### Workshop on Random Dynamical Systems and Anomalous Dynamics

### Scientific Coordinators: Yuzuru Sato (Hokkaido), Rainer Klages (London), Jeroen Lamb (London)

ysato@math.sci.hokudai.ac.jp | r.klages@qmul.ac.uk | jsw.lamb@imperial.ac.uk

- London Mathematical Laboratory
- Royal Society of London UK-Japan grant
- Grant-in-Aid for Scientific Research, JSPS, Japan
- EU ITN CRITICS
- Research England GCRF UK-Vietnam grant.

### Workshop aim:

This workshop brings together theoretical physicists and mathematicians working on dynamical systems and stochastic processes. An emphasis is on the recent active fields of random dynamical systems and anomalous stochastic dynamics. The former refers to chaotic dynamical systems that are coupled to noise. This combination can create novel phenomena such as anomalous diffusion, stochastic bifurcations, and varieties of noise-induced phenomena. To describe them requires a new mathematical language defining objects such as random attractors, perturbed invariant measures, bifurcations, and rare event statistics. Anomalous dynamics denotes processes whose mean square displacement, unlike that of Brownian motion, grows non-linearly in time. They exhibit non-trivial properties related to non-Gaussian probability distributions, such as power-law decay of correlations, sub- or super-diffusion and ergodicity breaking. However, the emergence of such phenomena from deterministic dynamics is poorly understood. By bridging the gap between the dynamical systems and applied stochastic processes communities we aim to advance this understanding.

### Keywords:

random dynamical systems, stochastic bifurcations, anomalous stochastic processes, anomalous diffusion, non-ergodicity, ergodicity economics, fractional operators, non-Markovian dynamics

### Programme

All talks at Level 1 (one below street level) of 180 Queens Gate (Huxley building), Imperial College London.

**Wednesday 20 March 2019: lecture room Huxley 144**

**Random dynamical systems and time series analysis**

Time | Speaker, Talk and Abstract |
---|---|

13:30 – 13:35 | Opening Remarks |

13:35 – 14:00 | Yuzuru Sato (Hokkaido University)Stochastic chaos in a turbulent swirling flowWe report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely, the number of quasistationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can be recovered neither using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low-dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasistationary states. |

14:00 – 14:25 | Takumi Chihara (Hokkaido University)Stochastic bifurcation in a turbulent swirling flowThe von Karman flow is a swirling flow in a cylinder with two upper/lower propellers. Changing system symmetry the flow shows transition to turbulence. Collective motion in a turbulent von Karman flow has been studied as stochastic chaos in a stochastic Duffing equations. We study bifurcation in a random map extracted from the experimental time series of the von Karman flow. The random map exhibits qualitatively same stochastic bifurcation structure as the experimentally observed transitions. |

14:25 – 15:10 | Davide Faranda (CNRS)Minimal dynamical systems model of the northern hemisphere jet stream via embedding of climate dataWe derive a minimal dynamical model for the northern hemisphere mid-latitude jet dynamics by embedding atmospheric data, and investigate its properties (bifurcation structure, stability, local dimensions) for different atmospheric flow regimes. We derive our model according to the following steps: i) obtain a 1-D description of the mid-latitude jet-stream by computing the position of the jet at each longitude using the ERA-Interim reanalysis, ii) use the embedding procedure to derive a map of the local jet position dynamics, iii) introduce the coupling and stochastic effects deriving from both atmospheric turbulence and topographic disturbances to the jet. We then analyze the dynamical properties of the model in different regimes: i) one that gives the closest representation of the properties extracted from real data, ii) one featuring a stronger jet (strong coupling), iii) one featuring a weaker jet (low coupling), iv) modified topography. We argue that such a simple model provides a useful description of the dynamical properties of the atmospheric jet. |

15:10 – 15:40 | Break |

15:40 – 16:25 | Yoshito Hirata (University of Tokyo)Identifying nonlinear stochastic systems via a set of two hypothesis testsHistorically, nonlinear time series analysis has been developed to distinguish nonlinear deterministic systems from linear stochastic systems. But, potentially, there are many nonlinear stochastic systems. Here we propose how to identify nonlinear stochastic systems by introducing two hypothesis tests. The proposed set of methods is demonstrated with toy examples. |

