arXiv subjects

Yueheng Lan

Publications and source records attributed to Yueheng Lan.

At least 19 recordsLinked to original sources

arXiv2026

The Symbolic Partition of Chaotic Flows Based on Ordinal Patterns

As a crucial tool in the field of chaotic systems study, Symbolic dynamics prompts extensive research into various methods for symbolic partitioning. The limitations of the majority of these methods are usually heuristic and empirical for partitioning the multivariate chaotic state space. Fortunately, we successfully take KA method and obtain primary coarse symbolic boundary and refine the symbolic boundary via GKA method on chaotic map in our previous studies. However, this method fails when applied to continuous chaotic flows. The trajectories of complex continuous chaotic flows are complex, and while their mechanisms are quite different from those of chaotic maps, they are by no means completely distinct -- after all, both are governed by the same fundamental laws of chaos. In response to the aforementioned challenges, a modified approach should be developed to overcome the failure of the existing method for continuous chaotic flows. In this study, we extend the Koopman-analysis-based symbolic partitioning approach to continuous chaotic flows. For general chaotic flows, Koopman analysis is employed to identify suitable Poincare sections. In this work, we construct different candidate Poincare sections based on ordinal patterns. By combining this ordinal-pattern-based Poincare section construction method with the Koopman-analysis-based return-map approach, we achieve effective symbolic partitioning of continuous chaotic flows. Noise perturbation tests are also conducted, demonstrating the robustness of the proposed method. This ordinal-pattern-based analysis is applied to Rossler system, Lorenz system, Lu system and Chen system. This study, with the aid of ordinal patterns, successfully introduces an effective symbolic partition into continuous systems, achieving a faithful transfer of the method from maps to continuous flows.

nlin.CD↗

arXiv2026

Data-driven characterization of spatiotemporal chaos using ensemble reservoir computing

Spatiotemporal chaotic systems are difficult to characterize in a model-free manner because of their high dimensionality, strong nonlinearity, and sensitivity to initial conditions. Coupled map lattices, as a representative class of extended nonlinear systems, exhibit diverse regimes such as frozen random pattern, defect chaotic diffusion, and fully developed turbulence. In this work, we propose an ensemble version of multiplexing local reservoir computing for the data-driven characterization of spatiotemporal chaos. By constructing multiple base learners with randomized hyperparameters and combining their outputs, the method improves prediction robustness and quantifies predictive uncertainty through ensemble spread. More importantly, we show that this uncertainty contains direct dynamical information. It identifies frozen positions in frozen random pattern, supports the estimation of defect diffusion coefficients in defect chaotic diffusion, and provides an effective indicator of chaotic intensity in fully developed turbulence. Analyses of the spatial power spectrum and Lyapunov exponent spectrum further support the consistency between the uncertainty field and the intrinsic dynamical properties of the system. These results show that ensemble reservoir computing can serve not only as a prediction tool but also as a data-driven framework for the dynamical characterization of high-dimensional nonlinear systems.

nlin.CD↗

arXiv2025

Renormalization group based implicit function approach to connecting orbits

Connecting orbits are important invariant structures in the state space of nonlinear systems and various techniques are designed for their computation. However, a uniform analytic approximation of the whole orbit seems rare. Here, based on renormalization group, an implicit function scheme is designed to effectively represent connections of disparate types, where coefficients of the defining function satisfy a set of linear algebraic equations, which greatly simplifies their computation. Unknown system parameters are conveniently determined by minimizing an error function. Symmetry may be profitably utilized to reduce the computation load. Homoclinic or heteroclinic connections are found in five popular examples approximately or exactly, demonstrating the effectiveness of the new scheme.

