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 (mackay@maths.warwick.ac.uk).

1. Adding new road capacity can increase travel time for everyone;

2. Self-sustaining clusters can emerge even when it appears death rate exceeds reproduction;

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 nonlinearly;

4. Indecomposable spatially extended deterministic systems (as well as stochastic ones) can exhibit non-unique probabilistic behaviour and sensitivity to boundary conditions.

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.

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).

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.