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Robert MacKay

University of Warwick, Mathematics, Faculty Member

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Robert MacKay FRS CPhys FInstP CMath FIMA is a Professor in the Mathematics Institute of the University of Warwick and Director of Mathematical Interdisciplinary Research at Warwick. He was Director of Warwick’s Centre for Complexity Science from 2007-15 and President of the (UK) Institute of Mathematics and its Applications for 2012-13. He has made many contributions to the theory and applications of Nonlinear Dynamics. He has extensive research leadership and management experience.
Supervisors: J.M.Greene and M.D.Kruskal

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Steven J. Miller

Williams College

The University of New South Wales

Oleg Yu Vorobyev

Siberian Federal University

Alexandre Borovik

Armando Marques-Guedes

UNL - New University of Lisbon

Udayan Prajapati

St. Xavier's College, Ahmedabad

Higher Institute of Applied Sciences and Technology (HIAST)

Umberto Cerruti

Università degli Studi di Torino

Yildiz Technical University

Tomasz Szemberg

University of the National Education Commission in Krakow

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Papers by Robert MacKay

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

Communications in Mathematical Physics, Oct 8, 2008

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-... more Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov-Schmidt reductions in Fourier space. In particular, existence of periodic/antiperiodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.

Example of simplest bifurcation diagram for a monotone family of vector fields on a torus

arXiv (Cornell University), Jul 8, 2022

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurc... more

Chaotic Properties of Subshifts Generated by a Nonperiodic Recurrent Orbit

International Journal of Bifurcation and Chaos, May 1, 2000

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit clos... more The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain non-periodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a non-periodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

An approximate renormalization for the break-up of invariant tori with three frequencies

Physics Letters, Aug 1, 1994

Use of Stokes’ theorem for plasma confinement

Philosophical Transactions of the Royal Society A, Jun 8, 2020

Stokes' theorem, in its original form and Cartan's generalization, is crucial for designing magne... more Stokes' theorem, in its original form and Cartan's generalization, is crucial for designing magnetic fields to confine plasma (ionized gas). The paper illustrates its use, in particular to address the question whether quasi-symmetric fields, those for which guidingcentre motion is integrable, can be made with little or no toroidal current. This article is part of the theme issue 'Stokes at 200

Ergodic pumping: A mechanism to drive biomolecular conformation changes

Physica D: Nonlinear Phenomena, Apr 1, 2006

We propose that a significant contribution to the power stroke of myosin and similar conformation... more We propose that a significant contribution to the power stroke of myosin and similar conformation changes in other biomolecules is the pressure of a single molecule (e.g. a phosphate ion) expanding a trap, a mechanism we call "ergodic pumping". We demonstrate the principle with a toy computer model and discuss the mathematics governing the evolution of slow degrees of freedom in large Hamiltonian systems. We indicate in detail how the mechanism could fit with known features of the myosin cycle. Many other biomolecular conformation changes could be driven in part by ergodic pumping. We suggest the use of ergodic pumping as a design principle in nanobiotechnology.

Nonlinear Structures and Thermodynamic Instabilities in a One-Dimensional Lattice System

Physical Review Letters, Dec 13, 2004

The equilibrium states of the discrete Peyrard-Bishop Hamiltonian with one end fixed are computed... more The equilibrium states of the discrete Peyrard-Bishop Hamiltonian with one end fixed are computed exactly from the two-dimensional nonlinear Morse map. These exact nonlinear structures are interpreted as domain walls (DW), interpolating between bound and unbound segments of the chain. The free energy of the DWs is calculated to leading order beyond the Gaussian approximation. Thermodynamic instabilities (e.g. DNA unzipping and/or thermal denaturation) can be understood in terms of DW formation.

Nonlinearity in complexity science

Nonlinearity, Nov 13, 2008

The role of nonlinearity in complexity science is discussed, and some nonlinear research problems... more

Quantifying Emergence in terms of Persistent Mutual Information

arXiv (Cornell University), Mar 15, 2010

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the pas... more We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long term prediction, which we propose as the primary signature of (Strong) Emergent Behaviour. The key feature of our definition of PMI is the omission of an interval of 'present' time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behaviour or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map we compare predicted with measured longtime PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalisation of the notion of ordered phases.