16:25 – 17:10 | Rainer Klages (Queen Mary University of London)Stochastic modeling of diffusion in dynamical systems: three examplesConsider equations of motion yielding dispersion of an ensemble of particles. For a given dynamical system an interesting problem is not only what type of diffusion is generated but also whether the resulting diffusive dynamics matches to a known stochastic process. I will discuss three examples of dynamical systems displaying different types of diffusive transport: The first model is fully deterministic but non-chaotic by showing a whole range of normal and anomalous diffusion under variation of a single control parameter [1]. The second model is a soft Lorentz gas where a point particles moves through repulsive Fermi potentials situated on a triangular periodic lattice [2]. It is fully deterministic by displaying an intricate switching between normal and superdiffusion under variation of control parameters. The third model randomly mixes in time chaotic dynamics generating normal diffusive spreading with non-chaotic motion where all particles localize [3]. Varying a control parameter the mixed system exhibits a transition characterised by subdiffusion. In all three cases I will show successes, failures and pitfalls if one tries to reproduce the resulting diffusive dynamics by using simple stochastic models. Joint work with all authors on the references cited below. [1] L. Salari, L. Rondoni, C. Giberti, R. Klages, Chaos, 25, 073113 (2015) [2] R.Klages, S.S.Gallegos, J.Solanpaa, M.Sarvilahti, E.Rasanen, preprint arXiv:1811.06976 (2018) [3] Y.Sato, R.Klages, preprint arXiv:1810.02674 (2018) |

**Thursday 21 March 2019: lecture room Huxley 130**

Stochastic process and diffusion

Time | Speaker, Talk and Abstract |
---|---|

09:30 – 10:15 | Alex Adamou (London Mathematical Laboratory)I argue no, because the time and ensemble averages of its position are different. In this talk I shall share some thoughts on terminology.Is Brownian motion ergodic? |

10:15 – 11:00 | Massimiliano Giona (La Sapienza Università di Roma)“Die welt als wille und (stochastische) vorstellung” – Some reflections on the representation and statistical approximation of stochastic processes Given a stochastic process, several statistical representations and approximations are possible depending on the time-scales, on the simplifying assumptions regarding the “internal degrees of freedom”, on the symmetries. This is e.g. the case of symmetric lattice random walk of independent particles, viewed as an approximation of a process evolving continuously in space and time, for which the diffusion equation represents the long-term hydrodynamic limit. Of course, other statistical descriptions are possible, and the hyperbolic hydrodynamic model represents a more ”gentle” description as regards the property of finite propagation. Starting from the hyperbolic hydrodynamic description of lattice random walk, expressed in terms of Poisson-Kac density waves, its spinorial statistical representation is analyzed, pointing out its necessity in order to describe completely a process possessing finite propagation velocity. The hyperbolic structure of the statistical description influences also the setting of proper boundary conditions. The case of interfacial boundary conditions in multiphases lattices are analyzes. Moreover, some fundamental and delicate issues associated with the partial wave representation of Levy walks and quantum stochastic processes (the meaning of which is introduced in the presentation) are briefly outlined and discussed. |

11:00 – 11:30 | Break |

11:30 – 12:15 | Thai Son Doan (Vietnam Academy of Science and Technology)On solutions of stochastic fractional differential equationsIn this talk, we establish an existence of solutions for stochastic fractional differential equations. We also show a lower bound on the asymptotic separation between solutions. In the last part, we show a coincidence between the classical solution and the mild solution for these systems by establishing a variation of constants formula. This is a joint work with P. T. Anh, P.T. Huong, P.E. Kloeden and H.T. Tuan. |

12:15 – 13:45 | Break |

13:45 – 14:30 | Sergei Fedotov (The University of Manchester)Anomalous superdiffusive transport and space-dependent variable order fractional equationsI will discuss Levy walks and Levy flights that are the fundamental notions in physics and biology with numerous applications including T-cell motility in the brain and active transport within living cells. I will present several new integro-differential equations for a Levy walk. I will also discuss the space-dependent variable order fractional equations and anomalous aggregation phenomenon |

