Phase Trajectories Research Articles

Individual differences in the evolution of causal illusions.

In this research, we investigated individual differences in the formation and persistence of causal illusions. In a re-analysis of existing data, we identified two clusters of participants - persistent and adjusting - based on their trajectories in learning from repeated exposure to null contingencies. The persistent cluster maintained stable causal illusions, while the adjusting cluster demonstrated a reduction over time. This re-analysis provided a nuanced understanding of individual differences in causal learning, emphasizing the differential role of probability estimations in predicting causal judgements. These findings were replicated in a subsequent study, highlighting the robustness of the identified effects. In a pre-registered study, we extended the paradigm to include a second phase (active phase) to assess how individual differences in causal illusion trajectories in the passive phase would manifest when participants had agency in the information gathering process. The results were consistent with those of the two previous studies and confirmed our primary hypothesis that the adjusting cluster would exhibit a lower tendency to introduce the candidate cause on learning trials, and would, therefore, observe a higher frequency of cause-absent trials. Together, these studies provide comprehensive insights into the underpinnings of causal illusion development and persistence, potentially informing de-biasing interventions.

Complex Periodic Motions and Bifurcations of a Forced Duffing Oscillator with Its Field-Programmable Gate Arrays Implementation

The dynamical behavior of a Duffing oscillator under periodic excitation is investigated using semi-analytical methods. Bifurcation trees with varying periodic excitation are constructed. The stability, saddle-node bifurcation and period-doubling bifurcation are revealed by assessing the eigenvalue of the model. From the bifurcation trees, we observed that saddle-node and period-doubling bifurcations occur when the excitation frequency and excitation amplitude vary to an appropriate value. The generation of periodic-doubling bifurcation leads to a change in the periodicity of periodic motion. The relationships among periodic-m motions are interconnected yet independent of each other. To satisfy the need of parameter selection for FPGA circuits, a dual-parameter map is calculated to study the periodic characteristics. Then, an FPGA circuit model is designed and implemented. The results show that the phase trajectory and waveform of the FPGA hardware circuit match the numerical model.

Uniaxial Compressive Failure Characteristics and Fractal Analysis of Mud–Sand Composite Rock Samples Based on Acoustic Emission Tests

In order to study the influence of rock combination types on their mechanical properties and failure characteristics, uniaxial compression tests of single rock samples and combined rock samples were conducted. Acoustic emission (AE) signals during the test process were collected, and the differences in AE signals of single rock samples and combined rock samples were studied based on the fractal theory. The results showed that the peak strength, elastic modulus, peak strain, and failure degree of the combined rock samples are all between those of the two single rock samples. The AE ringing count gradually increases with the loading process and suddenly increases to the maximum when the rock sample fails. During this process, the phase trajectory volume corresponding to the ringing count shows an evolution law of first decreasing and then increasing, while the correlation dimension corresponding to the ringing count signal shows an overall evolution law of first increasing and then decreasing. The results indicate that the phase trajectory volume, correlation dimension, and crack changes have a consistent dynamic change. Therefore, the phase trajectory and correlation dimension are effective tools to describe the pore change characteristics of rock combinations.

Design of parameters for running-in cylinder liner piston rings based on running-in attractor

Purpose The purpose of this study aims to shorten the running-in time and improve the running-in quality of cylinder liner piston rings (CLPRs), the running-in tests were carried out and running-in parameters of CLPRs were designed based on running-in attractor theory, which can guide the choice of optimal working conditions for other friction pairs. Design/methodology/approach The running-in state and time under different working conditions are identified by the evolution law of the running-in attractor phase trajectory and fractal and chaotic characteristic quantities. The CLPRs running-in tests under different conditions were conducted and the friction signals were collected. The constructed phase trajectories and calculated chaotic parameters of the running-in attractor are obtained and the running-in state and time are identified by the evolution law of phase trajectories and chaotic characteristic quantities. The running-in quality is obtained by the surface morphology fractal dimension and characteristic roughness parameters. Findings The running-in parameters for short running-in time and good running-in quality are designed based on the fractal and chaotic theory and the optimal solution method are used to verify the results through the single objective or multi-objective optimization, and the corresponding optimal running-in parameters are obtained. Originality/value The optimal working condition parameters obtained from the design have guiding significance for the selection of CLPR running-in parameters, and this work can provide ideas for the other friction pairs. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-05-2024-0179/

