报告人： Domenico Lippolis (Tokyo Metropolitan University)
Abstracts:
Lecture 1: Stretch and Fold (45min)
I begin by introducing the concept of Markov partitions for chaotic systems, and of symbolic dynamics applied to return maps.
Secondly, I explain the stretching and folding mechanism at the basis of chaotic behavior, and how to construct a discrete-time dynamical system from a set of ordinary differential equations such as Lorenz’s model. In the realm of Hamiltonian systems, chaotic maps can symbolically be described by the so-called Smale’s horseshoes, and so can the transition from regular to chaotic dynamics.
Lecture 2: Discrete Symmetry Reduction (45min)
In this lecture, I explore the role of discrete symmetries in chaotic dynamical systems.
The paradigmatic Lorenz `butterfly’ attractor and other representative nonlinear flows are reduced by a simple transformation, by leveraging a reflection symmetry. A more widely applicable treatment may be developed by switching to the Perron-Frobenius formalism, where one follows swarms of orbits rather than single trajectories. The latter approach is also more practical when dealing with discrete-time maps, as well as chaotic billiards, where in particular unstable periodic orbits are reduced and reordered. I will also present an example of discrete symmetry reduction in a coupled map lattice, a higher-dimensional system.
Lecture 3: Mapping Densities in a Noisy State Space (45min)
I will examine the interplay of nonlinear, deterministic dynamics, with weak noise, in the neighborhood of
unstable periodic orbits. The competition between deterministic contraction and smearing due to noise (forward in time), or
deterministic stretching (backward in time) and noise, results in a local distribution of trajectories that always exhibits memory effects.
In that sense, noise in nonlinear dynamics is never effectively uncorrelated. I will exemplify these findings on two-dimensional hyperbolic maps, where the evolution of densities is studied with the Fokker-Planck operator.
Lecture 4: Chaos with Background Noise and the Resolution of the State Space (45min)
The Fokker-Planck formalism explained in the previous lecture is wielded to partition the state space of a chaotic system with weak noise.
The strategy is that to consider the unstable periodic orbits of the noiseless system as the skeleton of the dynamics, and locally replace them with the eigenfunctions of the Fokker-Planck operator of the linearized system in the state space partition. The latter is found to have a limiting resolution, which effectively projects the transport operator to a finite-dimensional space. As a result, long-time averages of observables of interest, such as escape rate, diffusivities, or Lyapunov exponents, are more efficiently computed from graphs or finite transfer matrices.
Lecture 5: Thermodynamics of Chaos: from Equilibrium to Relaxation Processes (90min)
Strongly chaotic systems may relax to a state of statistical equilibrium, where expectation values are weighted by the asymptotic density distribution of the phase space. That is called natural measure or conditionally invariant density, depending on whether the system is closed or it admits escape.
In that regime, a relation between information entropy, averaged observables, and escape rate was derived some 40 years back, which resembles the equivalence of free energy, internal energy, and entropy in equilibrium thermodynamics. In this lecture I will review this formalism, that originates from large-deviations theory, and further extend it to chaotic systems that are out of statistical equilibrium, and in the process of relaxing to an asymptotic state.
Minicourse schedule:
19 Aug.
11:00 – Lecture 1: Stretch and Fold (45min)
Lunch break
13:00 – Lecture 2: Discrete Symmetry Reduction (45min)
20 Aug.
11:00 – Lecture 3: Mapping Densities in a Noisy State Space (45min)
Lunch break
13:00 – Lecture 4: Chaos with Background Noise and the Resolution of the State Space (45min)
21 Aug.
11:00-12:30 – Lecture 5 (with an intermission on 11:45-12:00): Thermodynamics of Chaos: from Equilibrium to Relaxation Processes (90min)
地点： Room 1710
Zoom Meeting ID: 844 0594 7424 (Passcode: 076895)
Introduction to the speaker: Professor Lippolis is an expert on the thermodynamics of chaos. After completing his PhD under the supervision of Predrag Cvitanovic at Georgia Institute of Technology, he worked as a postdoctoral scholar at Pusan National University and the Institute of Advanced Study at Tsinghua, following which he took a position at Jiangsu University. Currently, Professor Lippolis works at Tokyo Metropolitan University.
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