Complex Dynamical Phenomena Research Articles

Detecting dynamical causality via intervened reservoir computing

An abundance of complex dynamical phenomena exists in nature and human society, requiring sophisticated analytical tools to understand and explain. Causal analysis through observational time series data is essential in comprehending complex systems when controlled experiments are not feasible or ethical. Although data-based causal discovery methods have been widely used, there is still a lack of direct ways more aligned with the intuitive definition of causality, i.e., whether interventions on one element lead to changes in the subsequent development of others. To solve this problem, we propose the method of intervened reservoir computing (IRC) based on constructing a neural network replica of the original system and applying interventions to it. This approach enables controlled trials, thus observing the intervened evolution, in the digital twins of the underlying systems. Simulated and real-world data are used to test our approach and demonstrate its accuracy in inferring causal networks. Given the importance of causality in understanding complex dynamics, we anticipate that IRC could serve as a powerful tool for various disciplines to decipher the intrinsic mechanisms of natural systems from observational data.

Complex localization mechanisms in networks of coupled oscillators: Two case studies.

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

A Simple Model of Tumor-Immune Interaction: The Effect of Antigen Delay

We propose a simple model of tumor-immune interactions, which involves effector cells and tumor cells. In the model, the stimulation delay of tumor antigen in the immune system is incorporated. We investigate the dynamical behavior of the model via theoretic analysis and numerical simulations. The saddle-node bifurcation can occur in both cases with and without delay. In contrast to the case without delay, stimulation delay may result in some complex dynamical behaviors and biological phenomena. In the presence of delay, conditions on absolute/conditional stability of equilibria and the existence of Hopf bifurcations are obtained. We further discuss the effect of the tumor on the switch between absolute stability and conditional stability. Numerical simulations also show the existence of homoclinic bifurcation and the dependence of the asymptotic state of the tumor progression on initial conditions for different delay values. Effects of delay on the dynamics of the model and on the region of tumor extinction are illustrated by simulations with different sets of parameter values. Finally, the corresponding biological implications are demonstrated.

Bursting oscillation of simply supported fluid-conveying pipes

ABSTRACT The main purpose of this article is to explore an originality multiple-frequency bursting of simply-supported fluid-conveying pipes with initial micro-bending shape under different low-frequency excitation conditions. Based on the Galerkin's method and the fast-slow dynamics analysis method, rich vibration phenomena are observed with different excitation conditions, and the multi-valued characteristics are verified by the domain of attraction. The results demonstrate that low-frequency excitation can cause complex dynamical phenomena of the fluid-conveying pipes.

Stability analysis of a class of variable fractional-order uncertain neutral-type systems with time-varying delay

Variable fractional-order (VFO) differential equations are a beneficial tool for describing the nonlinear behavior of complex dynamical phenomena. In comparison with the constant FO derivatives, it describes the memory properties of such systems that can vary in the time domain and spatial location. This article investigates the stability and stabilization of VFO neutral systems in the presence of time-varying structured uncertainties and time-varying delay. FO Lyapunov theorem is adopted to achieve order-dependent and delay-dependent criteria for both nominal and uncertain VFO neutral delay systems. The obtained conditions are given in respect of linear matrix inequality by designing a delayed state feedback controller. Simulations verify the main results.

Investigations of complex systems’ dynamics, based on reduced amount of information: introduction to the method

Controlling dynamics of complex systems is one of the most important issues in science and engineering. Thus, there is continuous need to study and develop numerical algorithms of control methods. In this paper, we would like to present our introductory study of a new simple method of investigations of such systems based on vector field properties and reduced amount of applied information. Firstly, we present the basis of our approach for extraction of nonlinear indicators of two-dimensional systems. We show that basing on simplified analyses and exploiting half of commonly applied information, we can precisely estimate widely applied indicators. We prove that our method is simpler, more efficient and more accurate than commonly applied algorithms. After the introductory analysis, we extend our studies and apply the presented method in investigations of complex systems, based on the analysis that we discussed in the first part of the article and carried out in two-dimensional subspaces. We present simplicity and effectiveness of our approach and demonstrate how it simplifies investigations of complex dynamical phenomena. We verify our method studying the example of synchronization and chimera phenomena in the chosen set of coupled oscillators.

