Adjacent Orbits Research Articles

Diversity and transition of periodic motion of a periodically excited soft-impacting machinery

Dynamics of a periodically excited vibro-impact system with soft impacts is investigated. Essential features of period-one multi-impact motion group and correlated transition characteristics in low-frequency range are discussed in detail by the way of two-parameter bifurcation space providing qualitative domains for different periodic motions. The main focus is given to the effect of sensitive parameters including constraint stiffness k 0, clearance threshold b, and damping parameter ζ on the system response. The low-frequency characteristics in the finite-dimensional parameter space are particularly explored. It is found that the increase of k 0 induces multi-type bifurcation of period-one double-impact symmetrical motion, which induces a rich variety of periodic motions, and period-one multi-impact motion group orbit primarily exist in the small-clearance b and low-frequency ω zone. Based on the evolution irreversibility of adjacent period-one multi-impact orbit, the mechanism of singularies appearing in pairs and two different transition zones (hysteresis and liguliform zones) is studied, the result of which provides a theoretical reference value for the common low-frequency vibration instability phenomenon in the field of mechanical engineering. For small-damping coefficient ζ, period-one multi-impact motion has a large quantity, and the main bridge for the transition of adjacent period-one multi-impact motion is liguliform zone, which embraces period-one multi-impact asymmetrical motion and period-n multi-impact subharmonic motion and a certain chaotic zone. For large-damping coefficient ζ, the amount of period-one multi-impact motion group is reduced, and the main bridge for the transition of adjacent period-one multi-impact motion is hysteresis zone, where adjacent period-one multi-impact orbits can coexist according to initial conditions. As designing and renovating impact mechanical equipment, the reasonable matching law of dynamic parameters can be determined through two-parameter bifurcation space, which is conducive to making the system work in stable periodic motion and obtaining larger instantaneous impact velocity.

First Systematic Study Reporting the Changes in Eclipse Cutoff Frequency for Pulsar J1544+4937

We present results from long-term monitoring of frequency-dependent eclipses of the radio emission from PSR J1544+4937, which is a black widow spider millisecond pulsar (MSP) in a compact binary system. The majority of such systems often exhibit relatively long-duration radio eclipses caused by ablated material from their companion stars. With the wide spectral bandwidth of the upgraded Giant Metrewave Radio Telescope, we present the first systematic study of temporal variation of eclipse cutoff frequency. With decade-long monitoring of 39 eclipses for PSR J1544+4937, we notice significant changes in the observed cutoff frequency ranging from 343 ± 7 to ≥740 MHz. We also monitored changes in eclipse cutoff frequency on timescales of tens of days and observed a maximum change of ≥315 MHz between observations that were separated by 22 days. In addition, we observed a change of ∼47 MHz in eclipse cutoff frequency between adjacent orbits, i.e., on timescales of ∼2.9 hr. We infer that such changes in the eclipse cutoff frequency depict an eclipse environment for the PSR J1544+4937 system that is dynamically evolving, where, along with the change in electron density, the magnetic field could also be varying. We also report a significant correlation between the eclipse cutoff frequency and the mass-loss rate of the companion. This study provides the first direct evidence of mass-loss rate affecting the frequency-dependent eclipsing in a spider MSP.

Turing-like patterns induced by the competition between two stable states in a discrete-time predator–prey model

Patterns, typical spatiotemporal ordered structures, are widely present in a variety of systems, which are the results of the emergence of complex systems. The theory of Turing, occupying a long-range inhibitor and a short-range activator, provides an explanation for the diversity of Turing patterns. Turing-like patterns, caused by various non-Turing mechanisms, have also been investigated in various models. Patterns resulting from Turing or non-Turing mechanisms have predominantly focused on continuous-time models, however, for populations with non-overlapping generations, discrete-time models are more realistic than continuous-time ones. Here, we investigated a type of Turing-like patterns in a discrete-time model. The results show that the competition between two stable states can lead to the formation of Turing-like patterns. Specifically, we obtained a type of Turing-like patterns in a discrete-time model, which exist outside the parameter regions of Turing instability. Only by applying strong impulse noise to the homogeneous stable state can the patterns be excited. And the excitation threshold is related to the control parameter. Analysis shows that the Turing-like patterns are the results of competition between two stable states corresponding to adjacent orbits, and the excitation threshold is determined by the relative position between orbits. This findings exemplify the multitude of mechanisms observed in nature that give rise to the emergence of Turing or Turing-like patterns, and shed light on the pattern formation for Turing/Turing-like patterns.

