is known to involve intense turbulent dissipation events within the self-sustaining process, consistent with previous findings. Beyond the initial exponential phase, we observe a slow, sustained growth of perturbations over a long period – spanning tens of integral time scales – before eventual saturation. During this stage, the perturbation energy increases approximately linearly with time. While this behaviour resembles observations and predictions in other turbulent flows, the underlying physical process here is fundamentally different. Specifically, the perturbation field during this period is dominated by streaky structures and the growth mechanism is linked to the saturation of the wall-normal streak length scale at the largest dimension permitted by the flow geometry (i.e. the channel height). Finally, an evaluation of the dominant production term components reveals that the well-known lift-up effect is primarily responsible for the growth of these streaky perturbations.

We investigate the stability of the flow past two side-by-side square cylinders (at Reynolds number 200 and gap ratio 1) using tools from dynamical systems theory. The flow is highly irregular due to the complex interaction between the flapping jet emanating from the gap and the vortices shed in the wake. We first perform spectral proper orthogonal decomposition (SPOD) to understand the flow characteristics. We then conduct Lyapunov stability analysis by linearising the Navier–Stokes equations around the irregular base flow and find that it has two positive Lyapunov exponents. The covariant Lyapunov vectors (CLVs) are also computed. Contours of the time-averaged CLVs reveal that the footprint of the leading CLV is in the near-wake, whereas the other CLVs peak further downstream, indicating distinct regions of instability. SPOD of the two unstable CLVs is then employed to extract the dominant coherent structures and oscillation frequencies in the tangent space. For the leading CLV, the two dominant frequencies match closely with the prevalent frequencies in the drag coefficient spectrum and correspond to instabilities due to vortex shedding and jet-flapping. The second unstable CLV captures the subharmonic instability of the shedding frequency. Global linear stability analysis (GLSA) of the time-averaged flow identifies a neutral eigenmode that resembles the leading SPOD mode of the first CLV, with a very similar structure and frequency. However, while GLSA predicts neutrality, Lyapunov analysis reveals that this direction is unstable, exposing the inherent limitations of the GLSA when applied to chaotic flows.

We show that both temporal and spatial symmetry breaking in canonical K-type boundary layer transition arise as organised structures with quantifiable energetic pathways rather than unstructured noise. Before the skin-friction maximum, the flow is described by a periodic, spanwise-symmetric fundamental harmonic response (FHR) to the Tollmien–Schlichting wave. The FHR is spatially compact, produces hairpin packets and remains fully harmonic despite a turbulence-like appearance, thereby delimiting the deterministic regime. Past this point, a distinct regime change occurs: a hierarchy of quasi-periodic and aperiodic structures emerges, followed shortly by anti-symmetric structures that develop similarly despite no anti-symmetric inputs. We identify these structures as symmetry-decomposed spectral and space–time proper orthogonal modes that resolve the progression from deterministic harmonics to broadband dynamics. We introduce inter-modal and inter-symmetry energy budgets derived from symmetry-decomposed Navier–Stokes equations. They reveal a directed energy transfer from the FHR into the leading temporal and spatial symmetry breaking modes and, subsequently, into broadband residual fluctuations, showing that broadband dynamics grow only once inter-modal transfer is active, while inter-symmetry transfer also strongly amplifies broadband anti-symmetric fluctuations once asymmetry is present. These key insights support a view of laminar–turbulent transition as a sequence of symmetry breaking events, energetically driven by dominant space–time modes that route energy from harmonic flow to broadband turbulence.

We study a reaction–advection–diffusion model of a target–offender–guardian system designed to capture interactions between urban crime and policing. Using Crandall–Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime–policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighbourhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analysing non-linear interactions, bifurcations and pattern formation in multi-agent social systems.

We use the theory of spectral submanifolds (SSMs) to develop a low-dimensional reduced-order model for plane Couette flow restricted to the shift–reflect invariant subspace in the permanently chaotic regime at ${Re}=187.8$ studied by Kreilos & Eckhardt (2012, Chaos: Interdisciplinary J. Nonlinear Sci., vol. 22, 047505). Our three-dimensional model is obtained by restricting the dynamics to the slowest mixed-mode SSM of the edge state. We show that this results in a nonlinear model that accurately reconstructs individual trajectories, representing the entire chaotic attractor and the laminar dynamics simultaneously. In addition, we derive a two-dimensional Poincaré map that enables the rapid computation of the periodic orbits embedded in the chaotic attractor.

The growth of small perturbations in isotropic turbulence is studied using massive ensembles of direct numerical simulations. These ensembles capture the evolution of the ensemble-averaged flow field and the ensemble variance in the fully nonlinear regime of perturbation growth. Evolution equations for these two fields are constructed by applying the ensemble average operator to the Navier–Stokes equations and used to study uncertainty growth in scale and physical space. It is shown that uncertainty growth is described by a flux of energy from the ensemble-averaged flow to the ensemble variance. This flux is formally equivalent to the subgrid scale (SGS) energy fluxes of the turbulence cascade, and can be interpreted as an inverse uncertainty cascade from small to large scales. In the absence of information sources (measurements), the uncertainty cascade is unsteady and leads to the progressive filtering of the small scales in the ensemble-averaged flow, a process that represents the loss of predictability due to chaos. Similar to the kinetic energy cascade, the uncertainty cascade displays an inertial range with a constant average uncertainty flux, which is bounded from below by the average kinetic energy dissipation. Locally in space, uncertainty fluxes differ from the SGS energy fluxes at the same scale, but both have similar statistics and are significantly correlated with each other in space. This suggests that uncertainty propagation is partly connected to the energy cascade and that they share similar mechanisms. These findings open avenues to model uncertainty propagation in turbulence following an approach similar to the SGS models in large-eddy simulations. This is relevant not only to efficiently assess the reliability and accuracy of turbulence forecasts, but also to design uncertainty-robust reconstruction techniques for data assimilation or SGS modelling.

