Malaria remains a significant global health challenge, with sub-Saharan Africa bearing the majority of the burden. While vector control measures such as pyrethroid-based insecticidal nets and indoor residual spraying have significantly reduced malaria incidence, the emergence of insecticide resistance in Anopheles mosquito populations threatens these gains. Resistance develops through genetic mutations under prolonged selection pressure, complicating control efforts and necessitating a deeper understanding of its evolutionary dynamics. This study introduces a novel mathematical framework to investigate the emergence and spread of insecticide resistance in mosquito populations. By modelling insecticide resistance as a continuous (quantitative) trait influenced by multiple genes, we capture its variability and evolutionary transient dynamics. We propose an age-structured mosquito population model using integro-differential equations, where the resistance trait influences life-history parameters such as mortality and reproduction. Our approach provides new insights into how resistance emerges and spreads within mosquito populations over time. We analyse the model’s properties, including the existence of a unique maximal bounded semiflow, and derive conditions for the existence and stability of steady states. Through parameterization and simulations, we explore the transient and long-term dynamics of resistance evolution under different scenarios. The results offer valuable insights into the evolutionary mechanisms driving insecticide resistance and inform the design of sustainable vector control strategies.

We perform causal analysis on the low-dimensional Galerkin model for shear flow developed by Moehlis et al. (New J. Phys., vol. 6, 2004, 56). Our method integrates both equation-based analysis and the proposed Galerkin-based Granger causality (GGC) to investigate the effect of the nonlinear terms on the dynamics. Two types of quadratic interactions are identified: a fully triadic interaction and a modulated two-mode coupling. The propagation of these interactions through the nonlinear dynamics leads to a directed cause-and-effect network. Furthermore, the relative importance of each mode amplitude on the dynamics of the target mode is quantified. This analysis provides a deeper understanding of the nonlinear dynamics and distills control opportunities. To demonstrate the applicability of the proposed GGC to realistic flows where Galerkin projection is impractical, a turbulent lid-driven cavity flow is further studied. We foresee applications of the proposed causal analysis framework as valuable tools for Galerkin modelling – guiding investigations of modal causality, prediction uncertainty, model-order reduction and control design.

We investigate the incompressible flow inside a two-dimensional square cavity, driven by the sliding motion of its four lids, all at the same speed and with facing lids moving in opposite directions. The problem has three symmetries: two mirror symmetries with respect to the diagonals and a $\pi$ rotation invariance about the centre of the cavity. The base flow, a steady state that has all three symmetries, is the unique solution at sufficiently low values of the Reynolds number ($ \textit{Re}$) and acts as a global attractor. At higher $ \textit{Re}$, it has become unstable and shares the phase space with a globally attracting space–time symmetric periodic orbit that, in addition to the rotational invariance, is also invariant under evolution over half a period followed by reflection about either of the diagonals. In between, a wealth of solution branches and intervening bifurcations mediate the transition process. In particular, a pair of steady states that break the mirror symmetries but are mirror-symmetry images of each other regulate the appearance and disappearance of a second space–time symmetric periodic orbit and a pair of asymmetric periodic orbits that are also mirror images of each other. The catalogue of instabilities includes both local (two pitchfork, two Hopf, a saddle-node and a cyclic fold) and global (two heteroclinic and one homoclinic) bifurcations. The sequence of transitions is explained in terms of a one-dimensional path through the parameter space of a codimension-four bifurcation: the double zero bifurcation with Z$_2$ symmetry and degeneracy of the third order terms.