Phonon-mediated magneto-resonances in biased graphene layers

We explore the non-equilibrium transport regime in graphene using a large dc current in combination with a perpendicular magnetic field. The strong in-plane Hall field generated in the graphene channel results in Landau levels that are tilted spatially. The energy of cyclotron orbits in the bulk varies as a function of the spatial position of the guiding center, enabling us to observe a series of compelling features. While Shubnikov–de Haas oscillations are predictably suppressed in the presence of the Hall field, a set of fresh magneto resistance oscillations emerge near the charge neutrality point as a function of dc current. Two branches of oscillations with linear dispersions are evident as we vary carrier density and dc current, the velocities of which closely resemble the transverse acoustic (TA) and longitudinal acoustic (LA) phonon modes, suggestive of phonon-assisted intra-Landau level transitions between adjacent cyclotron orbits. Our results offer unique possibilities to explore non-equilibrium phenomena in two-dimensional materials and van der Waals heterostructures.

Constructing n-dimensional discrete non-degenerate hyperchaotic maps using QR decomposition

Lyapunov exponents (LEs) characterize the average exponential rate of convergence or divergence between adjacent orbits in phase space. Thus, the number of positive LEs can reflect the complexity of chaotic systems from a certain point of view. To resist the dynamic degradation of digital chaos, we propose a novel universal method that is based on QR decomposition for constructing non-degenerate hyperchaotic maps. A large number of positive LEs can be generated to increase the complexity of chaotic systems. Furthermore, we construct a 4-D discrete non-degenerate hyperchaotic map as an example to demonstrate the adaptability and efficacy of the proposed scheme. For the proposed method, the related control parameters can not only effectively adjust Lyapunov exponents, but also carry out chaotic regulation on the discrete map, such as amplitude control and offset boosting. In addition, a pseudorandom number generator (PRNG) is designed with desirable statistical properties. Then, a microcontroller-based platform was developed to implement the proposed chaotic map. This study is interesting and has a potential for real-world applications, such as secure communication and cryptography.

Understanding the Spatiotemporal Characteristics of Land Subsidence and Rebound in the Lianjiang Plain Using Time-Series InSAR with Dual-Track Sentinel-1 Data

The Lianjiang Plain, renowned for its position as ‘China’s textile hub’ and characterized by its high population density, has experienced considerable subsidence due to excessive groundwater extraction in recent years. Although some studies have investigated short-term subsidence in this plain, research on long-term subsidence and rebound remain understudied. In this paper, the characteristics of surface deformation in the Lijiang Plain during two periods (2015–2017 and 2018–2021) have been investigated using the time-series interferometric synthetic aperture radar (TS-InSAR) technique, and the correlation with the changes in groundwater level, geological factors, and urban construction are discussed. The InSAR-derived results are cross-validated with the adjacent orbit datasets. Large-scale and uneven subsidence ranging from −124 mm/year to +40 mm/year is observed from 2015 to 2017. However, a significant decrease in the subsidence rate during 2018–2021, with local rebound deformation up to +48 mm/year in three regions, is also observed. Groundwater level changes are found to be the major cause of the ground deformation, and the intercomparison between groundwater level and ground displacement time series from TS-InSAR measurements also indicates a clear relationship between them during 2018–2021. Geological factors control the range of deformation area over the study period. The impact of urban construction on surface subsidence is evident, contributing to high deformation. Our findings could improve the understanding of how deformation is affected by groundwater rebound and offer valuable insights into groundwater management, urban planning, and land subsidence mitigation.

Global-in-time existence of unbounded separatrices and blow-up time of solutions for Liénard systems whose restoring force is an odd power function

The existence, uniqueness, and number of limit cycles in Liénard dynamics have been intensely studied since the 1940s because they relate to Hilbert's 16th problem. Limit cycles sharply distinguish the nature of adjacent orbits. Moreover, they are essential for understanding dynamical systems. In contrast, the importance of unbounded separatrices is less well recognized even though they play an identical role. Therefore, this study focuses on the Liénard dynamical system and investigates whether all orbits, including unbounded separatrices, exist globally in time. The blow-up times of solutions that cannot be extended up to negative or positive infinity are also discussed. The result allows classification of all solutions by the time for which they exist. For example, for the van der Pol oscillator known to have the sole limit cycle, all solutions that pass through any point in the region bounded by that limit cycle and the two solutions that correspond to the unbounded separatrices continue to exist for all time, whereas the other solutions have positive or negative blow-up times. This fact cannot be judged solely from the shapes of the orbits obtained by simulation. Phase plane analysis is applied to prove the main theorem. Furthermore, several examples and their plane figures are provided to facilitate the understanding of the proof.

