Singularly perturbed dynamics of the tippedisk

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disc, being reminiscent of (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow–fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.

Stochastic rectification of fast oscillations on slow manifold closures

The problems of identifying the slow component (e.g., for weather forecast initialization) and of characterizing slow–fast interactions are central to geophysical fluid dynamics. In this study, the related rectification problem of slow manifold closures is addressed when breakdown of slow-to-fast scales deterministic parameterizations occurs due to explosive emergence of fast oscillations on the slow, geostrophic motion. For such regimes, it is shown on the Lorenz 80 model that if 1) the underlying manifold provides a good approximation of the optimal nonlinear parameterization that averages out the fast variables and 2) the residual dynamics off this manifold is mainly orthogonal to it, then no memory terms are required in the Mori–Zwanzig full closure. Instead, the noise term is key to resolve, and is shown to be, in this case, well modeled by a state-independent noise, obtained by means of networks of stochastic nonlinear oscillators. This stochastic parameterization allows, in turn, for rectifying the momentum-balanced slow manifold, and for accurate recovery of the multiscale dynamics. The approach is promising to be further applied to the closure of other more complex slow–fast systems, in strongly coupled regimes.

Vortex cluster arising from an axisymmetric inertial wave attractor

We present an experimental and numerical study of the nonlinear dynamics of an inertial wave attractor in an axisymmetric geometrical setting. The rotating ring-shaped fluid domain is delimited by two vertical coaxial cylinders, a conical bottom and a horizontal wave generator at the top: the vertical cross-section is a trapezium, while the horizontal cross-section is a ring. Forcing is introduced via axisymmetric low-amplitude volume-conserving oscillatory motion of the upper lid. The experiment shows an important result: at sufficiently strong forcing and long time scale, a saturated fully nonlinear regime develops as a consequence of an energy transfer draining energy towards a slow two-dimensional manifold represented by a regular polygonal system of axially oriented cyclonic vortices undergoing a slow prograde motion around the inner cylinder. We explore the long-term nonlinear behaviour of the system by performing a series of numerical simulations for a set of fixed forcing amplitudes. This study shows a rich variety of dynamical regimes, including a linear behaviour, a triadic resonance instability, a progressive frequency enrichment reminiscent of weak inertial wave turbulence and the generation of a slow manifold in the form of a polygonal vortex cluster confirming the experimental observation. This vortex cluster is discussed in detail, and we show that it stems from the summation and merging of wave-like components of the vorticity field. The nature of these wave components, the possibility of their detection under general conditions and the ultimate fate of the vortex clusters at even longer time scale remain to be explored.

Hamiltonian reduction of Vlasov–Maxwell to a dark slow manifold

We show that non-relativistic scaling of the collisionless Vlasov–Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov–Maxwell phase space. Vlasov–Maxwell dynamics restricted to the slow manifold recovers the Vlasov–Poisson and Vlasov–Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov–Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov–Maxwell solutions that do not excite light waves, and are therefore ‘dark’. We provide a heuristic lower bound for the time interval over which Vlasov–Maxwell solutions initialized optimally near the slow manifold remain dark. We also show how the dynamics on the slow manifold naturally inherits a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov–Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.

Existence of Solitary Waves in a Perturbed KdV-mKdV Equation

In this paper, we establish the existence of a solitary wave in a KdV-mKdV equation with dissipative perturbation by applying the geometric singular perturbation technique and Melnikov function. The distance of the stable manifold and unstable manifold is computed to show the existence of the homoclinic loop for the related ordinary differential equation systems on the slow manifold, which implies the existence of a solitary wave for the KdV-mKdV equation with dissipative perturbation.

Quasi-Steady-State and Partial-Equilibrium Approximations in Chemical Kinetics: One Stage Beyond the Elimination of a Fast Variable

<pre>Classical approximations in chemical kinetics, the quasi-steady-state approximation (QSSA) and the partial-equilibrium approximation (PEA), are used to reduce rate equations for the concentrations and the extents of the reaction steps, respectively. We make precise two conditions on the rate constants necessary and sufficient to eliminate a well-chosen variable in the vicinity of a steady state. The first condition expresses that dynamics admits a small characteristic time associated with a fast variable. The second condition ensures that the fast variable is a concentration for QSSA and an extent for PEA. Both approximations exploit the zeroth order of a singular perturbation method. Eliminating a fast variable does not mean that it has reached a steady state. The fast evolution is considered over and the slow evolution of the eliminated variable is governed by the slow variables. The evolution of the slow variables occurs on a slow manifold in the space of the concentrations or the extents. In some cases the dynamics of the slow variables can be associated with a reduced chemical scheme. QSSA and PEA are applied to three chemical schemes associated with linear and nonlinear dynamics. We find that QSSA cannot be identified with the elimination of a reactive intermediate. The nonlinearities of the rate equations induce a more complex behavior.</pre>