A Self-Oscillator Based on Liquid Crystal Elastomer Fiber Under Constant Voltage.

Self-oscillation is the phenomenon in which a system generates spontaneous, consistent periodic motion in response to a steady external stimulus, making it highly suitable for applications in soft robotics, motors, and mechatronic devices. In this paper, we present a self-oscillator based on liquid crystal elastomer (LCE) fiber under constant voltage. The system primarily consists of an LCE-liquid metal (LCE-LM) composite fiber, a metal mass sphere, and a straight rod featuring both conductive and insulating segments. Building upon an established dynamic LCE model, we derive the governing dynamic equations. Numerical calculations reveal two distinct motion regimes: a static regime and a self-oscillation regime. Furthermore, we provide the temporal behavior curves of electrothermal-induced contraction and tensile force, the phase trajectories variation curves of the equivalent driving force and damping force. These detailed studies elucidate that self-oscillation results from the contraction of the electrothermal-responsive LCE-LM fiber when the circuit is activated, with continuous periodic motion being sustained through the interplay between the metal mass sphere and a self-controlled dynamic circuit. We also investigate the threshold conditions necessary for initiating self-oscillation, as well as the key system parameters that influence its frequency and amplitude. Our self-oscillator demonstrates improved stability by reducing the effects of gravity and other disturbances. Additionally, the curved trajectory of the mass sphere can be achieved by replacing the straight rod with a curved one, resulting in a more flexible and easily controllable structure. Given these characteristics, a self-oscillator system based on LCE-LM fiber may be ideal for creating monitoring and warning devices, dynamic circuit systems, and for integrating actuators and controllers.

Learning global and local features of power load series through transformer and 2D-CNN: An image-based multi-step forecasting approach incorporating phase space reconstruction

As modern power systems continue to evolve, accurate power load forecasting remains a critical issue in energy management. The phase space reconstruction (PSR) method can effectively retain the inner chaotic property of power load from a system dynamics perspective and thus is a promising knowledge-based method for power load forecasting. To fully leverage the PSR method’s capability in modeling this high-dimensional, non-stationary characteristics of power load data, and to address the challenges faced by its classical mathematical prediction algorithms in effectively solving contemporary prediction scenarios characterized by massive volumes of data. This study proposes a novel learning-based multi-step forecasting approach that utilizes an image-based modeling perspective for the reconstructed phase trajectories. Firstly, the feature engineering approach that simultaneously utilizes dynamic evolution features and temporal locality features in the trajectory image is proposed. Through mathematical derivation, the equivalent characterization of the PSR method and another time series modeling approach, patch segmentation (PS), is demonstrated for the first time. Building on this prior knowledge, a novel image-based modeling perspective incorporating a global and local feature extraction strategy is introduced to fully leverage these valuable features. Subsequently, within this framework, a novel deep learning model, termed PSR-GALIEN, is designed for end-to-end processing. This model employs a Transformer Encoder and 2D convolutional neural networks (CNNs) to extract global and local patterns from the image, while a multi-layer perceptron (MLP)-based predictor is utilized for efficient correlation modeling. Extensive experiments on five real-world datasets show that PSR-GALIEN consistently outperforms six state-of-the-art deep learning models in short-term load forecasting scenarios with varying characteristics, demonstrating its great robustness. Ablation studies further confirm that the image-based modeling approach captures more relevant features than traditional sequential methods, improving the overall forecasting accuracy. Additionally, the attributions of the forecasting results of PSR-GALIEN can be explained through the proposed visualization-based method, which significantly enhances its interpretability.