We investigate the transitional flow regimes arising from the interaction between buoyancy and shear in Rayleigh–Bénard–Poiseuille (RBP) flows, considering both large and small domains. The transition boundaries between the bistable system consisting of spiral defect chaos (SDC) and ideal straight rolls (ISRs) in Rayleigh–Bénard convection, and subcritical turbulence in plane Poiseuille flows are not known. Using direct numerical simulations in a large spatial domain over a range of Rayleigh numbers, $Ra \in [0, 10000]$, Reynolds numbers, $\textit{Re} \in [0, 2000]$ and unit Prandtl number, we identify five distinct regimes: (i) bistable SDC and ISRs; (ii) ISRs; (iii) wavy rolls; (iv) intermittent rolls; and (v) shear-driven turbulence. The newly identified intermittent rolls state features longitudinal rolls that decay towards the laminar state before regenerating. In the turbulent regime, longitudinal rolls may coexist with turbulent–laminar bands, highlighting the role of longitudinal rolls in transitional RBP flows. To this end, we examine the unstable manifold of longitudinal rolls in a small domain, integrating along which led to turbulence. This turbulence occur transiently, decaying towards the unstable laminar base state where the longitudinal rolls can be excited again, forming a quasi-cyclic process referred to as the thermally assisted sustaining process (TASP). We further investigate the behaviour of TASP as $\textit{Re}$ and $Ra$ are varied, revealing a stable periodic orbit around the longitudinal roll and the laminar state, and a pathway towards turbulence above a certain $\textit{Re}$ threshold. Finally, we provide a state space sketch of the dynamical processes, emphasising the role of longitudinal rolls in transitional RBP in small domains, and discuss the potential connections to large domains.

The time evolution of Beltrami fields in the presence of a time-dependent background flow with spatially homogeneous velocity gradient is analysed using the barotropic vorticity equation. For backgrounds comprising a time-dependent isotropic expansion/contraction and a time-dependent solid-body rotation, we show that every scalar Laplacian eigenfunction generates an unsteady solution of the nonlinear vorticity equation in which the non-background component remains a time-dependent Beltrami field. We derive the evolution law for the background angular velocity in the presence of time-dependent deviatoric strain and velocity divergence, and we generalise the Chandrasekhar–Kendall construction to obtain unsteady Beltrami velocity fields. When the background deformation is a similarity (vanishing deviatoric strain), the Beltrami field is frozen into an advecting flow that differs from the background only by a spatially homogeneous, time-dependent drift. In general, deviatoric strain breaks the Beltrami property, but in regimes where departures are small, we introduce a ‘Beltrami field approximation’. Because the background velocity gradient has nine time-dependent degrees of freedom, three of which are constrained by the vorticity equation, six remaining functions may be prescribed to drive the Beltrami field. We illustrate the approach by describing elastic scattering of Beltrami fields by a background-flow pulse.

Active filaments, such as microtubules with attached cargo-carrying motor proteins, are important dynamic structures for fluid transport in and around living cells. The mathematical models of active filaments appearing in the literature typically involve combinations of follower forces, compressive tangential forces, along the filament, and an opposite force on the fluid that generates an effective surface flow. In this paper, we present a comparative dynamical systems study of active filament models examining the differences in dynamic states that occur when actuation is through follower forces alone, or the effect of surface flows is also included. We consider cases where actuation is applied only at the filament tip, or distributed uniformly along the filament length. By varying actuation strength, we show that the first bifurcations that provide the transition between the upright, whirling and beating states appear in all models. At higher values of actuation, when beating becomes unstable, however, qualitative differences between the models emerge. Those with distributed actuation produce a single, time-dependent state, which for the surface flow model is reminiscent of a rotating helix that periodically changes handedness and rotation direction. Tip actuation, however, yields complex transitions that ultimately produce a chaotic state. We link the differences in dynamics between tip and distributed actuation to differences in their respective internal stress distributions – differences that appear as early as the first bifurcation, where they affect the shapes of the unstable modes.