14:30 – 15:15 | Andrea Cairoli (Imperial College London)The diffusion process followed by a tracer in a medium out of equilibrium typically exhibits anomalous diffusive characteristics that cannot be captured by Brownian motion. Modeling the tracer fluctuating dynamics is thus a challenging task that can provide fundamental insight into the rheological and thermodynamic properties of active systems. Prototypical active media are suspensions of swimming microorganisms, like algae and bacteria, where the tracer is dragged by the hydrodynamic ow generated by the swimmers. Several experiments have characterized the tracer diffusion in dilute conditions in terms of a greatly enhanced diffusion coefficient, non-Gaussian tails of the displacement statistics, and crossover phenomena from non-Gaussian to Gaussian scaling. Despite the abundance of experimental results, there is so far no comprehensive theory that can describe all the observed diffusional characteristics of the tracer. Here we present a theoretical framework of the enhanced tracer diffusion in active suspensions from first-principles, by coarse-graining the microscopic hydrodynamic interactions between the tracer and the active particles as a stochastic process. The random driving force in the Langevin equation of motion is a colored Levy Poisson process that induces power-law distributed position displacements. This theory predicts a non-monotonic transition of the scaling exponents of the displacement statistics at different timescales, manifest in the distribution tail becoming fatter from short to intermediate timescales and converging to a Gaussian scaling for long ones. Our framework not only provides the toolkit necessary to characterize theoretically the tracer diffusion in active suspensions but also paves the way to the study of the tracer stochastic thermodynamics.Dancing to the swimmers’ beat: Loopy Levy flights enhance tracer diffusion in active suspensions |

15:15 – 15:45 | Break |

15:45 – 16:30 | Enrico Scalas (University of Sussex)The Mathematics of Human ContactWe present a statistical analysis of high-resolution contact pattern data within primary and secondary schools as collected by the SocioPatterns collaboration. Students are graphically represented as nodes in a temporally evolving network, in which links represent proximity or interaction between students. We focused on link- and node-level statistics, such as the on- and off-durations of links as well as the activity potential of nodes and links. Parametric models are fitted to the on- and off-durations of links, inter-event times and node activity potentials and, based on these, we propose a number of theoretical models that are able to reproduce the collected data within varying levels of accuracy. By doing so, we aim to identify the minimal network-level properties that are needed to closely match the real-world data, with the aim of combining this contact pattern model with epidemic models in future work. This is joint work with Stephen Ashton, Nicos Georgiou and Istvan Zoltan Kiss. Relevant preprints: https://arxiv.org/abs/1502.04072 https://arxiv.org/abs/1802.07261 Published papers: [1]Georgiou, Nicos, Kiss, Istvan Z and Scalas, Enrico (2015) Solvable non-Markovian dynamic network. Physical Review E, 92 (4). 042801. ISSN 1539-3755. [2]Ashton, Stephen, Scalas, Enrico, Georgiou, Nicos and Kiss, István Zoltán (2018) The mathematics of human contact: developing a model for social interaction in school children. Acta Physica Polonica A, 133 (6). pp. 1421-1432. ISSN 0587-4246. |

16:30 – 17:15 | Nickolay Korabel (The University of Manchester)Memory effects and Levy walk dynamics in intracellular transport of cargoes The phenomenon of cumulative inertia is demonstrated in intracellular transport involving multiple motor proteins in human epithelial cells by measuring the empirical survival probability of cargoes on microtubules and their detachment rates. It is found the longer a cargo moves along a microtubule, the less likely it detaches from it. As a result, the movement of cargoes is non-Markovian and involves a memory. Memory effects are observed on the scale of up to 2 seconds. We provide a theoretical link between the measured detachment rate and the super-diffusive Levy walk-like cargo movement. |

19:30 | Workshop Dinner |

**Friday 22 March 2019: lecture room Huxley 145**

Random dynamical systems and fractals

**(supported by UK-Japan grant by Royal Society of London)**

Time | Speaker, Talk and Abstract |
---|---|

09:30 – 10:15 | Jeroen Lamb (Imperial College London)Conditioned Lyapunov exponents for random dynamical systemsWe introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. We illustrate its use in the analysis of local bifurcations in systems with unbounded noise. |

10:15 – 11:00 | Hiroki Sumi (Kyoto University)Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps and abstractWe consider i. i. d. random holomorphic dynamical systems on the Riemann sphere. Especially, we introduce a nice class of such systems and we show that generic random holomorphic dynamical systems of complex polynomials belong to the nice class. Also, regarding the random dynamical systems generated by a subfamily of the quadratic family Lz(1-z) | L in C/0, we classify the elements belonging to the above nice class. For the preprint, see H. Sumi, Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods. https://arxiv.org/abs/1608. |