Dynamical mode recognition of coupled flame oscillators by supervised and unsupervised learning approaches

Combustion instability in gas turbines and rocket engines, as one of the most challenging problems in combustion research, arises from the complex interactions among flames influenced by chemical reactions, heat and mass transfer, and acoustics. Identifying and understanding combustion instability is essential for ensuring the safe and reliable operation of many combustion systems, where exploring and classifying the dynamical behaviors of complex flame systems is a core task. To facilitate fundamental studies, the present work concerned dynamical mode recognition of coupled flame oscillators made of flickering buoyant diffusion flames, which have gained increasing attention in recent years but are not sufficiently understood. The time series data of flame oscillators were generated through fully validated reacting flow simulations. Due to the limitations of expertise-based models, a data-driven approach was adopted. In this study, a nonlinear dimensional reduction model of variational autoencoder (VAE) was used to project the high dimensional data onto a 2-dimensional latent space. Based on phase trajectories in the latent space, both supervised and unsupervised classifiers were proposed for datasets with and without well-known labeling, respectively. For labeled datasets, we established the Wasserstein-distance-based classifier (WDC) for mode recognition; for unlabeled datasets, we developed a novel unsupervised classifier (GMM-DTW) combining dynamic time warping (DTW) and Gaussian mixture model (GMM). Through comparing with conventional approaches for dimensionality reduction and classification, the proposed supervised and unsupervised VAE-based approaches exhibit a prominent performance across seven assessment metrics for distinguishing dynamical modes, implying their potential extension to dynamical mode recognition in complex combustion problems.

Research on the complex dynamical behavior of H-bridge inverter with RLC load

Complex dynamical behaviors such as bifurcation and chaos exist in H-bridge inverter with RLC load, and these nonlinear behaviors will greatly increase the harmonic content of the output current and reduce the stability and reliability of the system. In this paper, a PI controller is added to widen the stable operation domain of the system. The stroboscopic mapping theory is used to model the system, the nonlinear dynamic behavior of the inverter is investigated by the bifurcation diagram, the folding diagram and the phase trajectory diagram are used for comparative verification, and the TDFC method is introduced to inhibit the chaotic behavior of the inverter, which further improves the stable range of the system operation. The fast-change stability theorem is used to analyze the stability of the system theoretically and verify the correctness of the numerical simulation. Therefore, the conclusions of the study provide a reliable theoretical basis for the design of the inverter system, which has important theoretical significance and practical value.

Research on energy transfer and vibration suppression of nonlinear energy sink with nonlinear damping

The nonlinear energy sink is a nonlinear shock absorber that provides targeted energy transfer and plays a crucial role in structural vibration suppression. In this paper, the energy transfer and dynamic characteristics of nonlinear damping nonlinear energy sink systems are analyzed. Firstly, the theoretical model of the nonlinear damping NES system is described, and secondly, the equations of the system in the free state are derived by using the complex variable average method and the multi-scale method to obtain the energy transfer efficiency equation of the system. Through the study of the slow invariant manifold and energy transfer efficiency of the system, the influence of each system parameter on the energy transfer efficiency is analyzed. Then the equations of the system under harmonic excitation are processed in the same method to obtain the boundary equations of the system, through which the saddle-node bifurcation diagram of the system is plotted to study the influence of each system parameter on the saddle-node bifurcation characteristics of the system. Finally, the slow-varying equation of the system under harmonic excitation is derived, and the influence of each system parameter on the frequency detuning parameter interval of the strongly modulated response is obtained using the phase trajectory and a one-dimensional mapping diagram of the system, and the time-mileage diagram and Poincare section are used to prove the damping effect of the strongly modulated response.