math-ph↗

arXiv2025

The symbolic partition with generalized Koopman analysis

Symbolic dynamics serves as a crucial tool in the study of chaotic systems, prompting extensive research into various methods for symbolic partitioning. The limitations of these methods are heuristic and empirical for the partition the multivariate chaotic state space. Notably, the use of operator theory in partitioning the multivariable chaotic series into precise symbolic cells has been underexplored. In this paper, we propose a novel symbolic partition method, referred to as Koopman Analysis(KA) method, exploiting Koopman operator theory to address the symbolic partition, especially multivariate chaotic time series. We map the chaotic time series into the basis functions to obtain the approximate representation of the Koopman operator.Then we transpose the Koopman approximate matrix and subsequently perform spectral decomposition to obtain the Koopman left eigenfunctions. We apply KA method to one-dimensional unimodal chaotic map to obtain Koopman left eigenfunctions. Then we find some particular eigenfunctions whose eigenvalues are zero, some of which can be used to identify the symbolic boundary of region composed of chaotic series in that the oscillation coincides with the subregion where the theoretical symbolic boundary is located. We refer to the function as the Valid Left Eigenfunction with Zero(VLEZ). Based on the number of oscillations, we further classify VLEZ into two categories. Then, we modify the KA method applicable to chaotic localized subregion and further propose the Generalized Koopman Analysis (GKA) method. The KA method can be also applied to the multimodal maps, multivariate chaotic maps and hyperchaotic maps and their noisy version. The current work can be well further expand to higher dimensional and more complex time series due to its interpretability and availability.

nlin.CD↗

arXiv2024

Optimal Dynamical Gauge in the Quantum Rabi Model

In this paper, we investigate the gauge dependence of various physical observables in the quantum Rabi model (QRM) under different potential fields, arising from the Hilbert-space truncation of the atomic degree of freedom. We discover that in both the square-well potential and oscillator potential,the optimal gauges for the ground-state energy of the QRM vary with respect to the cavity frequency, with the dipole gauge being optimal in the low-frequency limit and the Coulomb gauge in the high-frequency limit of the cavity frequency. Additionally, for higher energy levels, the optimal gauge asymptotically approaches the dipole gauge. However, for the dynamical quantity out-time-order correlator (OTOC), we find the necessity to introduce an optimal dynamical gauge. We determine the optimal dynamical gauge by minimizing the mean error between the two-level OTOC and the full Hamiltonian one. We expect that this study will contribute to a more profound understanding of the subtle relation between gauge choice and the dynamics of QED systems.

quant-ph↗

arXiv2023

Perturbing Chaos with Cycle Expansions

Due to existence of periodic windows, chaotic systems undergo numerous bifurcations as system parameters vary, rendering it hard to employ an analytic continuation, which constitutes a major obstacle for its effective analysis or computation. In this manuscript, however, based on cycle expansions we found that spectral functions and thus dynamical averages are analytic, if symbolic dynamics is preserved so that a perturbative approach is indeed possible. Even if it changes, a subset of unstable periodic orbits (UPOs) can be selected to preserve the analyticity of the spectral functions. Therefore, with the help of cycle expansions, perturbation theory can be extended to chaotic regime, which opens a new avenue for the analysis and computation in chaotic systems.

nlin.CD↗

arXiv2023

Symbolic partition in chaotic maps

In this work, we only use data on the unstable manifold to locate the partition boundaries by checking folding points at different levels, which practically coincide with homoclinic tangencies (HTs). The method is then applied to the classic two-dimensional Henon map and a well-known three-dimensional map. Comparison with previous results is made in the Henon case and Lyapunov exponents are computed through the metric entropy based on the partition, to show the validity of the current scheme.

nlin.CD↗

arXiv2023

Reconstructing dynamics of complex systems from noisy time series with hidden variables

Reconstructing the equation of motion and thus the network topology of a system from time series is a very important problem. Although many powerful methods have been developed, it remains a great challenge to deal with systems in high dimensions with partial knowledge of the states. In this paper, we propose a new framework based on a well-designed cost functional, the minimization of which transforms the determination of both the unknown parameters and the unknown state evolution into parameter learning. This method can be conveniently used to reconstruct structures and dynamics of complex networks, even in the presence of noisy disturbances or for intricate parameter dependence. As a demonstration, we successfully apply it to the reconstruction of different dynamics on complex networks such as coupled Lorenz oscillators, neuronal networks, phase oscillators and gene regulation, from only a partial measurement of the node behavior. The simplicity and efficiency of the new framework makes it a powerful alternative to recover system dynamics even in high dimensions, which expects diverse applications in real-world reconstruction.