Langevin Equation for Slow Degrees of Freedom of Hamiltonian Systems

Understanding complex systems, 2010

A way is sketched to derive a Langevin equation for the slow degrees of freedom of a Hamiltonian ... more

Interaction of two charges in a uniform magnetic field: I. Planar problem

Nonlinearity, Jun 29, 2006

The interaction of two charges moving in R 2 in a magnetic field B can be formulated as a Hamilto... more The interaction of two charges moving in R 2 in a magnetic field B can be formulated as a Hamiltonian system with 4 degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with 2 degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non-integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion.

Energy Thresholds for Discrete Breathers in One-, Two-, and Three-Dimensional Lattices

Physical Review Letters, Feb 17, 1997

Discrete breathers are time-periodic, spatially localized solutions of equations of motion for cl... more Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degrees of freedom interacting on a lattice. They come in one-parameter families. We report on studies of energy properties of breather families in one-, two-and three-dimensional lattices. We show that breather energies have a positive lower bound if the lattice dimension of a given nonlinear lattice is greater than or equal to a certain critical value. These findings could be important for the experimental detection of discrete breathers.

Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks

Understanding complex systems, 2021

Renormalization and Quantum Scaling of Frenkel–Kontorova Models

Journal of Statistical Physics, Dec 1, 2005

We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel-Kontor... more We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel-Kontorova models studied by Aubry and others to non-zero Planck's constant and temperature. This analysis is based on the study of a renormalization operator for the case of irrational mean spacing using Feynman's functional integral approach. We show how existing classical results extend to the quantum regime. In particular we extend MacKay's renormalization approach for the classical statistical mechanics to deduce scaling of low frequency effects and quantum effects. Our approach extends the phenomenon of hierarchical melting studied by Vallet, Schilling and Aubry to the quantum regime.

Variational properties of a nonlinear elliptic equation and rigidity

Duke Mathematical Journal, May 1, 2000

We consider in this paper elliptic equations which are perturbations of Laplace's equation by a c... more We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon.

Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori

Physics Letters, Oct 1, 2000

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

Journal of Nonlinear Science, Apr 29, 2008

The interaction of two charges moving in R 3 in a magnetic field B can be formulated as a Hamilto... more The interaction of two charges moving in R 3 in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either "bounce-back", "pass-through", or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into "free" and "trapped" parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories which come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.

Differential forms for plasma physics

Journal of Plasma Physics, Jan 13, 2020

Differential forms provide a coordinate-free way to express many quantities and relations in math... more

Multistability in networks of weakly coupled bistable units

Physica D: Nonlinear Phenomena, Apr 1, 1995

We study the stationary states of networks consisting of weakly coupled bistable units. We prove ... more We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a 'propagation failure' phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples.

A new stochastic framework for ship capsizing

arXiv (Cornell University), Jun 7, 2021

We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands... more We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs.

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

Communications in Mathematical Physics, Oct 8, 2008

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-... more Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov-Schmidt reductions in Fourier space. In particular, existence of periodic/antiperiodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.

Example of simplest bifurcation diagram for a monotone family of vector fields on a torus

arXiv (Cornell University), Jul 8, 2022

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurc... more

Chaotic Properties of Subshifts Generated by a Nonperiodic Recurrent Orbit

International Journal of Bifurcation and Chaos, May 1, 2000

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit clos... more The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain non-periodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a non-periodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

An approximate renormalization for the break-up of invariant tori with three frequencies

Physics Letters, Aug 1, 1994

Use of Stokes’ theorem for plasma confinement

Philosophical Transactions of the Royal Society A, Jun 8, 2020

Stokes' theorem, in its original form and Cartan's generalization, is crucial for designing magne... more Stokes' theorem, in its original form and Cartan's generalization, is crucial for designing magnetic fields to confine plasma (ionized gas). The paper illustrates its use, in particular to address the question whether quasi-symmetric fields, those for which guidingcentre motion is integrable, can be made with little or no toroidal current. This article is part of the theme issue 'Stokes at 200

Ergodic pumping: A mechanism to drive biomolecular conformation changes

Physica D: Nonlinear Phenomena, Apr 1, 2006

We propose that a significant contribution to the power stroke of myosin and similar conformation... more We propose that a significant contribution to the power stroke of myosin and similar conformation changes in other biomolecules is the pressure of a single molecule (e.g. a phosphate ion) expanding a trap, a mechanism we call "ergodic pumping". We demonstrate the principle with a toy computer model and discuss the mathematics governing the evolution of slow degrees of freedom in large Hamiltonian systems. We indicate in detail how the mechanism could fit with known features of the myosin cycle. Many other biomolecular conformation changes could be driven in part by ergodic pumping. We suggest the use of ergodic pumping as a design principle in nanobiotechnology.

