# Dr Michele Bartuccelli

Reader

Laurea in Theoretical Physics Cum Laude (Italy), PhD (Denmark)

### Research

### Research interests

- Dissipative Partial Differential Equations
- Chaos and Turbulence
- Applied Functional Analysis
- Mathematical Modelling (Population Dynamics, Ecosystems)
- Dynamics of Time-Dependent Nonlinear Oscillators
- Hamiltonian Dynamics

Further details can be found on my personal web page.

### My publications

### Publications

Bartuccelli MV (2014) Sharp constants for the l°°-norm on the torus and applications to dissipative partial differential equations, Differential and Integral Equations 27 (1-2) pp. 59-80

Sharp estimates are obtained for the constants appearing in the Sobolev embedding theorem for the L°° norm on the d-dimensioned torus for d = 1,2,3. The sharp constants are expressed in terms of the Riemann zeta-function, the Dirichlet beta-series and various lattice sums. We then provide some applications including the two dimensional Navier-Stokes equations.

Bartuccelli MV, Berretti A, Deane JHB, Gentile G, Gourley SA (2008) Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator, NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS 9 (5) pp. 1966-1988 PERGAMON-ELSEVIER SCIENCE LTD

Bartuccelli MV, Deane JHB, Gentile G (2014) The high-order Euler method and the spin?orbit model.: A fast algorithm for solving differential equations with small, smooth nonlinearity, Celestial Mechanics and Dynamical Astronomy 121 (3) pp. 233-260

© 2014, Springer Science+Business Media Dordrecht.We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin?orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an excellent measure of agreement between analytical predictions and the proposed fast algorithm.

Bartuccelli M, Deane J, Gentile G (2012) Attractiveness of periodic orbits in parametrically forced systemswith

time-increasing friction, Journal of Mathematicsl Physics 53 (10) 102703 pp. 102703-1-102703-27 American Institute of Physics

time-increasing friction, Journal of Mathematicsl Physics 53 (10) 102703 pp. 102703-1-102703-27 American Institute of Physics

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.

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.

Wright JA, Bartuccelli M, Gentile G (2014) The effects of time-dependent dissipation on the basins of attraction

for the pendulum with oscillating support, NONLINEAR DYNAMICS

for the pendulum with oscillating support, NONLINEAR DYNAMICS

We consider a pendulum with vertically oscillating support and time-dependent

damping coefficient which varies until reaching a finite final value. The sizes

of the corresponding basins of attraction are found to depend strongly on the

full evolution of the dissipation. In order to predict the behaviour of the

system, it is essential to understand how the sizes of the basins of attraction

for constant dissipation depend on the damping coefficient. For values of the

parameters in the perturbation regime, we characterise analytically the

conditions under which the attractors exist and study numerically how the sizes

of their basins of attraction depend on the damping coefficient. Away from the

perturbation regime, a numerical study of the attractors and the corresponding

basins of attraction for different constant values of the damping coefficient

produces a much more involved scenario: changing the magnitude of the

dissipation causes some attractors to disappear either leaving no trace or

producing new attractors by bifurcation, such as period doubling and

saddle-node bifurcation. For an initially non-constant damping coefficient,

both increasing and decreasing to some finite final value, we numerically

observe that, when the damping coefficient varies slowly from a finite initial

value to a different final value, without changing the set of attractors, the

slower the variation the closer the sizes of the basins of attraction are to

those they have for constant damping coefficient fixed at the initial value. If

during the variation of the damping coefficient attractors appear or disappear,

remarkable additional phenomena may occur. For instance, a fixed point

asymptotically may attract the entire phase space, up to a zero measure set,

even though no attractor with such a property exists for any value of the

damping coefficient between the extreme values.

damping coefficient which varies until reaching a finite final value. The sizes

of the corresponding basins of attraction are found to depend strongly on the

full evolution of the dissipation. In order to predict the behaviour of the

system, it is essential to understand how the sizes of the basins of attraction

for constant dissipation depend on the damping coefficient. For values of the

parameters in the perturbation regime, we characterise analytically the

conditions under which the attractors exist and study numerically how the sizes

of their basins of attraction depend on the damping coefficient. Away from the

perturbation regime, a numerical study of the attractors and the corresponding

basins of attraction for different constant values of the damping coefficient

produces a much more involved scenario: changing the magnitude of the

dissipation causes some attractors to disappear either leaving no trace or

producing new attractors by bifurcation, such as period doubling and

saddle-node bifurcation. For an initially non-constant damping coefficient,

both increasing and decreasing to some finite final value, we numerically

observe that, when the damping coefficient varies slowly from a finite initial

value to a different final value, without changing the set of attractors, the

slower the variation the closer the sizes of the basins of attraction are to

those they have for constant damping coefficient fixed at the initial value. If

during the variation of the damping coefficient attractors appear or disappear,

remarkable additional phenomena may occur. For instance, a fixed point

asymptotically may attract the entire phase space, up to a zero measure set,

even though no attractor with such a property exists for any value of the

damping coefficient between the extreme values.