In this study, we explore the effect of basis functions on the performance and convergence of the Galerkin projection-based reduced-order model (ROM) in the minimal flow unit of Couette flow. POD (proper orthogonal decomposition) modes obtained from direct numerical simulation, and controllability and balanced truncation modes from the linearised Navier–Stokes equations (LNSEs) with different base flows (laminar base flow and turbulent mean flow) and an eddy viscosity model are considered. In the neighbourhood of the laminar base state, the ROMs based on the modes from the LNSEs with the laminar base flow and molecular viscosity are found to perform very well as they are able to capture the linear stability of the laminar base flow for each plane Fourier component only with a single degree of freedom. In particular, the ROM based on the balanced truncation modes models the linear dynamics involving transient growth around the laminar base flow most effectively, consistent with previous studies. In contrast, for turbulent state, the ROM based on POD modes is found to reproduce its statistics and coherent dynamics most effectively. The ROMs based on the modes from the LNSE with turbulent mean flow and an eddy viscosity model performs better compared with any other ROMs using the modes from the LNSE. These observations suggest that the performance and convergence of a ROM are highly state-dependent. In particular, this state dependence is strongly correlated with the information and dynamics that each of the basis functions contain. Discussions supporting these observations are also provided in relation to the flow physics involved and the form of coherent structures in Couette flow.

In Navier–Stokes (NS) turbulence, large-scale turbulent flows inevitably determine small-scale flows. Previous studies using data assimilation with the three-dimensional (3-D) NS equations indicate that employing observational data resolved down to a specific length scale, $\ell ^{\rm 3\text{-}D}_{\ast }$, enables the successful reconstruction of small-scale flows. Such a length scale of ‘essential resolution of observation’ for reconstruction $\ell ^{\rm 3\text{-}D}_{\ast }$ is close to the dissipation scale in three-dimensional NS turbulence. Here, we study the equivalent length scale in two-dimensional (2-D) NS turbulence, $\ell ^{\rm 2\text{-}D}_{\ast }$, and compare with the three-dimensional case. Our numerical studies using data assimilation and conditional Lyapunov exponents reveal that, for Kolmogorov flows with Ekman drag, the length scale $\ell ^{\rm 2\text{-}D}_{\ast }$ is actually close to the forcing scale, substantially larger than the dissipation scale. Furthermore, we discuss the origin of the significant relative difference between the length scales, $\ell ^{\rm 2\text{-}D}_{\ast }$ and $\ell ^{\rm 3\text{-}D}_{\ast }$, based on inter-scale interactions, ‘cascades’ and orbital instabilities in turbulence dynamics.

The triadic interactions and nonlinear energy transfer are investigated in a subsonic turbulent jet at $Re = 450\,000$. The primary focus is on the role of these interactions in the formation and attenuation of streaky structures. To this end, we employ bispectral mode decomposition, a technique that extracts coherent structures associated with dominant triadic interactions. A strong triadic correlation is identified between Kelvin–Helmholtz (KH) wavepackets and streaks: interactions between counter-rotating KH waves generates streamwise vortices, which subsequently give rise to streaks through the lift-up mechanism. The most energetic streaks occur at azimuthal wavenumber $m = 2$, with the dominant contributing triad being $[m_1, m_2, m_3] = [1, 1, 2]$. The spectral energy budget reveals that the net effect of nonlinear triadic interactions is an energy loss from the streaks. As these streaks convect downstream, they engage in further nonlinear interactions with other frequencies, which drain their energy and ultimately lead to their attenuation. Further analysis identifies the dominant scales and direction of energy transfer across different spatial regions of the jet. While the turbulent jet exhibits a forward energy cascade in a global sense, the direction of energy transfer varies locally: in the shear layer near the nozzle exit, triadic interactions among smaller scales dominate, resulting in an inverse energy cascade, whereas farther downstream, beyond the end of the potential core, interactions among larger scales prevail, leading to a forward cascade.

We derive effective Boussinesq and Korteweg–de Vries equations governing nonlinear wave propagation over a structured bathymetry using a three-scale homogenization approach. The model captures the anisotropic effects induced by the bathymetry, leading to significant modifications in soliton dynamics. Homogenized parameters, determined from elementary cell problems, reveal strong directional dependencies in wave speed and dispersion. Our results provide new insights into nonlinear wave propagation in structured shallow-water environments, and consequently motivate further fundamental and applied studies in wave hydrodynamics and coastal engineering.