Elastic turbulence can lead to increased flow resistance, mixing and heat transfer. Its control – either suppression or promotion – has significant potential, and there is a concerted ongoing effort by the community to improve our understanding. Here we explore the dynamics of uncertainty in elastic turbulence, inspired by an approach recently applied to inertial turbulence in Ge et al. (J. Fluid Mech., vol. 977, 2023, A17). We derive equations for the evolution of uncertainty measures, yielding insight on uncertainty growth mechanisms. Through numerical experiments, we identify four regimes of uncertainty evolution, characterised by (i) rapid transfer to large scales, with large-scale growth rates of $\tau ^{6}$ (where $\tau$ represents time), (ii) a dissipative reduction of uncertainty, (iii) exponential growth at all scales and (iv) saturation. These regimes are governed by the interplay between advective and polymeric contributions (which tend to increase uncertainty), viscous, relaxation and dissipation effects (which reduce uncertainty) and inertial contributions. In elastic turbulence, reducing Reynolds number increases uncertainty at short times, but does not significantly influence the growth of uncertainty at later times. At late times, the growth of uncertainty increases with Weissenberg number, with decreasing polymeric diffusivity and with the logarithm of the maximum length scale, as large flow features adjust the balance of advective and relaxation effects. These findings provide insight into the dynamics of elastic turbulence, offering a new approach for the analysis of viscoelastic flow instabilities.

The recent detection of high-redshift supermassive black holes with JWST has renewed interest in the processes driving black hole growth. At the same time, both simulations and observations point to a widespread population of off-center intermediate-mass black holes in dwarf galaxies. Their ability to merge outside galactic centers may play a key role in shaping black hole mass assembly. Here, we investigate the dynamics of off-center black holes in dwarf galaxies hosting cored dark matter haloes, where long dynamical friction timescales and core stalling naturally arise. By embedding off-center black holes into an idealized galactic potential and scanning a wide range of orbital configurations, we assess the likelihood of close interactions through Jacobi capture. We find that captures occur in about 13% of cases. Such captures, possibly sustained within compact stellar systems like stripped nuclei or globular clusters, represent a crucial first step toward assembling massive black hole binaries beyond galactic centers.

We derive the scale-by-scale uncertainty energy budget equation and demonstrate theoretically and computationally the presence of a self-similar equilibrium cascade of decorrelation in an inertial range of scales during the time range of power-law growth of uncertainty in statistically stationary homogeneous turbulence. This cascade is predominantly inverse and driven by compressions of the reference field’s relative deformation tensor and their alignments with the uncertainty velocity field. Three other subdominant cascade mechanisms are also present, two of which are forward and also dominated by compressions and one of which, the weakest and the only nonlinear one of the four, is inverse. The uncertainty production and dissipation scalings which may follow from the self-similar equilibrium cascade of decorrelation lead to power-law growths of the uncertainty integral scale and the average uncertainty energy which are also investigated. Compressions are key not only to chaoticity, as previously shown, but also to stochasticity.

When a liquid film on a horizontal plate is driven in motion by a shear stress, surface waves are easily generated. This paper studies such flow at moderate Reynolds numbers, where the surface tension and inertial force are equally important. The governing equations for two-dimensional flows are derived using the long-wave approximation along with the integral boundary-layer theory. For small disturbances, the dispersion relation and neutral curves are determined by the linear stability analysis. For finite-amplitude perturbations, the numerical simulation suggests that the oscillations generated by the perturbation in a certain place continuously spread to the surrounding areas. When the effects of surface tension and gravity reach equilibrium, steady-state solutions will emerge, which include two cases: solitary waves and periodic waves. The former have heteroclinic trajectories between two stationary points, while the latter include five patterns at different parameters. In addition, there are also periodic waves that do not converge after a long period of time. During these evolution processes, strange attractors appear in the phase space. By examining the Poincaré section and the sensitivity to initial values, we demonstrate that these waves can be divided into two types: quasi-periodic and chaotic solutions. The specific type depends on parameters and initial conditions.

Feigenbaum universality is shown to occur in subcritical shear flows. Our testing ground is the counter-rotation regime of the Taylor–Couette flow, where numerical calculations are performed within a small periodic domain. The accurate computation of up to the seventh period-doubling bifurcation, assisted by a purposely defined Poincaré section, has enabled us to reproduce the two Feigenbaum universal constants with unprecedented accuracy in a fluid flow problem. We have further devised a method to predict the bifurcation diagram up to the accumulation point of the cascade based on the detailed inspection of just the first few period-doubling bifurcations. Remarkably, the method is applicable beyond the accumulation point, with predictions remaining valid, in a statistical sense, for the chaotic dynamics that follows.