nlin.AO↗

arXiv2023

Inferring synchronizability of networked heterogenous oscillators with machine learning

In the study of network synchronization, an outstanding question of both theoretical and practical significance is how to allocate a given set of heterogenous oscillators on a complex network in order for improving the synchronization performance. Whereas methods have been proposed to address this question in literature, the methods are based on accurate models describing the system dynamics, which, however, are normally unavailable in realistic situations. Here we show that this question can be addressed by the model-free technique of feed-forward neural network (FNN) in machine learning. Specifically, we measure the synchronization performance of a number of allocation schemes and use the measured data to train a machine. It is found that the trained machine is able to not only infer the synchronization performance of any new allocation scheme, but also find from a huge amount of candidates the optimal allocation scheme for synchronization.

nlin.AO↗

arXiv2022

Wielding Intermittency with Cycle Expansions

As periodic orbit theory works badly on computing the observable averages of dynamical systems with intermittency, we propose a scheme to cooperate with cycle expansion and perturbation theory so that we can deal with intermittent systems and compute the averages more precisely. Periodic orbit theory assumes that the shortest unstable periodic orbits build the framework of the system and provides cycles expansion to compute dynamical quantities based on them, while the perturbation theory can locally analyze the structure of dynamical systems. The dynamical averages may be obtained more precisely by combining the two techniques together. Based on the integrability near the marginal orbits and the hyperbolicity in the part away from the singularities in intermittent systems, the chief idea of this paper is to revise intermittent maps and maintain the natural measure produced by the original maps. We get the natural measure near the singularity through the Taylor expansions and periodic orbit theory captures the natural measure in the other parts of the phase space. We try this method on 1-dimensional intermittent maps with single singularity, and more precise results are achieved.

nlin.CD↗

arXiv2022

A reduced variational approach for searching cycles in high-dimensional systems

Searching recurrent patterns in complex systems with high-dimensional phase spaces is an important task in diverse fields. In the current work, an improved scheme is proposed to accelerate the recently designed variational approach for finding periodic orbits in systems with chaotic dynamics based on the existence of inertial manifold widely observed in various spatially extended systems, especially those with high dimensions. On the premise of keeping exponential convergence of the variational method, an effective loop evolution equation is derived to greatly reduce the storage and computing time. With repeated modification of local coordinates and evolution of the guess loop being carried out alternately, the rapid convergence and the stability of the reduction scheme are effectively achieved. The dimension of local coordinate subspaces is generally larger than the number of nonnegative Lyapunov exponents to ensure the exponential convergence. The proposed scheme is successfully demonstrated on several well-known examples and expected to supply a powerful tool in the exploration of high-dimensional nonlinear systems.

nlin.CD↗

arXiv2021

Criticality in Reservoir Computer of Coupled Phase Oscillators

Accumulating evidences show that the cerebral cortex is operating near a critical state featured by power-law size distribution of neural avalanche activities, yet evidence of this critical state in artificial neural networks mimicking the cerebral cortex is lacking. Here we design an artificial neural network of coupled phase oscillators and, by the technique of reservoir computing in machine learning, train it for predicting chaos. It is found that when the machine is properly trained, oscillators in the reservoir are synchronized into clusters whose sizes follow a power-law distribution. This feature, however, is absent when the machine is poorly trained. Additionally, it is found that despite the synchronization degree of the original network, once properly trained, the reservoir network is always developed to the same critical state, exemplifying the "attractor" nature of this state in machine learning. The generality of the results is verified in different reservoir models and by different target systems, and it is found that the scaling exponent of the distribution is independent on the reservoir details and the bifurcation parameter of the target system, but is modified when the dynamics of the target system is changed to a different type. The findings shed lights on the nature of machine learning, and are helpful to the design of high-performance machine in physical systems.