The motionless conducting state of liquid-metal convection with an applied vertical magnetic field confined in a vessel with insulating sidewalls becomes linearly unstable to wall modes through a supercritical pitchfork bifurcation. Nevertheless, we show that the transition proceeds subcritically, with stable finite-amplitude solutions with different symmetries existing at parameter values beneath this linear stability threshold. Under increased thermal driving, the branch born from the linear instability becomes unstable and solutions are attracted to the most subcritical branch, which follows a quasiperiodic route to chaos. Thus, we show that the transition to turbulence is controlled by this subcritical branch and hence turbulent solutions have no connection to the initial linear instability. This is further quantified by observing that the subcritical equilibrium solution sets the spatial symmetry of the turbulent mean flow and thus organises large-scale structures in the turbulent regime.

The Stokes boundary layer (SBL) is the oscillating flow above a flat plate. Its laminar flow becomes linearly unstable at a Reynolds number of $\textit{Re} = U_0 \sqrt {T_0/\nu } \approx 2511$, where $U_0$ is the amplitude of the oscillation, $T_0$ is the period of oscillation and $\nu$ is the fluid’s kinematic viscosity, but turbulence is observed subcritically for $\textit{Re} \gtrsim 700$. The state space consists of laminar and turbulent basins of attraction, separated by a saddle point (the ‘edge state’) and its stable manifold (the ‘edge’). This work presents the edge trajectories for the transitional regime of the SBL. Despite linear dynamics disallowing the lift-up mechanism in the laminar SBL, edge trajectories are dominated by coherent structures as in other canonical shear flows: streaks, rolls and waves. Stokes boundary layer structures are inherently periodic, interacting with the oscillating flow in a novel way: streaks form near the plate, migrate upward at a speed $2\sqrt {\pi }$ and dissipate. A streak-roll-wave decomposition reveals a spatiotemporally evolving version of the self-sustaining process (SSP): (i) rolls lift fluid near the plate, generating streaks (via the lift-up mechanism); (ii) streaks can only persist in regions with the same sign of laminar shear as when they were created, defining regions that moves upward at a speed $2 \sqrt {\pi }$; (iii) the sign of streak production reverses at a roll stagnation point, destroying the streak and generating waves; (iv) trapped waves reinforce the rolls via Reynolds stresses; (v) mass conservation reinforces the rolls. This periodic SSP highlights the role of flow oscillations in sustaining transitional structures in the SBL, providing an alternative picture to ‘bypass’ transition, which relies on pre-existing free stream turbulence and spanwise vortices.

We analyse the long-time dynamics of trajectories within the stability boundary between laminar and turbulent square duct flow. If not constrained to a symmetric subspace, the edge trajectories exhibit a chaotic dynamics characterised by a sequence of alternating quiescent phases and intense bursting episodes. The dynamics reflects the different stages of the well-known near-wall streak–vortex interaction. Most of the time, the edge states feature a single streak with a number of flanking vortices attached to one of the four surrounding walls. The initially straight streak undergoes a linear instability and eventually breaks in an intense bursting event. At the same time, the downstream vortices give rise to a new low-speed streak at one of the neighbouring walls, thereby causing the turbulent activity to ‘switch’ from one wall to the other. If the edge dynamics is restricted to a single or twofold mirror-symmetric subspace, the bursting and wall-switching episodes become self-recurrent in time, representing the first periodic orbits found in square duct flow. In contrast to the chaotic edge states in the non-symmetric case, the imposed symmetries enforce analogue bursting cycles to simultaneously appear at two parallel opposing walls in a mirror-symmetric configuration. Both the localisation of turbulent activity to one or two walls and the wall-switching dynamics are shown to be common phenomena in marginally turbulent duct flows. We argue that such episodes represent transient visits of marginally turbulent trajectories to some of the edge states detected here.

