Session P47: Nonlinear Dynamics and Chaos

Nonlinear compact periodic solutions in flat band networks

Linear wave equations on translationally invariant flatband (FB) networks exhibit one or more dispersionless bands in their Bloch spectrum. These macroscopically degenerate bands exist due to local symmetries and destructive interference on the lattice. Short-range hopping FB networks host compact localized (eigen)states (CLS) with nonzero amplitudes restricted to a finite volume. We consider the presence of local nonlinear terms in the wave equations. We study the continuation of CLS into the nonlinear domain while keeping their compactness and renormalizing their frequency. We then study the stability of these nonlinear CLS in terms of resonances with extended and compact localized states.   

Chaos in the Band Structure of a Soft Sinai Lattice

We study the effect of broken spatial and dynamical symmetries on the band structure of two lattices with unit cells that are soft versions of the classic Sinai billiard. We find significant signatures of chaos in the band structure of these lattices, in energy regimes where the underlying classical unit cell undergoes a transition to chaos. Broken dynamical symmetries and the presence of chaos can diminish the feasibility of changing and controlling band structure in a wide variety of two-dimensional lattice-based devices, including two-dimensional solids, optical lattices, and photonic crystals.   

Transient Fractality as a Mechanism for Emergent Irreversibility in Chaotic Hamiltonian Dynamics

Boltzmann refuted the Loschmidt irreversibility paradox by arguing that the number of events with a positive entropy production should be infinitely more than that with a negative entropy production despite the one-to-one correspondence between them. His idea was later verified by numerical simulations of reversible dissipative equations of motion, where it was confirmed that one should sample states exactly on a zero-volume fractal attractor to decrease the entropy. Thus, the probability for a negative entropy production vanishes although it is allowed by the equations of motion. We here address the question of whether this fractal picture applies to chaotic Hamiltonian dynamics. Although the Liouville theorem excludes fractality in the long-time limit, we find that a fractal structure transiently emerges in the Bunimovich billiard. Moreover, this transient fractality can be reformulated in light of the flucutuation theorem. As a result, we give a lower bound for an information-theoretic irreversibility, which is determined by the transient fractality.   

Investigation of Nonlinearity Effect During Storm Time Disturbance.

We examine the nonlinearity effect in the Disturbed Storm Time (D_st ) Signals during negligible geomagnetic storm and high geomagnetic storm for monthly (January-December) D_st index time series each year from 2008-2016 using Recurrence Plot, Recurrence Quantification Analysis and Approximate Entropy. A distinct and noticeable trend between negligible geomagnetic Storm and high geomagnetic storm was observed. It was also observed that during negligible geomagnetic storm, the dynamics of the D_st signals behave in stochastic path while the increasing geomagnetic storm (high) tends to behave in deterministic path. As the influence of geomagnetic storm increases, the recurrence plot showed more deterministic structure indicating that that the recurrence plot posseses the potential to detect geomagnetic storm even at infinitesimal influence of geomagnetic storm. Also as the influence of geomagnetic storm varies, the recurrence rate also reflects the dynamical changes and its influence on determinism. From the Observed results, nonlinear dynamics is seen to perform excellently as a diagnostic model to geomagnetic storm disturbance.   

Using a fluctuation analysis of limit cycle oscillations in inner ear hair bundles as a new test of low dimensional dynamical models

The transduction of sound in the inner ear is an active process involving elastic elements, molecular motors, and ion channels. A manifestation of this process in inner ear hair cells is the presence of spontaneous bundle oscillation, observed in vitro in some species. This limit cycle oscillation has been modelled using both a simple two-dimensional Hopf oscillator and more complex variants that explicitly incorporate the physiological variables. The models include stochastic noise in one or more variables representing implicitly the effect of other degrees of freedom.
We study the effects of noise on these stable limit cycles using both analytic calculations and numerical simulations. We show that d-dimensional noisy oscillators generically display phase diffusion at long time scales and (d-1) degrees of freedom that behave roughly as overdamped oscillators. On shorter time scales, we show that noise in the dependent variables which are hidden from measurement, e.g., motor activity, are coupled back to observable variables in ways that allow one to use noise measurements to access otherwise “hidden” features of the system. We compare our theoretical results to data from Rana catesbeiana hair cells.
  