A mesh refinement method for low‐thrust orbit transfer problems

AbstractThis paper researches the fuel‐optimal geocentric orbit transfer problem using a low‐thrust, electric propulsion. First, the transfer trajectory is divided into N burn‐coast arcs via introducing (N−1) intermediate orbits between the initial and the target orbits. By restricting the time duration of each burn, the changes in orbital states caused by one time of ignition can be represented by a discrete velocity impulse; therefore, a considerable large N, a great many optimization variables, and a months and revolutions long trajectory will be calculated to solve the optimization problem. By proposing a new transfer coordinate system, the performance index is analytically calculated, and strict constraints derived from the continuous condition between adjacent orbits are removed from the mathematical optimization model. Meanwhile, an improved particle swarm optimization (PSO) method and a mesh refinement method are proposed to address the optimization problem. The mesh refinement algorithm begins with a simple bi‐impulse transfer problem, automatically increases/decreases N, and finally converges to the optimal solution. Numerical simulation shows the efficiency of the method with some classical and representative scenarios.

Comparison of orbital volume measured using computed tomography and fluid displacement: A comparative study using dry skull

Objectives: The purpose of the study was as follows: (1) To compare orbital volume calculated using computed tomography (CT) scans with real-time measurements. (2) To evaluate if the orbital volume of the adjacent orbit can be taken as a tool of comparison for orbital volume correction in orbital fractures. Materials and Methods: A prospective study was conducted on 30 orbital cavities with intact orbital walls. The foramina and fissures were sealed using molding wax. Alginate impressions of the orbital cavity were taken to determine the orbital volume using the Archimedes principle. GE light speed 64 slices CT scanner was used to scan the orbits with 0.8 mm sections. Orbital volume was calculated using the CT software GE ADW 4.5 workstation. Results: The comparison of CT measurement and Archimedes principle measurement shows a mean and standard error of 0.2333 mm and 0.09794, respectively, on the right orbits and a mean and standard error of 0.10011 mm and 0.0981, respectively, on the left orbits. The comparison of the right and left orbits shows a mean and standard error of 0.4067 mm and 0.2771 mm, respectively, using the Archimedes principle and a mean and standard error of 0.3300 mm and 0.2775, respectively, using CT scans. Conclusion: It was found that the difference in orbital volume between CT and real-time measurement is very negligible. The study shows that CT is a reliable tool for the orbital volume measurement and the adjacent orbit can be taken as a comparative tool for the orbital volume corrections.

An effective interpolation strategy for mitigating the day boundary discontinuities of precise GNSS orbit products

A precise Global Navigation Satellite System (GNSS) satellite orbit is an essential prerequisite for precise point positioning (PPP). Due to the daily orbit discontinuities (DODs) between adjacent daily orbit products provided by the International GNSS Service, the interpolation accuracy at the epochs near the day boundaries cannot be as high as expected and can significantly degrade the positioning accuracy. This is especially for the emerging Beidou and Galileo satellites with larger DODs. In this study, a new interpolation strategy was proposed to mitigate the DOD effect by estimating the zero- to second-order biases between the orbits of two adjacent days. The results show that the proposed interpolation strategy outperforms the traditional interpolation strategies. The 3D orbit interpolation error for the last 10 min on each day is reduced from almost a few centimeters or even meters to millimeters, an improvement of approximately 86–99% for all satellites. PPP validation also provides concrete evidence of this improvement, as the DOD-induced satellite phase residuals are reduced significantly. Thus, the new interpolation strategy mitigates the DOD effect and helps multi-GNSS users obtain the highest interpolation accuracy at the day boundaries, which can be comparable to that at the non-day boundaries.

Spheno-orbital Lesions-A Major Review of Nonmeningioma Causes.

To review the imaging features of the broad range of nonmeningioma lesions of the greater wing of the sphenoid (GWS) bone and adjacent orbit to assist clinicians in differentiating these lesions from each other and from the most common lesion of the GWS and adjacent orbit, meningioma. We reviewed 32 cases of spheno-orbital lesions involving the GWS from our own practice, as well as 109 published cases (total 141), with emphasis on available imaging features on computerized tomography (CT) and MRI. Features that might assist in differentiating meningioma from its mimics were analyzed for each lesion, including the presence of an osteoblastic or hyperostotic response, bone erosion or osteolysis, homogeneous hypo- or hyperintensity on T2-weighted MRI, leptomeningeal involvement, and the absence of a "dural tail" on contrast-enhanced MRI. The clinical and imaging features were also briefly summarized for each diagnostic group. The largest diagnostic group was metastasis (67 cases, 47.5%). The most useful imaging features that helped differentiate meningioma from its mimics were the presence of bone erosion and the absence of a "dural tail." Other features were helpful in a small minority of cases only. Metastatic prostate cancer was the largest single group (21 cases), and 18 (85.7%) of these were osteoblastic and most closely mimicked meningioma. Prostate cancer patients were generally older than males with GWS meningioma. Almost all other (44/46, or 95.7%) metastatic lesions showed evidence of bone erosion. Almost half (30 of 61, 49.2%) of patients with metastasis presented without a known diagnosis of malignancy. Among children 16 years of age and less, Langerhans cell histiocytosis (10 cases), dermoid cyst (5), and Ewing's sarcoma (5) were the most common diagnoses. A combination of a careful history and both CT and MRI gives information, which can best guide the management of patients with spheno-orbital lesions. Metastatic prostate cancer to the GWS most closely mimics GWS meningioma but can in most cases be differentiated on clinical and imaging features. Older males with hyperostotic lesions of the GWS should be investigated for prostate cancer. Other metastatic lesions and primary tumors of the GWS, as well as benign and structural lesions can readily be differentiated from meningioma on clinical and imaging features.