Large numbers of relative periodic orbits (RPOs) have been found recently in doubly periodic, two-dimensional Kolmogorov flow at moderate Reynolds numbers ${\textit{Re}} \in \{40, 100\}$. While these solutions lead to robust statistical reconstructions at the ${\textit{Re}}$ values where they were obtained, it is unclear how their dynamical importance changes with ${\textit{Re}}$. Arclength continuation on this library of solutions reveals that large numbers of RPOs quickly become dynamically irrelevant, reaching dissipation values either much larger or smaller than the values typical of the turbulent attractor at high ${\textit{Re}}$. The scaling of the high-dissipation RPOs is shown to be consistent with a direct connection to solutions of the unforced Euler equation, and is observed for a wide variety of states beyond the ‘unimodal’ solutions considered in previous work (Kim & Okamoto, Nonlinearity vol. 28, 2015, p. 3219). However, the weakly dissipative states have properties indicating a connection to exact solutions of a forced Euler equation. The dynamical irrelevance of many solutions leads to poor statistical reconstruction at higher ${\textit{Re}}$, raising serious questions for the future use of RPOs for estimating probability densities. Motivated by the Euler connection of some of our RPOs, we also show that many of these states can be well described by exact relative periodic solutions in a system of point vortices. The point vortex RPOs are converged via gradient-based optimisation of a scalar loss function which (i) matches the dynamics of the point vortices to the turbulent vortex cores and (ii) insists the point vortex evolution is itself time-periodic.

Stochastic resonance (SR) is universal phenomenon, where noise amplifies a weak periodic signal in bistable nonlinear systems, with wide applications in biology, climate science, engineering etc., although in fluid dynamics it remains underexplored. Recently, we unexpectedly found SR above non-modal elastic instability onset in an inertialess viscoelastic channel flow, where it emerges on the top of a chaotic streamwise velocity power spectrum $E_u$ due to its interaction with white-noise spanwise velocity power spectrum $E_w$ and weak elastic waves. These three conditions necessary for SR emergence differ from those required for the classical SR emergence mentioned above. Here, we consider SR in an inertialess viscoelastic channel flow with a smoothed inlet causing order of magnitude lower noise intensity than in our former studies. Our observations reveal that SR appears at the same conditions mentioned above, where SR is found just upon the instability onset in a lower subrange of a transition regime, in contrast, here, SR persists across all flow regimes – transition, elastic turbulence and drag reduction. Furthermore, we provide experimental evidence that SR, presented by a sharp peak in $E_u$, manifests as either a standing or propagating wave in the $x$-direction, with a rather uniform amplitude of streamwise velocity fluctuations and zero propagation velocity in the $z$-direction. These findings reveal a new mechanism underpinning the transition to a chaotic channel flow of viscoelastic fluids and establish SR as a robust framework for understanding complex flow dynamics. This work opens new avenues for exploring SR in other nonlinear systems and practical applications such as mixing enhancement and flow control in industrial and biological contexts.

The classical water-wave theory often neglects water compressibility effects, assuming acoustic and gravity waves propagate independently due to their disparate spatial and temporal scales. However, nonlinear interactions can couple these wave modes, enabling energy transfer between them. This study adopts a dynamical systems approach to investigate acoustic–gravity wave triads in compressible water flow, employing phase-plane analysis to reveal complex bifurcation structures and identify steady-state resonant configurations. Through this framework, we identify specific parameter conditions that enable complete energy exchange between surface and acoustic modes, with the triad phase (also known as the dynamical phase) playing a crucial role in modulating energy transfer. Further, incorporating spatial dependencies into the triad system reveals additional dynamical effects that depend on the wave velocity and resonance conditions: we observe that travelling-wave solutions emerge, and their stability is governed by the Hamiltonian structure of the system. The phase-plane analysis shows that, for certain velocity regimes, the resonance dynamics remains similar to the spatially independent case, while in other regimes, bifurcations modify the structure of resonant interactions, influencing the efficiency of energy exchange. Additionally, modulated periodic solutions appear, exhibiting changes in wave amplitudes over time and space, with implications for wave-packet stability and energy localisation. These findings enhance the theoretical understanding of acoustic–gravity wave interactions, offering potential applications in geophysical phenomena such as oceanic microseisms.