Sensitive detection schemes for small variations in the damping coefficient based on the Duffing-Holmes oscillator

In this work we proposed two detection schemes based on the non-linear properties of the Duffing-Holmes oscillator for the detection of small variations in the damping coefficient. Theoretically, a variation in the damping coefficient up to 0.001% with the possibility to be pushed further can be detected based on our model. A potential on-off magnetic sensor suitable for biomedical applications is suggested by implementing these two schemes with the Giant magnetoresistance based magnetic sensors.   

Dynamical Derivation of Heat Transport

The theoretical derivation of the empirical Fourier law is still an open subject of investigation. We plan to contribute to the solution of this problem moving along the lines of the earlier work of [1]. A set of non-linearly interacting classical oscillators with energy E large enough generates chaos turning classical dynamics into thermodynamics. Following [1] we adopt the popular model proposed by Fermi, Pasta and Ulam (FPU), with a chain of N oscillators, in the condition where the Boltzmann principle S = k ln W applies, and we define temperature T according to the thermodynamic prescription 1/T = dS/dE. We add two additional oscillators, with vanishing kinetic energy, to the right and left end of the FPU chain and we study theoretically and numerically the equilibration process. The kinetic energy of the two additional oscillators is used as an indicator of the local temperature at the border of the FPU chain. We plan to apply iteratively this process to establish, with the help of the central limit theorem, the emergence of Fourier law.   

Phononic Frequency Combs

Optical frequency combs, by its manifestation as a frequency ruler, has revolutionized optical frequency metrology and spectroscopy. Only very recently, the phononic analogue of such frequency combs was experimentally demonstrated1 in a microelectromechanical resonator confirming predictions made by numerical simulations of a Fermi-Pasta-Ulam system2. This initial experimental demonstration was followed up in further studies3-5 where variants of phononic frequency combs have been systematically studied using similar device testbeds over a range of drive conditions. The generation of phononic frequency combs in each case is inherently related to nonlinear modal coupling in such structures with exquisite control of geometry, material properties and conditions required for engineering application. In addition to the general scientific interest, phononic frequency combs also enable a unique approach to tracking the resonant frequency of a micromechanical resonator without an external feedback loop with the potential for improvements in frequency stability compared to conventional feedback oscillators6.
1 PRL 118 3 033903 2017
2 PRL 112 7 075505 2014
3 APL 111 064101 2017
4 arXiv:1704.08008
5 arXiv:1708.07563
6 arXiv:1710.07058   

Seeded localized nonlinear excitations in the Fermi-Pasta-Ulam-Tsingou system – Stability, delocalization and journey to equilibrium

It is well known that the Fermi-Pasta-Ulam-Tsingou system admits localized nonlinear excitations. We show that these excitations exhibit nearly periodic behavior at early times but delocalize by leaking energy in the form of solitary waves and other metastable excitations in the absence of phonons. Further, we show that the excitations share qualitative features with the strongly nonlinear Duffing oscillator. After the delocalization is complete, the system enters a quasi-equilibrium phase characterized by a Gaussian velocity distribution, high kinetic energy fluctuations and no equipartitioning of energy. Finally, at late times, we calculate the specific heat capacity of the system and compare it to analytical results to show that the system transitions past the quasi-equilibrium phase to equilibrium.   

Amplitude Death in a Ring of Nonlinear Oscillators with Unidirectional Coupling

It has been widely studied that the interaction among subsystems can lead to the collective behavior such as synchronization and amplitude death. Among these collective behaviors in coupled oscillators, the amplitude death refers to a situation where individual oscillators cease to oscillate when they are coupled, which is a useful control mechanism for stabilizing systems to steady states. Most of the previous studies on amplitude death have dealt with bidirectional coupling cases. Here, we studied the amplitude death in a ring of coupled nonlinear oscillators with unidirectional couplings and discuss the differences between amplitude death behavior in unidirectional and bidirectional coupling cases.
  

