Dr Jonathan Deane

In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In the presence of dissipation, most of the quasi-periodic solutions disappear and one expects, at most, only a few of them to survive together with the periodic attractors. However, to prove that a quasi-periodic solution really exists, usually one assumes that the frequencies still satisfy a Diophantine condition and, furthermore, that some external parameters of the system are suitably tuned with them. In this paper we consider a class of systems on the one-dimensional torus, subject to a periodic perturbation and in the presence of dissipation, and show that, however small the dissipation, if the perturbation is a trigonometric polynomial in the angles and the unperturbed frequencies satisfy a non-resonance condition of finite order, depending on the size of the dissipation, then a quasi-periodic solution exists with slightly perturbed frequencies provided the size of the perturbation is small enough. If on the one hand the maximal size of the perturbation is not uniform in the degree of the trigonometric polynomial, on the other hand all but finitely many frequencies are allowed and there is no restriction arising from the tuning of the external parameters. A physically relevant case, where the result applies, is the spin-orbit model, which describes the rotation of a satellite around its own axis, while revolving on a Keplerian orbit around a planet, in the case in which the dissipation is taken into account through the MacDonald torque.

We consider a parametrically-driven nonlinear ODE, which encompasses a simple model of an electronic circuit known as a parametric amplifier, whose linearisation has a zero eigenvalue. By adopting two different approaches we obtain conditions for the origin to be a global attractor which is approached (a) non-monotonically and (b) monotonically. In case (b), we obtain an asymptotic expression for the convergence to the origin. Some further numerical results are reported.

A new model which comprehensively explains the working principles of contact-mode Triboelectric Nanogenerators (TENGs) based on Maxwell’s equations is presented. Unlike previous models which are restricted to known simple geometries and derived using the parallel plate capacitor model, this model is generic and can be modified to a wide range of geometries and surface topographies. We introduce the concept of a distance-dependent electric field, a factor not taken in to account in previous models, to calculate the current, voltage, charge, and power output under different experimental conditions. The versatality of the model is demonstrated for non-planar geometry consisting of a covex-conave surface. The theoretical results show excellent agreement with experimental TENGs. Our model provides a complete understanding of the working principles of TENGs, and accurately predicts the output trends, which enables the design of more efficient TENG structures.

We consider a class of differential equations, x¨ + γ x˙ + g(x) = f (ωt), with ω ∈ Rd , describing onedimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x) = x2p+1, p ∈ N, we show that, when the dissipation coefficient is large enough, there is only one such trajectory and that it describes a global attractor. In the case of more general nonlinearities, including g(x) = x2 (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor.

The Wirtz pump is not only an excellent example of alternative technology, using as it does the kinetic energy of a stream to raise a proportion of its water, but its mathematical modelling also poses several intriguing problems. We give some history of the Wirtz pump and describe its operation. Taking a novel dynamical systems approach, we then derive a discrete mathematical model in the form of a mapping that describes its hydrostatic behaviour. Our model enables us to explain several aspects of the behaviour of the pump as well as to design one that gives approximately maximal, and maximally constant, output pressure.

In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model which has been recently introduced in a series of papers mainly by Efroimsky and Makarov. We present numerical and analytical results concerning the nature of the librations of Mercury’s spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time, consisting of a slow oscillation, with an amplitude of order of arcminutes, superimposed on the 88-day libration. This contrasts with recent astronomical observations and hence suggests that the 3:2 resonance in which Mercury has been trapped might have been originally described by a large-amplitude quasi-periodic libration which, only at a later stage, with the formation of a molten core, evolved into the small-amplitude libration which is observed nowadays.

We consider dissipative one-dimensional systems subject to a periodic force and study numerically how a time-varying friction affects the dynamics. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction. We find that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. From a quantitative point of view, the scenario depends both on the final value and the growth rate of the damping coefficient. The relevance of the results for the spin-orbit model are discussed in some detail.

In this paper we give an explicit sufficient condition for the affine map uλ(x):=λx to be the global energy minimizer of a general class of elastic stored-energy functionals I(u)=∫ΩW(∇u)dx in three space dimensions, where W is a polyconvex function of 3×3 matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of I(⋅) at λ1, where 1 is the 3×3 identity matrix. Our approach relies on arguments involving null Lagrangians (in this case, affine combinations of the minors of 3×3 matrices), on the previous work Bevan & Zeppieri, 2015, and on a careful numerical treatment to make the calculation of certain constants tractable. We also derive a new condition, which seems to depend heavily on the smallest singular value λ1(∇u) of a competing deformation u, that is necessary for the inequality I(u)

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.

We consider, from a mathematical perspective, the power generated by a contact-mode triboelectric nanogenerator, an energy harvesting device that has been thoroughly studied recently. We encapsulate the behaviour of the device in a differential equation, which although linear and of f1rst order, has periodic coeffcients, leading to some interesting mathematical problems. In studying these, we derive approximate forms for the mean power generated and the current waveforms, and describe a procedure for computing the Fourier coefficients for the current, enabling us to compute the power accurately and show how the power is distributed over the harmonics. Comparisons with numerics validate our analysis.