Quasi-Steady-State and Partial-Equilibrium Approximations in Chemical Kinetics: One Stage Beyond the Elimination of a Fast Variable

<pre>Classical approximations in chemical kinetics, the quasi-steady-state approximation (QSSA) and the partial-equilibrium approximation (PEA), are used to reduce rate equations for the concentrations and the extents of the reaction steps, respectively. We make precise two conditions on the rate constants necessary and sufficient to eliminate a well-chosen variable in the vicinity of a steady state. The first condition expresses that dynamics admits a small characteristic time associated with a fast variable. The second condition ensures that the fast variable is a concentration for QSSA and an extent for PEA. Both approximations exploit the zeroth order of a singular perturbation method. Eliminating a fast variable does not mean that it has reached a steady state. The fast evolution is considered over and the slow evolution of the eliminated variable is governed by the slow variables. The evolution of the slow variables occurs on a slow manifold in the space of the concentrations or the extents. In some cases the dynamics of the slow variables can be associated with a reduced chemical scheme. QSSA and PEA are applied to three chemical schemes associated with linear and nonlinear dynamics. We find that QSSA cannot be identified with the elimination of a reactive intermediate. The nonlinearities of the rate equations induce a more complex behavior.</pre>

Quasi-Steady-State and Partial-Equilibrium Approximations in Chemical Kinetics: One Stage Beyond the Elimination of a Fast Variable

<pre>Classical approximations in chemical kinetics, the quasi-steady-state approximation (QSSA) and the partial-equilibrium approximation (PEA), are used to reduce rate equations for the concentrations and the extents of the reaction steps, respectively. We make precise two conditions on the rate constants necessary and sufficient to eliminate a well-chosen variable in the vicinity of a steady state. The first condition expresses that dynamics admits a small characteristic time associated with a fast variable. The second condition ensures that the fast variable is a concentration for QSSA and an extent for PEA. Both approximations exploit the zeroth order of a singular perturbation method. Eliminating a fast variable does not mean that it has reached a steady state. The fast evolution is considered over and the slow evolution of the eliminated variable is governed by the slow variables. The evolution of the slow variables occurs on a slow manifold in the space of the concentrations or the extents. In some cases the dynamics of the slow variables can be associated with a reduced chemical scheme. QSSA and PEA are applied to three chemical schemes associated with linear and nonlinear dynamics. We find that QSSA cannot be identified with the elimination of a reactive intermediate. The nonlinearities of the rate equations induce a more complex behavior.</pre>

Asymptotic analysis of self-excited and forced vibrations of a self-regulating pressure control valve

Pressure vibrations in hydraulic systems are a widespread problem and can be caused by external excitation or self-exciting mechanisms. Although vibrations cannot be completely avoided in most cases, at least their frequencies must be known in order to prevent resonant excitation of adjacent components. While external excitation frequencies are known in most cases, the estimation of self-excited vibration amplitudes and frequencies is often difficult. Usually, numerical studies have to be executed in order to elaborate parameter influences, which is computationally expensive. The same holds true for the prediction of forced oscillation amplitudes. This contribution proposes asymptotic approximations of forced and self-excited oscillations in a simple hydraulic circuit consisting of a pump, an ideal consumer and a pressure control valve. Two excitation mechanisms of practical interest, namely pump pulsations (forced vibrations) and valve instability (self-excited vibrations), are analyzed. The system dynamics are described by a singularly perturbed third-order differential equation. By separating slow and fast variables in the system without external excitation, a first-order approximation of the slow manifold is computed. The flow on the slow manifold is approximated by an averaging procedure, whose piecewise defined zero-order solution maps the valve’s switching property. A modification of the procedure allows for the asymptotic approximation of the system’s forced response to an external excitation. The approximate solutions are validated within a realistic parameter range by comparison with numerical solutions of the full system equations.