In this work, we introduce the escape measure, a finite-time version of the natural measure, to investigate the transient dynamics of escape orbits in open Hamiltonian systems. In order to numerically calculate the escape measure, we cover a region of interest of the phase space with a grid and we compute the visitation frequency of a given orbit on each box of the grid before the orbit escapes. Since open systems are not topologically transitive, we also define the mean escape measure, an average of the escape measure on an ensemble of initial conditions. We apply these concepts to study two physical systems: the single-null divertor tokamak, described by a two-dimensional map; and the Earth-Moon system, as modeled by the planar circular restricted three-body problem. First, by calculating the mean escape measure profile, we visually illustrate the paths taken by the escape orbits within the system. We observe that the choice of the ensemble of initial conditions may lead to distinct dynamical scenarios in both systems. Particularly, different orbits may experience different stickiness effects. After that, we analyze the mean escape measure distribution and we find that these vary greatly between the cases, highlighting the differences between our systems as well. Lastly, we define two parameters: the escape correlation dimension, that is independent of the grid resolution, and the escape complexity coefficient, which takes into account additional dynamical aspects, such as the orbit's escape time. We show that both of these parameters can quantify and distinguish between the diverse transient scenarios that arise.

We devise a machine learning technique to solve the general problem of inferring network links that have time-delays. The goal is to do this purely from time-series data of the network nodal states. This task has applications in fields ranging from applied physics and engineering to neuroscience and biology. To achieve this, we first train a type of machine learning system known as reservoir computing to mimic the dynamics of the unknown network. We formulate and test a technique that uses the trained parameters of the reservoir system output layer to deduce an estimate of the unknown network structure. Our technique, by its nature, is non-invasive, but is motivated by the widely-used invasive network inference method whereby the responses to active perturbations applied to the network are observed and employed to infer network links (e.g., knocking down genes to infer gene regulatory networks). We test this technique on experimental and simulated data from delay-coupled opto-electronic oscillator networks. We show that the technique often yields very good results particularly if the system does not exhibit synchrony. We also find that the presence of dynamical noise can strikingly enhance the accuracy and ability of our technique, especially in networks that exhibit synchrony.

In the present work, we explore the influence of habitat complexity on the activities of prey and predator of a spatio-temporal system by incorporating self diffusion. First we modify the Rosenzweig-MacArthur predator-prey model by incorporating the effects of habitat complexity on the carrying capacity and fear effect of prey and predator functional response. We establish conditions for the existence and stability of all feasible equilibrium points of the non-spatial model and later we prove the existence of Hopf and transcritical bifurcations in different parametric phase-planes analytically and numerically. The stability of the spatial system is studied and we discuss the conditions for Turing instability. Selecting suitable control parameter from the Turing space, the existence conditions for stable patterns are derived using the amplitude equations. Results obtained from theoretical analysis of the amplitude equations are justified by numerical simulation results near the critical parameter value. Further, from numerical simulation, we illustrate the effect of diffusion of the dynamical system in the spatial domain by different pattern formations. Thus our model clearly shows that the fear effect of prey and predator's functional response make an anti-predator behaviour including habitat complexity which helps the prey to survive in the spatio-temporal domain through diffusive process.

We derive a set of identities for the theta functions on compact Riemann surfaces which generalize the famous trisecant Fay identity. Using these identities we obtain quasiperiodic solutions for a multidimensional generalization of the Hirota bilinear difference equation and for a multidimensional Toda-type system.

We present exact bright, dark and rogue soliton solutions of generalized higherorder nonlinear Schrodinger equation, describing the ultrashort beam propagation in tapered waveguide amplifier, via a similarity transformation connected with the constant-coefficient Sasa-Satsuma and Hirota equations. Our exact analysis takes recourse to identify the allowed tapering profile in conjunction with appropriate gain function which corresponds to PT-symmetric waveguide. We extend our analysis to study the effect of tapering profiles and higher-order terms on the evolution of self-similar waves and thus enabling one to control the self-similar wave structure and dynamical behavior.

Recent advances in acquisition equipment is providing experiments with growing amounts of precise yet affordable sensors. At the same time an improved computational power, coming from new hardware resources (GPU, FPGA, ACAP), has been made available at relatively low costs. This led us to explore the possibility of completely renewing the chain of acquisition for a fusion experiment, where many high-rate sources of data, coming from different diagnostics, can be combined in a wide framework of algorithms. If on one hand adding new data sources with different diagnostics enriches our knowledge about physical aspects, on the other hand the dimensions of the overall model grow, making relations among variables more and more opaque. A new approach for the integration of such heterogeneous diagnostics, based on composition of deep \textit{variational autoencoders}, could ease this problem, acting as a structural sparse regularizer. This has been applied to RFX-mod experiment data, integrating the soft X-ray linear images of plasma temperature with the magnetic state. However to ensure a real-time signal analysis, those algorithmic techniques must be adapted to run in well suited hardware. In particular it is shown that, attempting a quantization of neurons transfer functions, such models can be modified to create an embedded firmware. This firmware, approximating the deep inference model to a set of simple operations, fits well with the simple logic units that are largely abundant in FPGAs. This is the key factor that permits the use of affordable hardware with complex deep neural topology and operates them in real-time.