Topological EEG Nonlinear Dynamics Analysis for Emotion Recognition

Emotional recognition through exploring the electroencephalography (EEG) characteristics has been widely performed in recent studies. Nonlinear analysis and feature extraction methods for understanding the complex dynamical phenomena are associated with the EEG patterns of different emotions. The phase space reconstruction is a typical nonlinear technique to reveal the dynamics of the brain neural system. Recently, the topological data analysis (TDA) scheme has been used to explore the properties of space, which provides a powerful tool to think over the phase space. In this work, we proposed a topological EEG nonlinear dynamics analysis approach using the phase space reconstruction (PSR) technique to convert EEG time series into phase space, and the persistent homology tool explores the topological properties of the phase space. We perform the topological analysis of EEG signals in different rhythm bands to build emotion feature vectors, which shows high distinguishing ability. We evaluate the approach with two well-known benchmark datasets, the DEAP and DREAMER datasets. The recognition results achieved accuracies of 99.37% and 99.35% in arousal and valence classification tasks with DEAP, and 99.96%, 99.93%, and 99.95% in arousal, valence, and dominance classifications tasks with DREAMER, respectively. The performances are supposed to be outperformed current state-of-art approaches in DREAMER (improved by 1% to 10% depends on temporal length), while comparable to other related works evaluated in DEAP. The proposed work is the first investigation in the emotion recognition oriented EEG topological feature analysis, which brought a novel insight into the brain neural system nonlinear dynamics analysis and feature extraction.

TimeSOAP: Tracking high-dimensional fluctuations in complex molecular systems via time variations of SOAP spectra.

Many molecular systems and physical phenomena are controlled by local fluctuations and microscopic dynamical rearrangements of the constitutive interacting units that are often difficult to detect. This is the case, for example, of phase transitions, phase equilibria, nucleation events, and defect propagation, to mention a few. A detailed comprehension of local atomic environments and of their dynamic rearrangements is essential to understand such phenomena and also to draw structure-property relationships useful to unveil how to control complex molecular systems. Considerable progress in the development of advanced structural descriptors [e.g., Smooth Overlap of Atomic Position (SOAP), etc.] has certainly enhanced the representation of atomic-scale simulations data. However, despite such efforts, local dynamic environment rearrangements still remain difficult to elucidate. Here, exploiting the structurally rich description of atomic environments of SOAP and building on the concept of time-dependent local variations, we developed a SOAP-based descriptor, TimeSOAP (τSOAP), which essentially tracks time variations in local SOAP environments surrounding each molecule (i.e., each SOAP center) along ensemble trajectories. We demonstrate how analysis of the time-series τSOAP data and of their time derivatives allows us to detect dynamic domains and track instantaneous changes of local atomic arrangements (i.e., local fluctuations) in a variety of molecular systems. The approach is simple and general, and we expect that it will help shed light on a variety of complex dynamical phenomena.

Bifurcations driven by generalist and specialist predation: mathematical interpretation of Fennoscandia phenomenon.

In this paper, we revisit a predator-prey model with specialist and generalist predators proposed by Hanski et al. (J Anim Ecol 60:353-367, 1991) , where the density of generalist predators is assumed to be a constant. It is shown that the model admits a nilpotent cusp of codimension 4 or a nilpotent focus of codimension 3 for different parameter values. As the parameters vary, the model can undergo cusp type (or focus type) degenerate Bogdanov-Takens bifurcations of codimension 4 (or 3). Our results indicate that generalist predation can induce more complex dynamical behaviors and bifurcation phenomena, such as three small-amplitude limit cycles enclosing one equilibrium, one or two large-amplitude limit cycles enclosing one or three equilibria, three limit cycles appearing in a Hopf bifurcation of codimension 3 and dying in a homoclinic bifurcation of codimension 3. In addition, we show that generalist predation stabilizes the limit cycle driven by specialist predators to a stable equilibrium, which clearly explains the famous Fennoscandia phenomenon.

Relative Controllability for a Class of Linear Impulsive Systems

Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.

Bifurcation, geometric constraint, chaos, and its control in a railway wheelset system

In this paper, an improved railway wheelset system is presented. The dynamical behaviors of the system are investigated, including dissipativity and invariance of the system, stability of zero‐equilibrium point, and the bifurcation characteristics of railway wheelset system at zero equilibrium point. Furthermore, the chaotic behaviors of the motion of railway wheelset and the dynamical behaviors of the railway wheelset system under geometric constraint are studied. It shows that the railway wheelset system has complex dynamical phenomena owing to nonlinear factor and the railway wheelset system may have different complex dynamical behaviors with different nonlinear parameters. In addition, the motion of railway wheelset also has chaotic behaviors under geometric constraint. Finally, the chaos control of the railway wheelset system is achieved by using linear feedback control method. The numerical simulations are carried out in order to analyze the complex phenomena of the railway wheelset system.

Synchronization of the internal dynamics of optical soliton molecules

Compact bound states of light pulses in ultrafast lasers are known as optical soliton molecules. They constitute nonlinear superstructures of choice to investigate complex dynamical phenomena that manifest similarly in a wide range of nonlinear systems. Akin to matter molecules, optical soliton molecules can feature vibrational motions between their internal constituents. However, these vibrations are intrinsically nonlinear, with oscillation frequencies sensitive to system parameters. Therefore, vibrating soliton molecules present an opportunity for control. We here investigate the precise control of their oscillation frequencies through the universal mechanism of synchronization between master and slave oscillators. Self-oscillating soliton molecules are prepared within a passively mode-locked fiber laser. We experimentally demonstrate the synchronization of the internal vibrations of soliton molecules through the optical injection of a master oscillator signal. Direct observation of the synchronization process is enabled by balanced optical cross-correlation detection, a technique allowing real-time detection of intramolecular separation with femtosecond temporal resolution. We show efficient sub-harmonic, fundamental, and super-harmonic synchronization, forming a pattern of Arnold tongues with respect to the injection strength. Numerical simulations support experimental observations. By retrieving these universal synchronization features, the role of the soliton molecule as a nonlinear dynamical system of chief importance is further highlighted.

Hybrid physics and neural network model for lateral vehicle dynamic state prediction

The physical modeling-based approaches tend to be over-simplistic and cannot forecast the complex dynamical phenomena, thus leading to non-negligible errors. It is not easy to measure some parameters precisely, and they are usually approximated roughly. However, this approximation reduces the modeling accuracy of the physical model, which is a common problem in complex systems research. It is well-known that neural networks are capable of encoding dynamic information. The vehicle can be accurately modeled by collecting data during its motion. However, purely data-driven approaches have low interpretability and cannot be used in commercial applications. In this work, we present a new hybrid modeling architecture. Based on the physical model, the deep learning method is introduced to expand the incomplete dynamics described by differential equations. Compared with the physical modeling-based and purely data-driven approaches, the proposed technique has lower modeling error and higher interpretability. We evaluate the performance of the hybrid model based on the collected data. The test results show that the proposed architecture successfully captures the vehicle dynamics and reduces the error caused by multi-step prediction compared to the data-driven models. The results also show that the proposed method has value for significant research and practical application.

Complex dynamics in the improved Koren–Feingold cloud–rain system

In this paper, we study the improved aerosol–cloud–precipitation system. Since in our previous article, we guessed that the delayed term y(t−τ) in the denominator tends to cause singularity. So we remove the denominator by time transformation, the original cloud–rain model is transformed into a simpler polynomial system under certain conditions, namely, the improved Koren–Feingold cloud–rain system. Fortunately, the new model meets our expectations. The main work of the present paper is to study the mechanism of cloud and rain interaction by double Hopf bifurcation analysis via the methods of multiple scales. We find some complicated dynamical behaviors of the system via analytical method, such as stable quasi-periodic solutions and chaos. The numerical simulations agree well with those of theoretical analysis, which shows that we are improving the model in the right direction, meanwhile perhaps it suggests that the MMS employed in this paper provides us an effective and accurate method of analyzing double Hopf bifurcation. The above complex dynamical phenomena are of great significance for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And it is also significant to ecosystem and climate control.

Self‐Excited and Hidden Chaotic Attractors in Matouk’s Hyperchaotic Systems

Self-excited and hidden chaotic attractors are interesting complex dynamical phenomena. Here, Matouk’s hyperchaotic systems are shown to have self-excited and hidden chaotic attractors, respectively. Two case studies of hidden chaotic attractors are provided which are examined with orders 3.08 and 3.992, respectively. Moreover, self‐excited chaotic attractors are found in the fractional‐order system and its integer‐order counterpart. The existence of one‐eyed face self‐excited chaotic attractors is also reported in this work. Our results show that the fractional derivative affects the appearances of hidden chaotic attractors in this system.

Augmenting physical models with deep networks for complex dynamics forecasting* *This article is an updated version of: Yin Y, Le Guen V, Dona J, de Bezenac E, Ayed I, Thome N and Gallinari P 2021 Augmenting physical models with deep networks for complex dynamics forecasting Int. Conf. Learning Representations.

Forecasting complex dynamical phenomena in settings where only partial knowledge of their dynamics is available is a prevalent problem across various scientific fields. While purely data-driven approaches are arguably insufficient in this context, standard physical modeling-based approaches tend to be over-simplistic, inducing non-negligible errors. In this work, we introduce the APHYNITY framework, a principled approach for augmenting incomplete physical dynamics described by differential equations with deep data-driven models. It consists of decomposing the dynamics into two components: a physical component accounting for the dynamics for which we have some prior knowledge, and a data-driven component accounting for errors of the physical model. The learning problem is carefully formulated such that the physical model explains as much of the data as possible, while the data-driven component only describes information that cannot be captured by the physical model; no more, no less. This not only provides the existence and uniqueness for this decomposition, but also ensures interpretability and benefit generalization. Experiments made on three important use cases, each representative of a different family of phenomena, i.e. reaction–diffusion equations, wave equations and the non-linear damped pendulum, show that APHYNITY can efficiently leverage approximate physical models to accurately forecast the evolution of the system and correctly identify relevant physical parameters. The code is available at https://github.com/yuan-yin/APHYNITY.

Synchronization transitions in a hyperchaotic SQUID trimer.

The phenomena of intermittent and complete synchronization between two out of three identical, magnetically coupled Superconducting QUantum Interference Devices (SQUIDs) are investigated numerically. SQUIDs are highly nonlinear superconducting oscillators/devices that exhibit strong resonant and tunable response to applied magnetic field(s). Single SQUIDs and SQUID arrays are technologically important solid-state devices, and they also serve as a testbed for exploring numerous complex dynamical phenomena. In SQUID oligomers, the dynamic complexity increases considerably with the number of SQUIDs. The SQUID trimer, considered here in a linear geometrical configuration using a realistic model with experimentally accessible control parameters, exhibits chaotic and hyperchaotic behavior in wide parameter regions. Complete chaos synchronization as well as intermittent chaos synchronization between two SQUIDs of the trimer is identified and characterized using the complete Lyapunov spectrum of the system and appropriate measures. The passage from complete to intermittent synchronization seems to be related to chaos-hyperchaos transitions as has been conjectured in the early days of chaos synchronization.

Chaos and bifurcations in a discretized fractional model of quasi-periodic plasma perturbations

Abstract The nonlinear dynamics of a discretized form of quasi-periodic plasma perturbations model (Q-PPP) with nonlocal fractional differential operator possessing singular kernel are investigated. For example, the conditions for the stability and occurrence of Neimark–Sacker (NS) and flip bifurcations in the proposed discretized equations are provided. Moreover, analysis of nonlinearities such as the existence of chaos in this map is proved numerically via bifurcation diagrams, Lyapunov exponents and analytically via Marotto’s Theorem. Also, some simulation results are utilized to confirm the theoretical results and to show that the obtained map exhibits double routes to chaos: one is via flip bifurcation and the other is via NS bifurcation. Furthermore, more complex dynamical phenomena such as existence of closed invariant curves, homoclinic orbits, homoclinic connections, period 3 and period 4 attractors are shown. This kind of research is useful for physicists who work with tokamak models.

Periodic, Quasi-Periodic and Chaotic Oscillations in Two Heterogeneous AIMD/RED Network Congestion Models with State-Dependent Round-Trip Delays

AIMD and RED are two dominant algorithms for controlling Internet congestion. So this paper explores the periodic solutions and complex dynamical phenomena in the state-dependent round-trip delayed AIMD/RED network congestion model with heterogeneous flows, and its improved model. We first use the semi-analytical and semi-numerical method, known as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, to derive the analytical approximations of periodic oscillations of the system. The obtained results are compared with the numerical results by WinPP, and they show good consistency. At the same time, this suggests that the method used in this paper is correct and valid. Then for the sake of making the system more realistic, we improve the model by using the hyperbolic tangent function. We obtain the approximate solutions, and find some rich dynamical behaviors of this delayed heterogeneous system, including Period-1 to torus, Period-1 to Period-2 to Period-3 motions and two kinds of mechanisms of chaos, i.e. the windows of Period-2 and Period-3 orbits to chaos, where to the best knowledge of the authors, the former route has never been reported. The periodic oscillations may induce synchronization and further congestion, where chaotic oscillation usually means that the system is unstable and may even collapse. Hence, we need to avoid these abundant dynamics discovered in this paper because they are undesirable and harmful. The derived results can help researchers better understand the performance of the AIMD/RED system, and they can be a guide for choosing parameters in a suitable range in order to maintain the network stability and optimize system performance.

Chaotic Attractors Generated by a Memristor-Based Chaotic System and Julia Fractal

Abstract A chaotic system has many applications in random number generator, image encryption, and secure communication. It is essential to present abundant chaotic attractors in a single chaotic system. Thus a novel 3D chaotic system based on memristor is designed and analyzed. The system's key advantages are as follows: firstly, both the multi-scroll and extended chaotic attractors are produced by merely modifying system parameters. The complex dynamical phenomena, such as coexisting multiple multi-scroll chaotic attractors, transient phenomenon and state transformation, can be observed; secondly, the proposed system combines with the classical Julia fractal and Julia deformation forms, respectively, and fascinating and various circular chaotic attractors are generated, which take good advantage of the combination of a memristor-based chaotic system and Julia fractal. Meanwhile, by applying both dynamic analysis and numerical simulation, basic behaviors of the system, such as stability of equilibrium point, dissipation and symmetry, Poincare maps, instantaneous phase, 0-1 test and power spectrum are employed to illustrate the superiority of the system. Some rather impressive and promising applications of the proposed chaotic system to chaotic confidential communication and image encryption will be achieved later.