q-bio.NC↗

arXiv2020

Accelerating the computation of quantum brachistochrone

Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating redundant complexity involved in previous formulations. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically equivalent classes of paths and may be utilized to remove neutral directions in the Jacobian of the evolution. In the 'ground state' solution among the set of optimal paths, the time-reversal symmetry of the system shows up, which is expected to be universal for the symmetry-related initial and final state.

quant-ph↗

arXiv2020

Koopman analysis in oscillator synchronization

Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems while a global statistical approach is overly cursory. Koopman operator technique seems to well balance the two approaches. In this paper, we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighbouring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.

nlin.CD↗

arXiv2020

Koopman Operator and Phase Space Partition of Chaotic Maps

Koopman operator describes evolution of observables in the phase space, which could be used to extract characteristic dynamical features of a nonlinear system. Here, we show that it is possible to carry out interesting symbolic partitions based on properly constructed eigenfunctions of the operator for chaotic maps. The partition boundaries are the extrema of these eigenfunctions, the accuracy of which is improved by including more basis functions in the numerical computation. The validity of this scheme is demonstrated in well-known 1-d and 2-d maps. It seems no obstacle to extend the computation to nonlinear systems of high dimensions, which provides a possible way of dissecting complex dynamics.

nlin.CD↗

arXiv2019

A generating function approach to the growth rate of random matrix products

Random matrix products arise in many science and engineering problems. An efficient evaluation of its growth rate is of great interest to researchers in diverse fields. In the current paper, we reformulate this problem with a generating function approach, based on which two analytic methods are proposed to compute the growth rate. The new formalism is demonstrated in a series of examples including an Ising model subject to on-site random magnetic fields, which seems very efficient and easy to implement. Through an extensive comparison with numerical computation, we see that the analytic results are valid in a regime of considerable size.

cond-mat.stat-mech↗

arXiv2018

Model-Free Information Extraction in Enriched Nonlinear Phase-Space

Detecting anomalies and discovering driving signals is an essential component of scientific research and industrial practice. Often the underlying mechanism is highly complex, involving hidden evolving nonlinear dynamics and noise contamination. When representative physical models and large labeled data sets are unavailable, as is the case with most real-world applications, model-dependent Bayesian approaches would yield misleading results, and most supervised learning machines would also fail to reliably resolve the intricately evolving systems. Here, we propose an unsupervised machine-learning approach that operates in a well-constructed function space, whereby the evolving nonlinear dynamics are captured through a linear functional representation determined by the Koopman operator. This breakthrough leverages on the time-feature embedding and the ensuing reconstruction of a phase-space representation of the dynamics, thereby permitting the reliable identification of critical global signatures from the whole trajectory. This dramatically improves over commonly used static local features, which are vulnerable to unknown transitions or noise. Thanks to its data-driven nature, our method excludes any prior models and training corpus. We benchmark the astonishing accuracy of our method on three diverse and challenging problems in: biology, medicine, and engineering. In all cases, it outperforms existing state-of-the-art methods. As a new unsupervised information processing paradigm, it is suitable for ubiquitous nonlinear dynamical systems or end-users with little expertise, which permits an unbiased excavation of underlying working principles or intrinsic correlations submerged in unlabeled data flows.

cs.LG↗

arXiv2017

Explosive synchronization transition in a ring of coupled oscillators

Explosive synchronization(ES), as one kind of abrupt dynamical transition in nonlinearly coupled systems, is currently a subject of great interests. Given a special frequency distribution, a mixed ES is observed in a ring of coupled phase oscillators which transit from partial synchronization to ES with the increment of coupling strength. The coupling weight is found to control the size of the hysteresis region where asynchronous and synchronized states coexist. Theoretical analysis reveals that the transition varies from the mixed ES, to the ES and then to a continuous one with increasing coupling weight. Our results are helpful to extend the understanding of the ES in homogenous networks.

nlin.CD↗