A new two-parameter controllable multi-scroll 4D Hamiltonian conservative hyperchaotic system with improved nested COS-PWL function

A new two-parameter controllable multi-scroll Hamiltonian conservative hyperchaotic system is proposed in this paper. By utilizing an improved nested cosine-piecewise linear (COS-PWL) saturation function in combination with a 4D Hamiltonian conservative system, a 4D Hamiltonian conservative hyperchaotic system that can regulate multi-scroll according to two parameters is proposed. Compared with the nested Sine-PWL function, the system can produce hyperchaos. And the control parameter is more than just the saturation value E, the gain p is more sensitive to the wider control range of the system’s multi-scroll. The effect of the variation of the saturation value E and gain p of the control parameters on the phase trajectories of the multi-scroll is consistent with the behavior of the Lyapunov exponents (LEs) and bifurcations. Finally, we successfully constructed the system via the DSP hardware platform, demonstrating its practicality.

Исследование многоуровневых моделей управления вагонопотоками в адрес припортовых станций с применением теории дифференциальных уравнений

Purpose: To consider the issue of sustainability of multi-level control systems for car flows to port railway stations in the conditions of intellectualization of transportation management processes. To assess the influence of three-level and two-level control systems for the operational work of port roads on the quality of planning the supply of trains to port stations under various modes of operational operation of railways. Determine the conditions for the stability of the car flow control system by constructing and studying the corresponding hard and soft mathematical models described by autonomous systems of ordinary linear differential equations. Methods: In this study, the methods of the theory of stability of solutions of differential equations, as well as the theory of control and dynamical systems are used. A phase space is used in a visual representation, a qualitative and quantitative analysis of the behavior of phase trajectories in it is performed, corresponding to undisturbed and perturbed solutions of systems of differential equations describing the constructed models of control systems. Results: A comparison of the characteristics of multistage car traffic control systems is given. A study of the stability of multilevel models of car traffic management in the context of the intellectualization of management functions is proposed. In the environment of the analytical computing system, the conditions of asymptotic stability in time of a two-level wagon traffic control system are found and studied. Practical importance: The results obtained by computational experiment make it possible to assess the stability of the functioning of port-side transport and technological systems in the context of changes in the organization of production and the intellectualization of transportation management processes. Computer mathematics systems make it possible to implement the heuristic component of research and obtain theoretically sound and visual solutions to problems of optimizing cargo and wagon traffic control modes.

Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics

This paper focuses on finding exact soliton solutions and further examining the qualitative characteristics of these solutions for a nonlinear partial differential equation known as the (3+1)-dimensional integrable Kadomtsev–Petviashvili equation that portrays a unique dispersion effect. The (3+1)-dimensional integrable Kadomtsev–Petviashvili equation models the evolution of weakly nonlinear dispersive waves in three spatial dimensions “x,y,z” and time “t”. It balances nonlinearity and dispersion, leading to solitary wave solutions that are stable and localized. This equation is crucial in understanding complex wave phenomena in fields like fluid dynamics, plasma physics, and optics. Initially, the traveling wave transformation is employed to convert the nonlinear partial differential equations into the non-linear ordinary differential equations. Then, we use the Riccati equation approach to find solitary wave solutions for the resulting ordinary differential equation. To better comprehend the physical significance of these derived solutions, we illustrate them using various visual representations. Diverse novel collections of solitary wave solutions are obtained such as bright solitons, dark solitons, and dark singular solitons. Second, we convert the nonlinear ordinary differential equation into a linear dynamical system using the planar dynamical transformation. Then, we undertake a qualitative analysis of the dynamical system to study its chaotic characteristics and bifurcation phenomena. Phase portraits of bifurcation are observed at the equilibrium points of the planner dynamical system. We discuss various tools to identify the chaos (random and unpredictable behavior) in dynamical systems. Graphical visualization of qualitative analysis of the system is also depicted in 3-D phase portraits, 2-D phase portraits, Poincaré mapping, and time series. Additionally, the Runge–Kutta technique is used to do a thorough sensitivity analysis of the dynamical system. Using this analytical procedure, it is verified that the stability of the solution has negligible impact while there is little change in the beginning circumstances. This paper aims to provide vital views to scientists and researchers seeking to advance their experimental operations. Furthermore, this research might be expanded by incorporating solitons and situations and investigating wave dynamics. The methods employed provide a direct and uncomplicated approximation of all solutions when compared to the previously established techniques. Various combinations and values of the physical parameters are utilized to explore the optical soliton solutions of the resultant system.