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping. In the limit of large damping, under some generic nondegeneracy condition on the force, there are quasiperiodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a twodimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension d, for the existence of a quasiperiodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasiperiodic solution is proved to describe a local attractor.

We consider the Modified Kuramoto–Sivashinky Equation (MKSE) in one and two space dimensions and we obtain explicit and accurate estimates of various Sobolev norms of the solutions. In particular, by using the sharp constants which appear in the functional interpolation inequalities used in the analysis of partial differential equations, we evaluate explicitly the sup-norm of the solutions of the MKSE. Furthermore we introduce and then compute the so-called crest factor associated with the above solutions. The crest factor provides information on the distortion of the solution away from its space average and therefore, if it is large, gives evidence of strong turbulence. Here we find that the time average of the crest factor scales like λ(2d−1)/8 for λ large, where λ is the bifurcation parameter of the source term and d=1,2 is the space dimension. This shows that strong turbulence cannot be attained unless the bifurcation parameter is large enough.

We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients varying linearly between two given values over many different time intervals: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins and discuss its range of applicability. Second, we observe that sometimes very slight changes in the time interval may produce abrupt large variations in the relative areas of the basins of attraction of the surviving attractors: we show how comparing the contracted phase space at a time after the final value of dissipation has been reached with the basins of attraction corresponding to that value of constant dissipation can explain the presence of such variations. Both procedures are illustrated by application to a pendulum with periodically oscillating support.

This is a theoretical and numerical study of a model of a rope fountain subject to a drag force that depends linearly on the rope velocity. A precise, analytical description of the long-term shape adopted by the rope is given, and various consequences are derived from it. Using parameters that naturally appear in the model, we distinguish between cases wherein energy is conserved by means of a constant tension far from the rope source (the ‘free’ case) and where energy conservation is a consequence of a non-constant tension (the ‘braked’ case). The model is used, among other things, to generate rope fountain shapes based on approximate experimental estimates of the parameter values and a careful numerical treatment.

For each natural number n and any bounded, convex domain Ω ⊂ R n we characterize the sharp constant C(n, Ω) in the Poincaré inequality ||f − ¯ f Ω || L ∞ (Ω;R) ≤ C(n, Ω)|||f || L ∞ (Ω;R). Here, ¯ f Ω denotes the mean value of f over Ω. In the case that Ω is a ball Br of radius r in R n , we calculate C(n, Br) = C(n)r explicitly in terms of n and a ratio of the volumes of the unit balls in R 2n−1 and R n. More generally, we prove that C(n, B r(Ω)) ≤ C(n, Ω) ≤ n n+1 diam(Ω), where B r(Ω) is a ball in R n with the same n−dimensional Lebesgue measure as Ω. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

We exhibit a family of convex functionals with infinitely many, equal-energy C 1 stationary points that (i) occur in pairs v± satisfying det v± = 1 on the unit ball B in R 2 and (ii) obey the boundary condition v± = id on ∂B. When the parameter upon which the family of functionals depends exceeds √ 2, the stationary points appear to 'buckle' near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps v±(x) and prove that they are proportional to (− 1//) ln |x| as x → 0 in B. The lowest-energy pairs v± are energy minimizers within the class of twist maps (see Taheri [30] or Sivaloganathan and Spector [22]), which, for each 0 ≤ r ≤ 1, take the circle {x ∈ B : |x| = r} to itself; a fortiori, all v± are stationary in the class of W 1,2 (B; R 2) maps w obeying w = id on ∂B and det w = 1 in B.

We show that the N-covering map, which in complex coordinates is given by uN(z):=z↦zN/N|z|N-1 and where N is a natural number, is a global minimizer of the Dirichlet energy D(v)=∫B|∇v(x)|2dx with respect to so-called inner and outer variations. An inner variation of uN is a map of the form uN∘φ, where φ belongs to the class A(B):={φ∈H1(B;R2):det∇φ=1a.e.,φ|∂B(x)=x} and B denotes the unit ball in R2, while an outer variation of uN is a map of the form ϕ∘uN, where ϕ belongs to the class A(B(0,1/N)). The novelty of our approach to inner variations is to write the Dirichlet energy of uN∘φ in terms of the functional I(ψ;N):=∫BN|ψR|2+1N|ψτ|2dy, where ψ is a suitably defined inverse of φ, and ψR and ψτ are, respectively, the radial and angular weak derivatives of ψ, and then to minimise I(ψ;N) by considering a series of auxiliary variational problems of isoperimetric type. This approach extends to include p-growth functionals (p>1) provided the class A(B) is suitably adapted. When 1

A differential equation, periodically driven with period T, defines the time evolution of the solution, a state vector x(t). The Poincaré, or time one, map is a function that relates to x(t + T)to x(t). For most second and higher order nonlinear differential equations, the Poincaré map is not available in a closed form; it can generally only be inferred from numerical calculations. In this paper, we derive an iterative representation of the Poincaré map for Duffing's equation . Our objectives are (a) to represent the mapping in as succinct a form as possible (compact enough to be published in this paper) and (b) to demonstrate that this map representation adequately reproduces the behaviour of Duffing's equation, for instance bifurcation diagrams, co-existing attractors and Poincaré sections. We succeed in these objectives, and our representation increases computation speed by a factor of 45 over traditional numerical calculations.

