Robert MacKay in Australia
Robert MacKay (Director of Mathematical Interdisciplinary Research at Warwick, UK) is pleased to be visiting COSnet and MASCOS, 2 Nov - 17 Dec.
Based primarily in Mathematics at UNSW, he will visit nodes and give talks as follows:
La Trobe, Melbourne, Mon 14 Nov, 2.05pm (Mathematics), "Examples of how mathematics is useful (even essential) for understanding complex systems"; host: Reinout Quispel
Adelaide, Wed 16 Nov, 3.10pm (Mathematics), "Some perfectly mixing fluid flows"; host: Jim Denier
Griffith, Brisbane, Fri 18 Nov, 11am (Nanoscale Science and Technology), "Ergodic pumping: a mechanism to drive biomolecular conformation change"; host: Debra Bernhardt
UNSW, Sydney, 3-11 Nov and 21 Nov - 9 Dec, host: John Roberts
Fri 25 Nov, 2pm, UNSW, "Examples of how mathematics is useful (even essential) for understanding complex systems (closed and open)"
Fri 2 Dec, 2pm, U Sydney, "The triple linkage: a mechanical Anosov system"
Tue 6 Dec, 3pm, UNSW, "Dynamics of a piecewise affine homeomorphism of a torus"
ANU, Canberra, Wed 14 - Fri 16 Dec, host: Bob Dewar,
Wed 14, 4pm: "Interaction of two charges in a magnetic field"
Thu 15, 11am: "Examples how mathematics is useful (even essential) for understanding complex systems (closed and open)"
Fri 16, 11am: "Ergodic pumping: a mechanism to drive biomolecular conformation change"
Departure from Sydney, 24 Dec
He will be pleased to make contact with any researchers in complex systems or dynamical systems and applications (email@example.com).
Examples of how mathematics is useful (even essential) for understanding complex systems
I will present four examples, from my own and others' work, to
demonstrate that mathematics is useful and essential for understanding
complex systems, both closed and open:
1. Adding new road capacity can increase travel time for everyone;
2. Self-sustaining clusters can emerge even when it appears death rate
3. Everlasting self-localised excitations can
occur even when linear theory would suggest they would radiate away, and
media which are insulating for linear theory can be conducting
4. Indecomposable spatially extended deterministic systems
(as well as stochastic ones) can exhibit non-unique probabilistic
behaviour and sensitivity to boundary conditions.
Some perfectly mixing fluid flows
Within the class of $C^3$ divergence-free vector fields on a domain of $R^3$ with smooth boundary, vanishing on the boundary, I make an example which is mixing and remains so for all small perturbations. Thus the common belief that small elliptic islands are unavoidable is false. I propose two other types of example.
Ergodic pumping: a mechanism to drive biomolecular conformation changes
Biomolecular machines are complex open systems par excellence.
Many of them turn the free energy of hydrolysis of ATP into useful functions, like shortening muscle, advancing a transcription bubble along DNA, and pumping ions across membranes.
Yet how can free energy decreases get turned into anything useful in an unconscious thermal bath of biomolecules?
It is proposed that a significant contribution to the power stroke of myosin and some conformation changes in other biomolecules is the osmotic pressure of a single molecule (e.g. a phosphate ion) expanding a trap.
Necessary conditions to achieve this efficiently are given, and the elements of a mathematical justification.
It is proposed as a design principle for nanobiotechnology.
Joint work with D.J.C.MacKay.
Dynamics of a piecewise affine homeomorphism of the torus
Cerbelli and Giona introduced an interesting example of a piecewise
affine area-preserving homeomorphism of the 2-torus and proved its dynamics have several chaotic properties, e.g. mixing and
positive Lyapunov exponent.
The chaos is non-uniform, however:
the invariant manifolds fold on themselves infinitely often in
opposite directions and fill out the torus in a singular way, so they
proposed this was different from standard types of chaos.
Nevertheless, I show it is "pseudo-Anosov", which allows one to quantify
the singular behaviour and deduce much more about its dynamics.
A mechanical Anosov system: the triple linkage
The triple linkage consists of three disks arranged in a triangle in a plane, free to rotate around their centres and each connected by a rod from an off-centre point to a common floating pivot. Thurston and Weeks introduced it to demonstrate how interesting manifolds can turn up: for the parameter regime of interest, its configuration space is a surface of genus 3.
Anosov systems are the mathematical paradigm of chaotic behaviour: the dynamics are equivalent to a slight generalisation of a finite-state Markov chain. Yet, as Anosov himself regrets, they never seem to turn up in studies of the real world.
Tim Hunt and I put this situation right: we proved that in a suitable parameter regime the free motion of the triple linkage is Anosov on each positive energy level. Furthermore, the dynamics are mixing, correlations of Holder functions decay exponentially, and the rotations of the disks perform a 3D random walk when viewed on a large scale compared to 2 pi.
Interaction of two charges in a magnetic field
A basic problem in plasma physics is how two charged particles interact in a magnetic field, yet it seems not to have been studied in detail, even for a uniform field. We show that the planar motion is integrable if e1/m1 = e2/m2, and has a strong form of chaos if e1.e2 < 0. Preliminary results will be given in 3D. Joint work with Diogo Pinheiro.
Poincare's second species orbits (not scheduled, but given recently at McMaster University, Canada, and offered in case of interest)
Poincare is well known for his contributions to celestial mechanics and his discovery that one should expect chaotic behaviour in typical Hamiltonian systems of more than one degree of freedom. In the example system he studied, however, the chaos he found was exponentially weak in a perturbation parameter. Remarkably, he was aware of another, much stronger, source of chaos but barely exploited it. This is what Bolotin and I have done.
In the three-body problem of celestial mechanics, Poincare proved existence of first species periodic orbits, ones which as the masses of the second and third bodies go to zero go to pairs of Kepler ellipses about the Sun with rational frequency ratio. He also proposed existence of second species periodic orbits, ones which go to periodic sequences of pairs of arcs of Kepler orbit joined at collisions. Although it is agreed that he did not provide a proof, the idea was mostly good. Indeed, extended to N bodies and aperiodic sequences, it is the basis for the design of solar system exploration missions. Nevertheless, a mathematical proof was not provided until 1995 and even that was for only one two-collision periodic orbit.
We extended this drastically, by proving existence of large topological Markov chains of second species orbits for the circular restricted three-body problem for small enough secondary mass, both planar and nonplanar ("restricted" means the third body is taken to have zero mass and "circular" means the secondary and primary bodies are assumed to move in circles around their centre of mass).
Renormalisation of Frenkel-Kontorova chains: from minimum energy to quantum statistical mechanics (not scheduled, but given recently at the Fields Institute, Canada, and offered in case of interest)
We extend a renormalisation operator for the depinning transition of incommensurate minimum energy states of Frenkel-Kontorova chains to one for the quantum statistical mechanics. The known fixed point for the former induces two for the latter, both at temperature zero but with hbar/T = 0 and infinity. Scaling exponents are deduced. Joint work with Nuno Catarino.
N+1 topics in Mechanics Abstract for poster given at Phil Holmes' recent 60th birthday conference in Montreal, which includes most of the above topics plus about 10 more.