We investigate the emergent three-dimensional (3-D) dynamics of a rapidly yawing spheroidal swimmer interacting with a viscous shear flow. We show that the rapid yawing generates non-axisymmetric emergent effects, with the active swimmer behaving as an effective passive particle with two orthogonal planes of symmetry. We also demonstrate that this effective asymmetry generated by the rapid yawing can cause chaotic behaviour in the emergent dynamics, in stark contrast to the emergent dynamics generated by rapidly rotating spheroids, which are equivalent to those of effective passive spheroids. In general, we find that the shape of the equivalent effective particle under rapid yawing is different to the average shape of the active particle. Moreover, despite having two planes of symmetry, the equivalent passive particle is not an ellipsoid in general, except for specific scenarios in which the effective shape is a spheroid. In these scenarios, we calculate analytically the equivalent aspect ratio of the effective spheroid. We use a multiple scales analysis for systems to derive the emergent swimmer behaviour, which requires solving a non-autonomous nonlinear 3-D dynamical system, and we validate our analysis via comparison to numerical simulations.

The interaction between acoustic and surface gravity waves is generally neglected in classical water-wave theory due to their distinct propagation speeds. However, nonlinear dynamics can facilitate energy exchange through resonant triad interactions. This study focuses on the resonant triad interaction involving two acoustic modes and a single gravity wave in water of finite and deep depths. Using the method of multiple scales, amplitude equations are derived to describe the spatio-temporal behaviour of the system. Energy transfer efficiency is shown to depend on water depth, with reduced transfer in deeper water and enhanced interaction in shallower regimes. Numerical simulations identify parameter ranges, including resonant gravity wavenumber, initial acoustic amplitude and wave packet width, where the gravity-wave amplitude is either amplified or reduced. These results provide insights into applications such as tsunami mitigation and energy harnessing.

We study the onset of spontaneous dynamics in the follower force model of an active filament, wherein a slender elastic filament in a viscous liquid is clamped normal to a wall at one end and subjected to a tangential compressive force at the other. Clarke et al. (Phys. Rev. Fluids, vol. 9, 2024, 073101) recently conducted a thorough investigation of this model using methods of computational dynamical systems; inter alia, they showed that the filament first loses stability via a supercritical double-Hopf bifurcation, with periodic ‘planar-beating’ states (unstable) and ‘whirling’ states (stable) simultaneously emerging at the critical follower-force value. We complement their numerical study by carrying out a weakly nonlinear analysis close to this unconventional bifurcation, using the method of multiple scales. The main outcome is an ‘amplitude equation’ governing the slow modulation of small-magnitude oscillations of the filament in that regime. Analysis of this reduced-order model provides insights into the onset of spontaneous dynamics, including the creation of the nonlinear whirling states from particular superpositions of linear planar-beating modes as well as the selection of whirling over planar beating in three-dimensional scenarios.

We explore the transition to chaos in a prototypical hydrodynamic oscillator, namely a globally unstable low-density jet subjected to external time-periodic forcing. As the forcing strengthens at an off-resonant frequency, we find that the jet exhibits a sequence of nonlinear states: period-1 limit cycle $\rightarrow $ quasiperiodicity $\rightarrow$ intermittency $\rightarrow$ low-dimensional chaos. We show that the intermittency obeys type-II Pomeau–Manneville dynamics by analysing the first return map and the scaling properties of the quasiperiodic lifetimes between successive chaotic epochs. By providing experimental evidence of the type-II intermittency route to chaos in a globally unstable jet, this study reinforces the idea that strange attractors emerge via universal mechanisms in open self-excited flows, facilitating the development of instability control strategies based on chaos theory.