We examine a model for a bandpass sigma-delta modulator introduced by Feely and co-workers (1996). This is shown to have the dynamics of a piecewise isometry of a union of convex polygons on the plane by an appropriate transformation of the linearised parts into normal form. Using these we show that the periodically coded regions form a packing of the phase space by circles and we link this system to a number of similar ones. In particular we conjecture that for typical values of the parameter theta there is a positive measure set of points that have aperiodic codings

A simple mapping is derived, which describes the behavior of a peak current-mode controlled boost converter operating chaotically. The invariant density of this mapping is calculated iteratively and, from this, the power density spectrum of the input current at the clock frequency and its harmonics are deduced. The calculation is presented, along with experimental verification. The possibility of a novel application of chaos-amelioration of power supply interference-is discussed.

A class of clocked/autonomous circuits is defined in which the behaviour is described by a one-dimensional mapping. For these circuits, the power density spectrum of the harmonics of the clock frequency can be calculated directly from the invariant density of the mapping. An algorithm for carrying out this calculation is derived and illustrated, using the clocked monostable multivibrator as an example. An application to EMC improvement is suggested and possible extensions to the work are discussed.

Irregular dynamics of simple chaotic systems containing noise sensitive features are considered. Some results are given on the statistics of the time development of the resulting bursty epochs in the state variable of the system. Crisis-driven intermittency may occur, where the behaviours are almost indistinguishable from those of the noise-sensitive features. Experimental results are presented which illustrate the theoretical studies.

In this paper we discuss the packing properties of invariant disks defined by periodic behavior of a model for a bandpass Sigma–Delta modulator. The periodically coded regions form a packing of the forward invariant phase space by invariant disks. For this one-parameter family of PWIs, by introducing codings underlying the map operations we give explicit expressions for the centers of the disks by analytic functions of the parameters, and then show that tangencies between disks in the packings are very rare; more precisely they occur on parameter values that are at most countably infinite. We indicate how similar results can be obtained for other plane maps that are piecewise isometries.

The commonest method of characterizing a cold field electron emitter is to measure its current-voltage characteristics, and the commonest method of analysing these characteristics is by means of a Fowler-Nordheim (FN) plot. This tutorial/review-type paper outlines a more systematic method of setting out the Fowler-Nordheim-type theory of cold field electron emission, and brings together and summarises the current state of work by the authors on developing the theory and methodology of FN plot analysis. This has turned out to be far more complicated than originally expected. Emphasis is placed in this paper on: (a) the interpretation of FN-plot slopes, which is currently both easier and of more experimental interest than the analysis of FN-plot intercepts; and (b) preliminary explorations into developing methodology for interpreting current-voltage characteristics when there is series resistance in the conduction path from the high-voltage generator to the emitter's emitting regions. This work reinforces our view that FN-plot analysis is best carried out on the raw measured current-voltage data, without pre-conversion into another data format, particularly if series resistance is present in the measuring circuit. Relevant formulae are given for extracting field-enhancement-factor values from such an analysis.

The concept of a Variable Structure System (VSS) ,in which the structure is determined by the dynamical response, is described and measurements and simulations on VSSs comprising electronic systems which display bursty chaotic or "intermittent" behaviour, and also on a network traffic transfer protocol, which is demonstrated to be a VSS in the same sense, are used to support the thesis that such bursty behaviour is common in such systems. An example of trapping in a two-centre system is given to show that the ideas can be extended to continuous-variable dynamical systems having piecewise-linear properties. The studies of these simple electronic systems provide insight for cases where similar behaviour of a time series is observed in other complex systems; some other VSSs are listed and their properties are considered.

Applications are beginning to be found for chaotic power converters. Using the peak current controlled boost converter as an example throughout, the paper reviews the theory of chaos, shows how it may be employed to improve the electromagnetic compatibility (EMC) of power supplies, and presents a targeting scheme that can make a chaotic converter jump rapidly between two stabilised modes of operation.

Targeting methods appropriate for systems with discontinuities are considered. The multivalued inverse function is used to generate multiple preimages of the target region which quickly cover the attractor. This method is applied to the current-controlled boost converter in order to jump between two controlled states. A significant reduction in the length of the target orbit is observed when compared with targeting methods for invertible maps.