Exploring nonlinear dynamics in brain functionality through phase portraits and fuzzy recurrence plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Dynamics Analysis and Adaptive Synchronization of a Class of Fractional-Order Chaotic Financial Systems

It is of practical significance to realize a stable and controllable financial system by using chaotic synchronization theory. In this paper, the dynamics and synchronization are studied for a class of fractional-order chaotic financial systems. First, the stability and dynamics of the fractional-order chaotic financial system are analyzed by using the phase trajectory diagram, time series diagram, bifurcation diagram, and Lyapunov exponential diagram. Meanwhile, we obtain the range of each parameter that puts the system in a periodic state, and we also reveal the relationship of the derivative order and the chaotic behaviors. Then, the adaptive control strategy is designed to achieve synchronization of the chaotic financial system. Finally, the theoretical results and control method are verified by numerical simulations.

Exploring Nonlinear Dynamics In Brain Functionality Through Phase Portraits And Fuzzy Recurrence Plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Passive Islanding Detection of Inverter-Based Resources in a Noisy Environment

Islanding occurs when a load is energized solely by local generators and can result in frequency and voltage instability, changes in current, and poor power quality. Poor power quality can interrupt industrial operations, damage sensitive electrical equipment, and induce outages upon the resynchronization of the island with the grid. This study proposes an islanding detection method employing a Duffing oscillator to analyze voltage fluctuations at the point of common coupling (PCC) under a high-noise environment. Unlike existing methods, which overlook the noise effect, this paper mitigates noise impact on islanding detection. Power system noise in PCC measurements arises from switching transients, harmonics, grounding issues, voltage sags and swells, electromagnetic interference, and power quality issues that affect islanding detection. Transient events like lightning-induced traveling waves to the PCC can also introduce noise levels exceeding the voltage amplitude by more than seven times, thus disturbing conventional detection techniques. The noise interferes with measurements and increases the nondetection zone (NDZ), causing failed or delayed islanding detection. The Duffing oscillator nonlinear dynamics enable detection capabilities at a high noise level. The proposed method is designed to detect the PCC voltage fluctuations based on the IEEE standard 1547 through the Duffing oscillator. For the voltages beyond the threshold, the Duffing oscillator phase trajectory changes from periodic to chaotic mode and sends an islanded operation command to the inverter. The proposed islanding detection method distinguishes switching transients and faults from an islanded operation. Experimental validation of the method is conducted using a 3.6 kW PV setup.

Bifurcation boundaries analysis of the thin-walled internal resonance energy harvester

The dynamic instability of the thin-walled 1: 2 internal resonant piezoelectric vibration energy harvester is investigated. Previously work (Nie et al., 2019) derived the nonlinear electromechanical-coupled governing equations and validated them experimentally. The modulation equations are performed by the method of multiple scales. The system stability is obtained from the eigenvalues of the Jacobi matrix. Then, three types of bifurcation boundaries of the system are determined analytically and simulated numerically via Routh–Hurwitz criterion. Results show that internal resonance complicates the dynamic properties of the bifurcation region. For a large external excitation, new saddle node bifurcation regions appear in the stability diagram. Moreover, the new saddle node and Hopf bifurcation regions may intertwine. Furthermore, as the acceleration excitation increases, the system can be observed to transition to stable–unstable–stable states. The system’s high and low branch responses are fed back by the system’s large and small initial conditions. The system’s periodic and quasi-periodic motions are characterized by time response, FFT, phase trajectory, and Poincaré map. The study of the stability boundary of the piezoelectric energy harvester contributes to the harvesting of energy from ambient vibrations over a broader frequency range.