Bartuccelli MV, Deane JHB, Gentile G, Marsh L (2006) Invariant sets for the varactor equation, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 462 (2066) pp. 439-457 ROYAL SOCIETY

Bartuccelli MV (2014) Explicit estimates on the torus for the sup-norm and the dissipative length scale of solutions of the Swift-Hohenberg Equation in one and two space dimensions, Journal of Mathematical Analysis and Applications 411 (1) pp. 166-176

In this work we have obtained explicit and accurate estimates of the sup-norm for solutions of the Swift-Hohenberg Equation (SHE) in one and two space dimensions. By using the best (so far) available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations, we have evaluated in an explicit manner the values of the sup-norm of the solutions of the SHE. In addition we have calculated the so-called time-averaged dissipative length scale associated to the above solutions. © 2013 .

Bartuccelli M, Wright JA, Gentile G (2013) On a class of Hill's equations having explicit solutions, Applied Mathematics Letters 26 (10) pp. 1026-1030

We present a class of Hill's equations possessing explicit solutions through elementary functions. In addition we provide some applications by using some of the paradigmatic systems of classical dynamics, such as the pendulum with variable length. © 2013 Published by Elsevier Ltd.

Bartuccelli MV, Gibbon JD (2011) Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality,Journal of Mathematical Physics 52 (9)

Bartuccelli MV, Deane JHB, Gentile G, Schilder F (2010) Arnol'd tongues for a resonant injection-locked frequency divider: Analytical and numerical results, NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS 11 (5) pp. 3344-3362 PERGAMON-ELSEVIER SCIENCE LTD

Gentile G, Bartuccelli MV, Deane JHB (2007) Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies, REVIEWS IN MATHEMATICAL PHYSICS 19 (3) pp. 307-348 WORLD SCIENTIFIC PUBL CO PTE LTD

Bartuccelli M, Deane J, Zelik S (2013) Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 143 A (3) pp. 445-482

We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained. Copyright © 2013 Royal Society of Edinburgh.

Bartuccelli MV, Deane JHB, Gentile G (2007) Bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 463 (2085) pp. 2351-2369 ROYAL SOCIETY

Wright J, Bartuccelli M, Gentile G (2017) Comparisons between the pendulum with varying length and the pendulum with oscillating support,Journal of Mathematical Analysis and Applications 449 pp. 1684-1707 Elsevier

We consider two forced dissipative pendulum systems, the pendulum with vertically oscillating support and the pendulum with periodically varying length, with a view to draw comparisons between their behaviour. We study the two systems for values of the parameters for which the dynamics are non-chaotic. We focus our investigation on the persisting attractive periodic orbits and their basins of attraction, utilising both analytical and numerical techniques. Although in some respect the two systems have similar behaviour, we find that even within the perturbation regime they may exhibit different dynamics. In particular, for the same value of the amplitude of the forcing, the pendulum with varying length turns out to be perturbed to a greater extent. Furthermore the periodic attractors persist under larger values of the damping coefficient in the pendulum with varying length. Finally, unlike the pendulum with oscillating support, the pendulum with varying length cannot be stabilised around the upward position for any values of the parameters.

Gentile G, Bartuccelli M, Deane JHB (2005) Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms,Journal of Mathematical Physics AIP Publishing

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.

Bartuccelli MV, Deane JHB, Gentile G (2007) Periodic attractors for the varactor equation, DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL 22 (3) pp. 365-377 TAYLOR & FRANCIS LTD

Bartuccelli MV, Deane JHB, Gentile G (2009) Frequency locking in an injection-locked frequency divider equation, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 465 (2101) pp. 283-306 ROYAL SOC

Kyrychko Y, Gourley Stephen, Bartuccelli Michele (2006) Dynamics of a stage-structured population model on an isolated finite lattice,SIAM JOURNAL ON MATHEMATICAL ANALYSIS 37 (5) pp. 1688-1708 Society for Industrial and Applied Mathematics (SIAM)

In this paper we derive a stage-structured model for a single species on a finite one-dimensional lattice. There is no migration into or from the lattice. The resulting system of equations, to be solved for the total adult population on each patch, is a system of delay equations involving the maturation delay for the species, and the delay term is nonlocal involving the population on all patches. We prove that the model has a positivity preserving property. The main theorems of the paper are comparison principles for the cases when the birth function is increasing and when the birth function is a nonmonotone function. Using these theorems we prove results on the global stability of a positive equilibrium.