Conventional neural networks are universal function approximators, but because they are unaware of underlying symmetries or physical laws, they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here we significantly expand the scope of such networks by demonstrating a simple way to train them with any set of generalised coordinates, including easily observable ones.

Exploration of the impact of synthetic material landscapes featuring tunable geometrical properties on physical processes is a research direction that is currently of great interest because of the outstanding phenomena that are continually being uncovered. Twistronics and the properties of wave excitations in moir\'e lattices are salient examples. Moir\'e patterns bridge the gap between aperiodic structures and perfect crystals, thus opening the door to the exploration of effects accompanying the transition from commensurate to incommensurate phases. Moir\'e patterns have revealed profound effects in graphene-based systems1,2,3,4,5, they are used to manipulate ultracold atoms6,7 and to create gauge potentials8, and are observed in colloidal clusters9. Recently, it was shown that photonic moir\'e lattices enable observation of the two-dimensional localization-to-delocalization transition of light in purely linear systems10,11. Here, we employ moir\'e lattices optically induced in photorefractive nonlinear media12,13,14 to elucidate the formation of optical solitons under different geometrical conditions controlled by the twisting angle between the constitutive sublattices. We observe the formation of solitons in lattices that smoothly transition from fully periodic geometries to aperiodic ones, with threshold properties that are a pristine direct manifestation of flat-band physics11.

Despite the importance of short-term memory in cognitive function, how the input information is encoded and sustained in neural activity dynamics remains elusive. Here, by training recurrent neural networks to short-term memory tasks and analyzing the dynamics, the characteristic of the short-term memory mechanism was obtained in which the input information was encoded in the amplitude of transient oscillation, rather than the stationary neural activities. This transient orbit was attracted to a slow manifold, which allowed for the discarding of irrelevant information. Strong contraction to the manifold results in the noise robustness of the transient orbit, accordingly to the memory. The generality of the result and its relevance to neural information processing were discussed.

Interactions of inhibitory neurons produce gamma oscillations (30--80 Hz) in the local field potential, which is known to be involved in functions such as cognition and attention. In this study, the modified theta model is considered to investigate the theoretical relationship between the microscopic structure of inhibitory neurons and their gamma oscillations under a wide class of distribution functions of tonic currents on individual neurons. The stability and bifurcation of gamma oscillations for the Vlasov equation of the model is investigated by the generalized spectral theory. It is shown that as a connection probability of neurons increases, a pair of generalized eigenvalues crosses the imaginary axis twice, which implies that a stable gamma oscillation exists only when the connection probability has a value within a suitable range. On the other hand, when the distribution of tonic currents on individual neurons is the Lorentzian distribution, the Vlasov equation is reduced to a finite dimensional dynamical system. The bifurcation analyses of the reduced equation exhibit equivalent results with the generalized spectral theory. It is also demonstrated that the numerical computations of neuronal population follow the analyses of the generalized spectral theory as well as the bifurcation analysis of the reduced equation.

A nonlinear stimulated Raman adiabatic passage (STIRAP) is a fascinating physical process that dynamically explores chaotic and non-chaotic phases. In a recent paper Phys. Rev. Res. 2, 042004 (R) (2020), such a phenomenon is realized in a cavity-QED platform. There, the emergence of chaos and its impact on STIRAP efficiency are mainly demonstrated in the semiclassical limit. In the present paper I treat the problem in a fully quantum many-body framework. With the aim of extracting quantum signatures of a classically chaotic system, it is shown that an out-of-time-ordered correlator (OTOC) measure precisely captures chaotic/non-chaotic features of the system. The prediction by OTOC is in precise matching with classical chaos quantified by Lyapunov exponent (LE). Furthermore, it is shown that the quantum route corresponding to the semiclassical followed state encounters a dip in single-particle purity within the chaotic phase, depicting a consequence of chaos. A dynamics through the chaotic phase is associated with spreading of many-body quantum state and an irreversible increase in the number of participating adiabatic eigenstates.

We analyze the memory capacity of a delay based reservoir computer with a Hopf normal form as nonlinearity and numerically compute the linear as well as the higher order recall capabilities. A possible physical realisation could be a laser with external cavity, for which the information is fed via electrical injection. A task independent quantification of the computational capability of the reservoir system is done via a complete orthonormal set of basis functions. Our results suggest that even for constant readout dimension the total memory capacity is dependent on the ratio between the information input period, also called the clock cycle, and the time delay in the system. Optimal performance is found for a time delay about 1.6 times the clock cycle

This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.