Nonlinear Structures and Thermodynamic Instabilities in a One-Dimensional Lattice System

Physical Review Letters, Dec 13, 2004

The equilibrium states of the discrete Peyrard-Bishop Hamiltonian with one end fixed are computed... more The equilibrium states of the discrete Peyrard-Bishop Hamiltonian with one end fixed are computed exactly from the two-dimensional nonlinear Morse map. These exact nonlinear structures are interpreted as domain walls (DW), interpolating between bound and unbound segments of the chain. The free energy of the DWs is calculated to leading order beyond the Gaussian approximation. Thermodynamic instabilities (e.g. DNA unzipping and/or thermal denaturation) can be understood in terms of DW formation.

Nonlinearity in complexity science

Nonlinearity, Nov 13, 2008

The role of nonlinearity in complexity science is discussed, and some nonlinear research problems... more

Quantifying Emergence in terms of Persistent Mutual Information

arXiv (Cornell University), Mar 15, 2010

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the pas... more We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long term prediction, which we propose as the primary signature of (Strong) Emergent Behaviour. The key feature of our definition of PMI is the omission of an interval of 'present' time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behaviour or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map we compare predicted with measured longtime PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalisation of the notion of ordered phases.

Langevin Equation for Slow Degrees of Freedom of Hamiltonian Systems

Understanding complex systems, 2010

A way is sketched to derive a Langevin equation for the slow degrees of freedom of a Hamiltonian ... more

Interaction of two charges in a uniform magnetic field: I. Planar problem

Nonlinearity, Jun 29, 2006

The interaction of two charges moving in R 2 in a magnetic field B can be formulated as a Hamilto... more The interaction of two charges moving in R 2 in a magnetic field B can be formulated as a Hamiltonian system with 4 degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with 2 degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non-integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion.

Energy Thresholds for Discrete Breathers in One-, Two-, and Three-Dimensional Lattices

Physical Review Letters, Feb 17, 1997

Discrete breathers are time-periodic, spatially localized solutions of equations of motion for cl... more Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degrees of freedom interacting on a lattice. They come in one-parameter families. We report on studies of energy properties of breather families in one-, two-and three-dimensional lattices. We show that breather energies have a positive lower bound if the lattice dimension of a given nonlinear lattice is greater than or equal to a certain critical value. These findings could be important for the experimental detection of discrete breathers.

Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks

Understanding complex systems, 2021

Renormalization and Quantum Scaling of Frenkel–Kontorova Models

Journal of Statistical Physics, Dec 1, 2005

We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel-Kontor... more We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel-Kontorova models studied by Aubry and others to non-zero Planck's constant and temperature. This analysis is based on the study of a renormalization operator for the case of irrational mean spacing using Feynman's functional integral approach. We show how existing classical results extend to the quantum regime. In particular we extend MacKay's renormalization approach for the classical statistical mechanics to deduce scaling of low frequency effects and quantum effects. Our approach extends the phenomenon of hierarchical melting studied by Vallet, Schilling and Aubry to the quantum regime.

Variational properties of a nonlinear elliptic equation and rigidity

Duke Mathematical Journal, May 1, 2000

We consider in this paper elliptic equations which are perturbations of Laplace's equation by a c... more We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon.

Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori

Physics Letters, Oct 1, 2000

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

Journal of Nonlinear Science, Apr 29, 2008

The interaction of two charges moving in R 3 in a magnetic field B can be formulated as a Hamilto... more The interaction of two charges moving in R 3 in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either "bounce-back", "pass-through", or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into "free" and "trapped" parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories which come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.

Differential forms for plasma physics

Journal of Plasma Physics, Jan 13, 2020

Differential forms provide a coordinate-free way to express many quantities and relations in math... more

Multistability in networks of weakly coupled bistable units

Physica D: Nonlinear Phenomena, Apr 1, 1995

We study the stationary states of networks consisting of weakly coupled bistable units. We prove ... more We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a 'propagation failure' phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples.

A new stochastic framework for ship capsizing

arXiv (Cornell University), Jun 7, 2021

We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands... more We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs.