Nonlinear Dynamics Perspective Research Articles

Complex nonlinear dynamics of a multidirectional energy harvester with hybrid transduction

Abstract Mechanical energy harvesting has increasing scientific and technological interests due to novel energetic challenges. A critical issue in classical cantilever-based mechanical energy harvesting systems is the lack of multidirectional energy conversion capabilities and, due to that, deviations from the excitation source can drastically reduce their performance. This limitation has led to the development of energy harvesters with attached pendula, serving as a direction coupling mechanism. Nevertheless, the pendulum structure itself can act as an energy absorber, drastically reducing the harvester performance in certain scenarios. In order to overcome this issue, a hybrid transduction multidirectional pendulum-based energy harvester is proposed integrating a piezoelectric element, to capture energy from the principal direction, and an electromagnetic transducer, to harness rotational energy from the pendulum. This paper presents an in-depth analysis of the hybrid multidirectional pendulum-based energy harvester using a nonlinear dynamics perspective to evaluate the energy harvesting performance. A reduced-order model is proposed to represent the essential characteristics of such systems. A parametric analysis using a nonlinear dynamics perspective is carried out to map the system dynamics and performance. The emergence of complex and rich dynamics is observed, including chaos and hyperchaos. Results reveal the most and least effective combinations of structural parameters in terms of energy conversion. Additionally, the dynamical responses and patterns associated with high performance are identified. These responses are often characterized by a blend of irregular complex behaviors, coupled with a mix of oscillatory and rotational patterns of motion, resulting in wider bandwidth systems.

Minimal motifs for habituating systems

Habituation-a phenomenon in which a dynamical system exhibits a diminishing response to repeated stimulations that eventually recovers when the stimulus is withheld-is universally observed in living systems from animals to unicellular organisms. Despite its prevalence, generic mechanisms for this fundamental form of learning remain poorly defined. Drawing inspiration from prior work on systems that respond adaptively to step inputs, we study habituation from a nonlinear dynamics perspective. This approach enables us to formalize classical hallmarks of habituation that have been experimentally identified in diverse organisms and stimulus scenarios. We use this framework to investigate distinct dynamical circuits capable of habituation. In particular, we show that driven linear dynamics of a memory variable with static nonlinearities acting at the input and output can implement numerous hallmarks in a mathematically interpretable manner. This work establishes a foundation for understanding the dynamical substrates of this primitive learning behavior and offers a blueprint for the identification of habituating circuits in biological systems.

High‐Dissolution Mechanism of Ti–48Al–2Cr–2Nb Alloy in Electrochemical Machining from a Nonlinear Dynamic Perspective

High‐rate dissolution process of Ti–48Al–2Cr–2Nb alloy in electrochemical machining exhibits strong nonlinear dynamic characteristics, but the quantitative relationship between these nonlinear characteristics and surface topography, as well as the electro‐dissolution mechanism, is unknown. Therefore, the fractal and recursive features of dissolution behavior and quantitative characterization of the dissolved surface topography were investigated. Additionally, an electro‐dissolution mechanism model in electrochemical machining process was presented. Applying a higher potential of 11 V can enhance the stability of the electrochemical system, resulting in a smoother and uniform surface. The saturation correlation dimension, surface roughness, and integral area of lacunarity initially decrease before increasing with higher electrolytic pressure, while laminarity and determinism show an initial increase followed by a decrease. These five characteristic parameters reach their optimal values at an electrolyte pressure of 0.4 MPa, indicating that achieving a stable and deterministic electrochemical system is crucial for obtaining a uniform smooth surface. These nonlinear dynamic characteristic parameters can be considered as in‐situ monitoring parameters for surface quality in ECM. The anodic dissolution process comprises three stages: initial passive film dissolution and breakdown, rapid dissolving with surface coarsening, and stable dissolving with polishing.

On the heart rhythm analysis using a nonlinear dynamics perspective: analytical study and electronic simulation

Biological systems are highly complex, so understanding them requires extensive analysis. Cardiac rhythms are one such analysis. These rhythms are linked to a complex dynamic system defined on the basis of the electrical activity of cardiac cells. This electrical activity is essential to human physiology, defining numerous behaviours that include normal or pathological rhythms, generally measured by the electrocardiogram (ECG). This article presents a mathematical model to describe the electrical activity of the heart, using a nonlinear dynamics perspective. The stability analysis of this model in its autonomous state, uni-directionally coupled, shows a very rich dynamical behaviour characterized by periodical regions of stability and unstability. The model studied makes it possible to construct synthetic ECGs. These ECGs demonstrate a variety of responses, including normal and pathological rhythms: ventricular flutter, ventricular fibrillation, ventricular tachycardia and ventricular extrasystole. A quantitative analysis of the model is also carried out using bifurcation diagrams and the corresponding maximum Lyapunov exponents. In addition, variations in sinus rhythm are described by a time-dependent frequency (a dynamic variable varying in a disordered manner or following a given law), representing transient disturbances. This type of situation can represent transitions between different pathological behaviours or between normal and pathological physiologies. In this respect, the perspective of nonlinear dynamics is used to describe cardiac rhythms, which makes it possible to represent normal or pathological behaviours. An electronic simulation performed with the OrCAD-Pspice software for a real implementation of the cardiac system is carried out. The results obtained are in agreement with those obtained numerically.

Nonlinear dynamic analysis of wind pressure variability on standard tall building

Understanding the variability of wind pressure on tall building surfaces is of essential importance regarding the control of wind-induced response. In this study, advanced time series analysis techniques, e.g., phase space reconstruction and recurrence analysis, were applied to diagnose the wind pressure variability from a nonlinear dynamic perspective. It is shown that, the wind pressure acting on tall building surfaces exhibit distinct chaotic nature. Due to the complex flow patterns around the building, the underlying dynamics of wind pressure is subject to pronounced region-to-region variability. For windward surface, the wind pressure dynamics at the stagnation region is more deterministic than those of downstream. For the side surfaces, the distribution of recurrence quantification analysis (RQA) indicators follows a clear pattern, where the values are larger at the near-front edge, and decreases towards the rear. The distribution of RQA indicators on the leeward surface is opposite to that of windward surface, in which larger values occur more often at near-ground level. Further upstream, the values are found to decrease. The outcomes provide valuable insights to the wind-structural interaction, thus could help wind-resistant design.

Nonlinear dynamics of a compact and multistable mechanical energy harvester

The use of smart materials as transducers in mechanical energy harvesting systems has gained significant attention in recent years. Despite the numerous proposed solutions in the literature, challenges still exist in terms of their implementation within limited spaces while maintaining optimal performance. This paper addresses these challenges through the concepts of compactness and space-efficient design, as well as the incorporation of nonlinear characteristics and additional degrees-of-freedom. A multistable dual beam nonlinear structure featuring two magnetic interactions and two piezoelectric transducers is presented. A reduced order model with 2-degrees-of-freedom is established based on the harvester structure in order to capture the essential qualitative characteristics of the system. Stability analysis demonstrates that the combination of two nonlinear magnetic interactions furnish unprecedented multistable characteristics to this type of harvester. A framework using a nonlinear dynamics perspective is established to analyze multistable systems based on energy harvesting purposes. Different dynamical and stability characteristics are determined by the differences in the system stiffness ratio. Parametric analyses are carried out classifying regions of high performance in the external excitation parameter space. These regions are associated with rich and complex dynamics. Finally, a comprehensive comparison is conducted between the proposed harvester and the classical bistable harvester, revealing improvements in performance across nearly all relevant conditions. These findings highlight the enhanced capabilities of the proposed harvester design, solidifying its potential of application in diverse energy harvesting scenarios.

Heart Rhythm Analysis Using Nonlinear Oscillators with Duffing-Type Connections

Heartbeat rhythms are related to a complex dynamical system based on electrical activity of the cardiac cells usually measured by the electrocardiogram (ECG). This paper presents a mathematical model to describe the electrical activity of the heart that consists of three nonlinear oscillators coupled by delayed Duffing-type connections. Coupling alterations and external stimuli are responsible for different cardiac rhythms. The proposed model is employed to build synthetic ECGs representing a variety of responses including normal and pathological rhythms: ventricular flutter, torsade de pointes, atrial flutter, atrial fibrillation, ventricular fibrillation, polymorphic ventricular tachycardia and supraventricular extrasystole. Moreover, the sinoatrial rhythm variations are described by time-dependent frequency, representing transient disturbances. This kind of situation can represent transitions between different pathological behaviors or between normal and pathological physiologies. In this regard, a nonlinear dynamics perspective is employed to describe cardiac rhythms, being able to represent either normal or pathological behaviors.

A novel signal representation in SEI: Manifold

Specific emitter identification (SEI), as an important problem in situational awareness, identifies emitters via unique characteristics. However, current SEI methods mostly suffer from appropriately setting the trade-off between comprehensiveness and efficiency when extracting fingerprint features. To address the issue, this paper provides a novel SEI framework with a separate representation module. Within the novel framework, manifolds are proposed to be signal representations and multi-level manifold features are extracted as fingerprint features. We first build the SEI model from the nonlinear dynamic perspective, where the SEI process identifies the nonlinear systems via a measurement sequence. Then, we demonstrate that manifolds can represent emitters equivalently and prove the one-to-one correspondence between manifolds and emitter individuals. Hence, manifolds can highlight unique nonlinear dynamic characteristics and simultaneously describe comprehensive system working processes. The coordinate delayed technique and manifold learning methods are employed to reconstruct the phase space and manifold, respectively. For accomplishing the identification task, multi-level manifold features, comprising intrinsic dimension, topological features, conformal features, and Riemannian metric features, are extracted from the reconstructed manifolds and input to an ensemble learning scheme, named Adaboost. Extensive simulation and real-world experiments agree with our analytical conclusions and confirm the proposed method’s efficiency. The results also demonstrate that the proposed method achieves a high recognition accuracy, outstanding adaptability, and strong robustness.

A Daisyworld Ecological Parable Including the Revenge of Gaia and Greenhouse Effect

The Daisyworld model illustrates the concept of biological homeostasis in the global environment by establishing a connection between the biota and environment, resulting in a single intertwined system known as Gaia. In essence, the Daisyworld model represents life by daisy populations whereas temperature represents the environment, establishing a population dynamics model to represent life–environment ecological interactions. The recent occurrence of extreme weather events due to climate change and the critical crises brought on by the COVID-19 pandemic are strengthening the arguments for the revenge of Gaia, a term used to describe the protective response of the global biota-environment system. This paper presents a novel Daisyworld parable to describe ecological life–environment interactions including the revenge of Gaia and the greenhouse effect. The revenge of Gaia refers to a change in the interplay between life and environment, characterized by the Gaia state that establishes the life-environment state of balance and harmony. This results in reaction effects that impact the planet’s fertile regions. On the other hand, the greenhouse effect is incorporated through the description of the interactions of greenhouse gases with the planet, altering its albedo. Numerical simulations are performed using a nonlinear dynamics perspective, showing different ecological scenarios. An investigation of the system reversibility is carried out together with critical life–environment interactions. This parable provides a qualitative description that can be useful to evaluate ecological scenarios.

Multifaceted characteristics of aridity changes and causal mechanisms in Chinese drylands

The water cycle is accelerating in the context of global warming. However, how the multifaceted characteristics of aridity, particularly atmospheric, hydrological, and ecological drying, change and interact with each other are largely unknown. A gap we bridged was discovering the causal relationships underlying the atmosphere-hydrosphere-biosphere nexus from the nonlinear dynamic system perspective based on convergent cross mapping (CCM). Dryland area in China has expanded since 1982, while the vegetation greenness indicated by leaf area index has been increasing during the same period. The results showed that the causality among atmospheric, hydrological, and ecological drying in different subtypes of drylands was different. In arid and semi-arid regions: vegetation changes were mainly driven by soil moisture (SM) and vapor pressure deficit (VPD), and VPD regulated SM. In hyper-arid and dry sub-humid areas: VPD dominated vegetation changes. VPD increases did not contribute to SM loss under the intense water stress in hyper-arid regions, as the soil water supply cannot meet the atmospheric water demand. In dry sub-humid areas, human disturbances have attenuated the dependence of vegetation changes on SM variability. This research pioneers complex nonlinear dynamic analyses on the multifaceted characteristics of ecosystems, which can deepen our understanding of atmosphere-vegetation-soil interactions in drylands and guide the sustainable management of dryland ecosystems in China and elsewhere.

Connecting Gaits in Energetically Conservative Legged Systems

In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus’ matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.

Self‐organisation in dialogical patterns of a patient with borderline personality disorder and their therapist: A case study from nonlinear dynamics perspective

AbstractObjectiveThe purpose of this study was to identify self‐organised dynamics in sessions with dialogical patterns (i.e. reflective interactions of self‐states associated with psychological change) within long‐term psychotherapy. One of the hallmarks of self‐organisation is the presence of an Inverse Power Law (IPL) in the frequency distribution of relevant events, which reflect better adaptation to internal and environmental changes, produced by psychotherapy.MethodsA case study of successful psychotherapy (44 weekly sessions) with a patient diagnosed with borderline personality disorder (BPD) and the therapist was conducted. The expressions of the subjectivity (i.e. voices) in the patient and therapist were identified through the Model of Analysis of Discursive Positioning in Psychotherapy (MAPP). State Space Grids (SSG) identified sessions that presented dialogical patterns acting as attractors (8 sessions) and those with non‐dialogical patterns acting as attractors (12 sessions). Finally, Orbital Decomposition (ORBDE) produced statistics of self‐organisation in both groups of sessions and the degree of fit to an IPL.ResultsBoth groups of sessions presented an acceptable fit to the IPL distributions (R2 = .853 and .938), and statistics placed them in different self‐organised regions.ConclusionsAlthough dialogical patterns are an emergent property of therapy, their structure is similar to sessions of other types, in that dynamic processes occur in them that can be empowered in patients to consolidate psychological change. The nonlinear and emergent character of these self‐organised processes and their clinical implications for activating the patients' psychological resources are discussed.

Complex causalities between the carbon market and the stock markets for energy intensive industries in China

The low-carbon development of energy intensive industries is of vital importance to achieve China's energy and climate targets in 2030 while carbon market is an important mechanism to promote carbon reduction. This paper investigates three types of causality, including positive, negative and dark causality, between China carbon prices and four energy intensive stock indexes using Pattern Causality method from a nonlinear symbolic dynamic perspective. Our findings show that there exists weak bidirectional causality between these two markets, which is manifested in that the fluctuation of 1% in one market approximately cause the fluctuation of 0.15%–0.3% in the other market. Moreover, we further analysis the impact of policies on the causalities between these two markets by dividing the whole timescale into seven stages. The results indicated that the document announcing the formal launch of China's carbon trading system prompted the dominant market of their causality shifting from carbon market to stock markets. Carbon markets gradually show stronger causal influence on the stock markets before December 2017, and the opposite after April 2018. And the Covid-19 has further exacerbated the weakening role of the carbon finance. Finally, the delay effect of carbon market on power industry stock market can be identified when unveiling the dark causality type.

A dynamical map to describe COVID-19 epidemics.

Nonlinear dynamics perspective is an interesting approach to describe COVID-19 epidemics, providing information to support strategic decisions. This paper proposes a dynamical map to describe COVID-19 epidemics based on the classical susceptible-exposed-infected-recovered (SEIR) differential model, incorporating vaccinated population. On this basis, the novel map represents COVID-19 discrete-time dynamics by adopting three populations: infected, cumulative infected and vaccinated. The map promotes a dynamical description based on algebraic equations with a reduced number of variables and, due to its simplicity, it is easier to perform parameter adjustments. In addition, the map description allows analytical calculations of useful information to evaluate the epidemic scenario, being important to support strategic decisions. In this regard, it should be pointed out the estimation of the number deaths, infection rate and the herd immunization point. Numerical simulations show the model capability to describe COVID-19 dynamics, capturing the main features of the epidemic evolution. Reported data from Germany, Italy and Brazil are of concern showing the map ability to describe different scenario patterns that include multi-wave pattern with bell shape and plateaus characteristics. The effect of vaccination is analyzed considering different campaign strategies, showing its importance to control the epidemics.

Attractor dynamics of elite performance: the high bar longswing

ABSTRACT Combining biomechanics and motor control, the aim of this study was to investigate the limit cycle dynamics during the high bar longswing across the UK elite gymnastics pathway age groupings. Senior, junior and development gymnasts (N = 30) performed three sets of eight consecutive longswings on the high bar. The centre of mass motion was examined through Poincaré plots and recurrence quantification analysis exploring the limit cycle dynamics of the longswing. Close to one-dimensional limit cycles were displayed for the senior (correlation dimension (CD) = 1.17 ± .08), junior (CD = 1.26 ± .08) and development gymnasts (CD = 1.33 ± .14). Senior elite gymnasts displayed increased recurrence characteristics in addition to longer longswing duration (p < .01) and lower radial angular velocity of the mass centre (p < .01). All groups of gymnasts had highly recurrent and predictable limit cycle characteristics. The findings of this research support the postulation that the further practice, experience and individual development associated with the senior gymnasts contribute to the refinement of the longswing from a nonlinear dynamics perspective. These findings support the idea of functional task decomposition informing the understanding of skill and influencing coaches’ decisions around skill development and physical preparation.

Effects of Zeros in Phase Space Reconstruction for Small and Large Solar Radiation Data Points during Wet and Dry Seasonal Modeling and Prediction

The effect of zeros in the behaviour of nature system has been a major global concern which have been reported to bias the output of the analysis. This study examines the effect of zeros on small and large solar radiation data points in Nsukka from a nonlinear dynamic perspective. The solar radiation data used were collected from National Research for Space and Development Agency (NARSDA) and covers the period of two years (January 2012–December 2013). The influence of zeros on average mutual information method for delay time (𝜏), False nearest neighbour (FNN) for embedding dimension (𝑚), and phase space reconstruction is investigated by considering two different cases (one hour and five minutes interval for small and large data points on monthly basis respectively). The results reveal that the phase space trajectories of the raw and non-zero small data points for dry and wet seasons show evidence of an attractorin a well-defined region while raw and non-zero large data points have no attractor like shape but regular patterns and well-defined shapes are visible in dry and wet seasons. These imply low-dimensional and deterministic chaotic nature of the underlying dynamics of raw and non-zero data for small and large data points during wet and dry seasons. It is observed that there is little or no significant difference in the phase space reconstruction of raw and non-zero data for both small and large data points due to the low percentage of zeros in the time series data.&#x0D; Keywords: mutual information method, Phase space reconstruction, False nearest neighbour, chaotic nature.

Examining the reversal of soil erosion decline in the hotspots of sandstorms: A non-linear ecosystem dynamic perspective

Land degradation neutrality (LDN) has been widely recognized as an important target to achieve goals for global sustainable development. Ecological restoration projects (ERPs) are considered a traditional approach to mitigate wind erosion by altering the structures and functions of the ecosystem. Thus, the efficiency of China's historical large-scale ERPs for wind erosion mitigation should be examined and optimized to better address this global environmental challenge. We assessed the change in wind erosion using a biophysical model and investigated the associated drivers identified. In addition, the efficiency of ERPs was examined by adopting the theory of non-linear ecosystem dynamics. The erosion modulus significantly declined from 1990 to 2017, but this declining trend reversed since 2009. The structural equation model and related analyses revealed that the declining wind speed, wetting soil moisture, and vegetation greening substantially reduced wind erosion, while the reversal of declining wind speed around 2009 aggravated wind erosion, particularly in the northern Inner Mongolia Sandstorm Area. The relatively arid climate restricted vegetation restoration and further offset afforestation efforts, resulting in the transition of declining erosion modulus. From the perspective of non-linear ecosystem dynamics, a wet climate provides a crucial window of opportunity for ERP in drylands, while the window of risks due to dry and windy climates may extend the time required for a shift change after ERP and may postpone the stage change toward the LDN goal. Hence, climate-induced windows of opportunity and risks are crucial in identifying the time for initiating ERPs, and existing environmental conditions and critical thresholds should be carefully considered.

Characterization of daily rainfall variability in Hong Kong: A nonlinear dynamic perspective

AbstractModelling and forecasting of rainfall behaviour are of essential importance in various hydrological and meteorological studies, which depends largely on properly understanding the underlying rainfall dynamics. In this paper, 20‐years (i.e., 2000–2019) measurements of daily rainfall data at 6 rain gauge stations in Hong Kong were analysed in the framework of nonlinear dynamic analysis. Phase space reconstruction (PSR), recurrence plot (RP) and recurrence quantification analysis (RQA) were carried out to identify and characterize the chaotic nature of rainfall dynamics. The results of this study underscore the complex processes and variability that is associated with Hong Kong's rainfall data. The existence of chaos in rainfall process was well recognized across various stations. The rainfall dynamics were further investigated with respect to its inter‐annual and monthly variability. The dependence of rainfall dynamics on different temporal scales was likewise examined, which indicates that the rainfall dynamics at finer resolution are less complex than those at coarser resolution, thus the consequent rainfall variability is less pronounced.

Dynamic heteroscedasticity of time series interpreted as complex networks.

Heteroscedasticity of time series is an important issue addressed in relation to the nonlinearity and complexity of time series. Previous studies have focused on time series heteroscedasticity during a long-term period but have rarely analyzed it from a nonlinear dynamic perspective. This paper proposes a new model for converting a time series into a complex network. Our proposed model can examine not only the heteroscedasticity of a short-term series but also the dynamic evolution process of this heteroscedasticity. Using four typical crude oil time series as sample data, we construct four networks. A network node denotes the types of fluctuation patterns corresponding to the symbolization of the heteroscedastic features of a short-term fluctuation series based on the autoregressive generalized autoregressive conditional heteroscedasticity model, and a weighted edge represents the evolution direction and frequency between two patterns. Our findings show that the choice of the length of a short-term period depends on the diversity of these patterns. The identification of the nodes with greater out-strength or greater betweenness centrality can help us to understand the different roles of fluctuation patterns in the evolution process. We propose a method for predicting the most probable target nodes from a source node. The analysis of clustering effects can help in detecting the fluctuation patterns between different clusters. This paper investigates the evolution dynamic mechanism of the heteroscedastic features of a short-term time series, which can help researchers and investors deeply understand the dynamic process of time series.

Spatial rainfall variability in peninsular India: a nonlinear dynamic approach

This study examines the spatial rainfall variability in peninsular India from a nonlinear dynamic perspective. The rainfall variability is determined by employing the false nearest neighbor (FNN) method, a nonlinear dynamic dimensionality-based method, to rainfall data across peninsular India. The implementation of the FNN method involves phase space reconstruction of the rainfall time series using a delay embedding procedure and identification of the false nearest neighbors in the reconstructed phase space using a neighbor search approach. The method is applied to monthly rainfall data over a period of 35 years (January 1971–December 2005) at each of 367 grids (of 0.5° × 0.5° in size) across nine river basins in peninsular India, and the dimensionality (i.e. FNN dimension) is determined. The influence of the delay time (τ) on the FNN dimension estimation is also investigated by considering four different cases of separation of data in phase space reconstruction: monthly (τ = 1) and annual (τ = 12) as well as delay time values from the autocorrelation function method and the average mutual information method. The results indicate that, considering all the four τ values, the dimensionality of the 367 rainfall time series ranges from as low as 3 to as high as 20. The FNN dimension is found to generally vary, sometimes significantly, both for the same τ value for different grids and for the different τ values for the same grid. A basin-wise analysis of the dimensionality results also indicates that the Tapi basin (in the northwest part) exhibits the highest spatial rainfall variability (dimension ranging from 3 to 20), while the Vamsadhara basin (in the northeast part) exhibits the lowest rainfall variability (dimension ranging from 4 to 8). In general, rainfall in the southern and eastern basins have relatively lower dimensionality, while rainfall in the northern and western basins have relatively higher dimensionality. While there is almost no variability in the minimum FNN dimension of rainfall between the nine basins studied, there is significant variability in the maximum FNN dimension of rainfall between the basins. These results facilitate more reliable identification of rainfall spatial characteristics based on complexity and their classification and, thus, have important implications for the identification of the appropriate complexity of rainfall models for an area, rainfall interpolation/extrapolation, and basin water assessment and management.