Chaotic dynamic analysis of electrical contact resistance measured in sliding current-carrying friction

This work explores the dynamic laws of electrical contact resistance (ECR) signals measured in the sliding current-carrying friction tests based on chaos theory. It is found that ECR time series have a chaotic nature, strongly verified by the positive largest Lyapunov exponent λ1. The chaos may originate from the combined friction effects on short and long timescales. Phase space analysis indicates that the ECR phase trajectory adheres to the “convergence–stability” evolution law over time, which pertains closely to energy dissipation. Hence, the chaotic attractor exists during current-carrying friction. Strikingly, wear surfaces confirm that the ECR phase trajectory evolution process corresponds to the stages of “formation–keeping” of the chaotic attractor, and also agrees with the electrical wear stages of “running-in–steady-state”. Further, the chaos quantifier variation is consistent with the phase space analysis results to identify wear transition quantitatively.

Exploring third-order KdV–SIdV families: Analytical solutions, conservation properties, and phase plane trajectories

This study focuses on the Scale-Invariant (SIdV) families of third-order equations, which are among the few known equations sharing the same solitary wave solution of the KdV equation in the form of sech2. The primary objective is to generalize existing or discover new third-order KdV–SIdV families that feature the analytical sech2 solution of the KdV equation. The study aims to identify all admissible advecting velocity functions ensuring that the KdV–SIdV families accommodate every asymptotically decaying KdV traveling wave with the same wave speed, independent of the advecting velocity. Furthermore, the investigation seeks to delineate their corresponding conserved properties, thereby enhancing our understanding of these intriguing nonlinear partial differential equations. It is shown that the discovered KdV–SIdV families, while sharing the same solitary wave solution, exhibit differing conservation properties. Additionally, a qualitative analysis is conducted using phase plane trajectories to examine the traveling wave solutions and their characteristics for a particular SIdV equation. The emphasis lies in comprehensively exploring potential bifurcations and phase portraits of the vector fields governed by the corresponding system in the parametric space. Through the analysis of distinct phase trajectories across various regions, we derive a new diverse range of solitonic wave solutions, including bell/anti-bell solitary waves, peakon waves, singular waves, and periodic singular waves.

Nonlinear Dynamics and Meshing State Transition Analysis of Spalling Defect Gears with Multi-Lubrication Conditions

The purpose of this paper is to analyze the dynamic characteristics and the meshing state transition characteristics of spalling defect gears under various lubrication conditions. The time-varying geometric parameters, motion parameters and load distribution of the gears in the state of positive meshing and reverse impact are analyzed. The time-varying meshing stiffness considering the time-varying friction coefficient under elastohydrodynamic lubrication (EHL), mixed lubrication and boundary lubrication are calculated, and the spalling defect is introduced in the stiffness calculation process. The equivalent lubrication model of the gear is established, the switching condition of the lubrication state is derived, and the distribution of the lubrication state in the torque-speed plane is discussed. The multimeshing state dynamic model of the spalling defect gear is established, and the evolution characteristics of the coexistence response and the transition of the meshing state under different lubrication conditions are simulated by using the attraction domain, phase trajectory, meshing force and bifurcation diagram. The results show that the phase trajectory of the gear with the spalling defect is deformed compared with that of the healthy gear. The change of lubrication state has little effect on the phase trajectory and attraction domain of the gear with spalling defect, but has a great influence on the meshing force. At the spalling boundary, the meshing force is changed abruptly, and the worse the lubrication state, the more obvious the mutation. Due to the influence of spalling on tooth stiffness, it will also affect the meshing force in the nonspalling area.