11:00 – 11:25 | Break |

11:25 – 11:50 | Takayuki Watanabe (Kyoto University)We consider random holomorphic dynamical systems of complex polynomials with “Markovian rule”, which are not i.i.d. RDS. We show that a generic system is stable in average by using complex analytic techniques. For such a generic system, almost every random Julia set is of measure zero and the function which represents the probability of tending to infinity is continuous on the whole space. This is a joint work with Hiroki Sumi (Kyoto Univ.). For the preprint see https://arxiv.org/abs/1810.09922.Non-i.i.d. random holomorphic dynamical systems of complex polynomials and the generic property |

11:50 – 12:15 | Kanji Inui (Kyoto University)Dimensions and measures of the limit sets of infinite conformal IFSs related to the generalized complex continued fractions Many famous fractal sets (for example, Cantor set, Sierpinski gasket and so on) are defined as the limit sets of contractive iterated function systems (for short IFSs) with finitely many mappings. But, recently D. Mauldin and M. Urbanski studied limit sets of conformal IFSs (for short CIFSs) with infinitely many mappings. And, they showed that there exists a CIFS such that the Hausdorff measure of the limit set corresponding to the Hausdorff dimension is zero and the packing measure of the limit set corresponding to the same dimension is positive. Note that the limit sets of CIFSs with finitely many mappings do not have the above properties. In this talk, we introduce an analytic family of CIFSs with infinitely many mappings related to complex continued fractions such that the limit set of each system in the family has the above strange properties and such that the Hausdorff dimension of the limit set is a real-analytic and subharmonic function of the parameter. This study is a joint work with Hiroki Sumi (Kyoto University) and Hikaru Okada (Osaka University). For the preprints, see https://arxiv.org/abs/ |

12:15 – 13:00 | Ale Jan Hombourg (University of Amsterdam)I’ll consider skew product systems $(y,x) \mapsto (g(y), f_y(x))$ with low dimensional maps $f_y$ as fiber maps, driven by shifts or by expanding circle maps $g$. The fiber maps $f_y$ will have a common fixed point and I will consider the case where Lyapunov exponents in the fiber direction vanish. I will explain that this can lead to intermittent dynamics.On-off intermittency |

13:00 – 14:30 | Break |

14:30 – 14:55 | Toru Sera (Kyoto University)Interval maps with indifferent fixed points are called intermittent interval maps. They have been studied as models of intermittent phenomena, such as intermittent transitions to turbulence in convective fluid. In this talk, we will present a functional limit theorem for joint laws of the occupations near and away from the indifferent fixed points, in the sense of strong distributional convergence. It is a functional and joint-distributional extension both of Darling–Kac type limit theorem and of Lamperti type generalized arcsine laws at the same time.Functional limit theorem for intermittent interval maps |

14:55 – 15:20 | Yuto Nakajima (Kyoto University)Self-similar fractals related to regular tetrahedron We introduce fractal regular tetrahedron. Fractal n-gons, which are self-similar sets related to regular n-gon, have been investigated in two dimensional space. We try to extend similar arguments to three dimensional setting. In particular, we consider self-similar sets related to regular tetrahedron. |

15:20 – 16:05 | Dmitry Turaev (Imperial College London)Energy transfer in slow-fast Hamiltonian systems. We demonstrate that if a Hamiltonian system is not ergodic for some range of parameter values, then a slow and time-periodic change of parameters may lead to a sustained energy growth. This principle extends to a general setting of slow-fast Hamiltonian systems: violation of ergodicity in the fast subsystem leads to a rapid equilibration in the slow-fast system as a whole. We have a similar phenomenon in the quantum-mechanical setting where we show that a periodic destruction and restoration of a quantum integral leads to an exponential energy growth. |

16:05 | Closing |

### Location

Imperial College London: All talks at Level 1 (one below street level) of 180 Queens Gate (Huxley building)

### Date & Time

Wednesday 20th – Friday 22nd March 2019

### Information

ysato@math.sci.hokudai.ac.jp

r.klages@qmul.ac.uk

jsw.lamb@imperial.ac.uk