Bartuccelli M, Deane JHB, Gentile G (2017) Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque,Monthly Notices of the Royal Astronomical Society 469 (1) pp. 127-150 Oxford University Press

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.

Bartuccelli M, Deane JHB, Gentile G (2007) Globally and locally attractive solutions for quasi-periodic ally forced systems,JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 328 (1) pp. 699-714 ACADEMIC PRESS INC ELSEVIER SCIENCE

We consider a class of differential equations, x¨ + ³ xÙ + g(x) = f (Ét), with É ? 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.

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.

Wright JA, Deane Jonathan, Bartuccelli Michele, Gentile G (2015) Basins of attraction in forced systems with time-varying dissipation,Communications in Nonlinear Science and Numerical Simulation 29 (1-3) pp. 72-87 Elsevier

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.

Gentile G, Bartuccelli M, Deane JHB (2006) Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems,JOURNAL OF MATHEMATICAL PHYSICS 47 (7) 072702 AMER INST PHYSICS

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.

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.

Bartuccelli M, Gentile G, Wright J (2016) Stable dynamics in forced systems with sufficiently high/low forcing frequency,Chaos 26 083108 American Institute of Physics

We consider a class of parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability of the dynamics when the frequency of the forcing is relatively high or low. We show that, provided the frequency of the forcing is sufficiently high, KAM theorem may be applied even when the forcing amplitude is far away from the perturbation regime. A similar result is obtained for sufficiently low frequency forcing, but in that case we need the amplitude of the forcing to be not too large; however we are still able to consider amplitudes of the forcing which are outside of the perturbation regime. Our results are illustrated by means of numerical simulations for the system of a forced cubic oscillator. In addition, we find numerically that the dynamics are stable even when the forcing amplitude is very large (beyond the range of validity of the analytical results), provided the frequency of the forcing is taken correspondingly low.

Bartuccelli M, Deane JHB, Gentile G (2017) Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity,Celestial Mechanics and Dynamical Astronomy 128 (4) pp. 453-473 Springer Verlag

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is C1 in the velocity. Such an ODE arises as a model of spin-orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on timescales of the order of 106{ 107 years. The proposed algorithm is based on the High-order Euler Method (HEM) which was described in [Bartuccelli et al. 2015], and it requires computer algebra to set up the code for its implementation. The pay-o is an overall increase in speed by a factor of about 7:5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.

Bartuccelli Michele V. (2019) On the nature of space fluctuations of solutions of dissipative partial differential equations,Applied Mathematics Letters 96 pp. 14-19 Elsevier

In this work we have analysed the nature of space fluctuations in dissipative Partial Differential Equations (PDEs). By taking a well known and much investigated dissipative PDE as our representative, namely the Swift?Hohenberg Equation, we estimated in an explicit manner the values of the crest factor of its solutions. We believe that the crest factor, namely the ratio between the sup-norm and the

*L*norm of solutions, is a suitable and proper measure of space fluctuations in solutions of dissipative PDEs. In particular it gives some information on the nature of ?soft? and ?hard? fluctuations regimes in the flows of dissipative PDEs.^{2}Bartuccelli Michele V., Deane Jonathan H., Gentile Guido (2019) Explicit Estimates on the Torus for the Sup-norm and the Crest Factor of Solutions of the Modified Kuramoto?Sivashinky Equation in One and Two Space Dimensions,Journal of Dynamics and Differential Equations Springer Verlag

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 »

^{(2d?1)}/8 for » large, where » 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.Bartuccelli Michele V. (2019) On the crest factor for dissipative partial differential equations,Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 (2229) Royal Society

In this work, we have introduced and then computed the so-called crest factor associated with solutions of dissipative partial differential equations (PDEs). By taking two paradigmatic dissipative PDEs, we estimated in an explicit and accurate manner the values of the crest factor of their solutions. We then analysed and compared the estimates as a function of the positive parameter which appears in the PDEs in space dimensions one and two. These estimates shed some light on the dynamics of the fluctuations of the solutions of the two model PDEs, and therefore provide a criterion for discerning between small and large potential excursions in space for the solution of any dissipative PDE. Being able to detect between small and large intermittent fluctuations is one of the hallmarks of turbulence. We believe that the crest factor is an appropriate tool for extracting space fluctuation features in solutions of dissipative PDEs.