Instantons in Self-Organizing Logic Gates

Self-organizing logic [1] is a recently-suggested framework that allows the solution of Boolean truth tables “in reverse,” i.e., it is able to satisfy the logical proposition of gates regardless to which terminal(s) the truth value is assigned. It can be realized if time non-locality (memory) is present. By employing a practical realization of such gates using electronic circuits with memory, we show, numerically, that SOLGs exploit elementary instantons to reach equilibrium points [2]. Instantons are classical trajectories of the non-linear equations of motion describing SOLGs that connect topologically distinct critical points in the phase space. Our work provides a physical understanding of, and can serve as an inspiration for, new models of bi-directional logic gates that are emerging as important tools in physics-inspired, unconventional computing.
[1] F. L. Traversa and M. Di Ventra, Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 023107, 2017
[2] S.R.B. Bearden, H. Manukian, F.L. Traversa, M. Di Ventra, arXiv:1708.08949   

The minimal “hidden” computer needed to implement a computation

We consider the problem of constructing a physical process to implement some desired (single-valued) function f over a set of “visible” states X. The physical process is represented as a time-inhomogeneous continuous-time Markov chain (CTMC), for example modeling the dynamics of a driven system connected to a heat bath. A prototypical example is a physical implementation of a logical gate in a circuit.

In general, there exists functions f which are not implementable by any possible CTMC, even approximately. However, we demonstrate that for any f, an implementation is always possible if the system has access to some additional "hidden" states not in X. We then consider a natural decomposition of any such CTMC-based implementation into a countable set of discrete steps, demarcated from one another by the raising or lowering of barriers restricting probability flow between states. We demonstrate a tradeoff between the minimal number of hidden states and the minimal number of such steps needed to implement any given f. This tradeoff is analogous to space / time tradeoffs in computational circuit complexity theory - except that it arises even within each gate in such a circuit.   

Exploiting Nonlinear Dynamics for Programmable Behavior in Microfluidic Networks

The development of microfluidic systems that do not rely on abundant external hardware, yet retain controllable, complex behavior has been a primary research goal for the past decade. Microfluidic systems are composed of a network of flow channels and are generally considered to be linear systems; however, by creating nonlinearities in the network we are able to produce a diverse range of flow dynamics. Here, I present a simple microfluidic network that exhibits spontaneous oscillations in the flow rate. It is possible to switch between an oscillating and steady flow state by only changing the driving pressure. This functionality does not rely on external hardware and may even serve as an on-chip memory or timing mechanism. Further, I demonstrate the ability to control the direction of flow though intermediate channels not directly connected to the controlled terminal. I use analytic models and rigorous fluid dynamics simulations to show these results. We expect this work to advance built-in control mechanisms in microfluidic systems towards the goal of designing portable systems that are as controllable as microelectronic circuits.   

Surface temperatures in New York City: Geospatial data enables the accurate prediction of radiative heat transfer

Three decades into the research seeking to derive the urban energy budget, the dynamics of the thermal exchange between the densely built infrastructure and the environment are still not well understood. We present a novel hybrid experimental-numerical approach for the analysis of the radiative heat transfer in New York City. The aim of this work is to contribute to the calculation of the urban energy budget, in particular the stored energy. Improved understanding of urban thermodynamics incorporating the interaction of the various bodies will have implications on energy conservation at the building scale, as well as human health and comfort at the urban scale. The platform presented is based on longwave hyperspectral imaging of nearly 100 blocks of Manhattan, and a geospatial radiosity model that describes the collective radiative heat exchange between multiple buildings. The close comparison of temperature values derived from measurements and the computed surface temperatures (including streets and roads) implies that this geospatial, thermodynamic numerical model applied to urban structures, is promising for accurate and high resolution analysis of urban surface temperatures.   

Chaotic Dynamics of Inner Ear Hair Cells

Hair cells of the auditory and vestibular systems are capable of detecting sounds that induce sub-nanometer vibrations of the hair bundle, far below the stochastic noise levels of the surrounding fluid. Hair cells of certain species are also known to oscillate without external stimulation. The role of these spontaneous oscillations is not yet understood, but they are believed to be a manifestation of an underlying active mechanism. A deeper understanding of spontaneous motility could impact our understanding of the extreme sensitivity of hearing. Our experiments suggest that chaos exists in the dynamical system of the active hair cell. We observe a transition from chaos to order as increasingly stronger stimulus is applied to the hair bundle. This transition is accompanied by an increase in information transmission from the stimulus to the hair bundle, indicative of signal detection. Further, we use a simple theoretical model to describe the observed chaotic dynamics. The model exhibits an enhancement of sensitivity to weak stimuli when the system is poised in the chaotic regime. We propose that chaos may play a role in the hair cell's ability to detect low-amplitude sounds.