Classification Method for Network Security Data Based on Multi-featured Extraction

A method of classifying network security data based on multi-featured extraction is proposed to address instability of a nonlinear time series in a network security threat. Cybersecurity information is divided in line with the principle of acquiring multiple attributes. On this basis, an adaptive adaptation estimation technology is optimized in analogue. With the proposed method, a cybersecurity information classification system is constructed according to the phase interval reconstruction principle so that a dynamic and autonomous adaptation estimation of the cybersecurity threat can be completed to ensure the feasibility of cybersecurity information classification. The experimental result proves that the cybersecurity information classification technology based on multi-attribute extraction can effectively guide chaos into adjacent orbits and reasonably control the training scale. Moreover, the accuracy of the estimation is guaranteed and the cybersecurity threat is estimated because of its high-speed convergence and strong proximity. Therefore, the proposed classification technology can assist professionals and backstage managers in guaranteeing security by facilitating receipt of information in a timely manner.

Generalized Composite Multiscale Diversity Entropy and Its Application for Fault Diagnosis of Rolling Bearing in Automotive Production Line

This paper considers the entropy based feature extraction method for the fault diagnosis of rolling bearings in automobile production line, where the fault information is difficult to identify due to the strong nonlinear and non-stationary characteristics of the fault vibration signals. In our work, a novel entropy based method called generalized composite multiscale diversity entropy (GCMDiEn) is developed. This method can effectively track the inside pattern changes of the time series by the description of cosine similarity between adjacent orbits. Unlike most of the existing entropy based results which concentrate on the static orderliness, we analyze the dynamic complexity characteristics of the arbitrary time series. Moreover, compared with the multiscale diversity entropy, GCMDiEn calculates all coarse-grained time series entropy value with the same scale and extends the first-order moment to second-order moment to mitigate the shortcomings of the short time series instability and losing part of the frequency region information. This paper proposed a new rolling bearing fault diagnosis framework which combined the GCMDiEn with the empirical wavelet transform (EWT), Laplacian score (LS), and particle swarm optimization-based support vector machine (PSO-SVM). Finally, the simulation results show the superiority of the GCMDiEn method over the multiscale diversity entropy method. The proposed framework has a higher fault recognition rate (99.38%) than the existing methods.

MESSENGER Observations of Planetary Ion Characteristics in the Vicinity of Kelvin‐Helmholtz Vortices at Mercury

AbstractThe MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft regularly observed the magnetospheric flanks of Mercury during its orbital phase. Data from the Magnetometer (MAG) and the Fast Imaging Plasma Spectrometer (FIPS) allow us to investigate the statistical properties of planetary ions (Na+) in the presence of Kelvin‐Helmholtz (KH) waves at the duskside magnetopause. We collect the data from orbits with clear signatures of KH waves under northward interplanetary magnetic field, as well as from adjacent orbits that do not have KH signatures, and we compare the energy characteristics between the KH and non‐KH events. Although low planetary counts in FIPS data make the comparison of these characteristics difficult, we find that in the presence of KH waves: (1) large counts of planetary ions are observed and (2) differences in Na+ energy spectra are only seen inside the magnetosphere, where they show a deceleration signature for ions with energies above 2.0 keV/e. These results suggest that planetary ions are not escaping from the magnetosphere and that electric field structures related to KH waves can decelerate planetary ions originating from the magnetotail region. The understanding of the energy distribution of planetary ions in the magnetospheric flanks of Mercury is important for a better understanding of plasma convection in the magnetosphere.

Multiscale Diversity Entropy: A Novel Dynamical Measure for Fault Diagnosis of Rotating Machinery

In this article, a fault diagnosis scheme based on multiscale diversity entropy (MDE) and extreme learning machine (ELM) is presented. First, a novel entropy method called diversity entropy (DE) is proposed to quantify the dynamical complexity. DE utilizes the distribution of cosine similarity between adjacent orbits to track the inside pattern change, resulting in better performance in complexity estimation. Then, the proposed DE is extended to multiscale analysis called MDE for a comprehensive feature description by combining with the coarse gaining process. Third, the obtained features using MDE are fed into the ELM classifier for pattern identification of rotating machinery. The effectiveness of the proposed MDE method is verified using simulated signals and two experimental signals collected from the bearing test and the dual-rotator of the aeroengine test. The analysis results show that our proposed method has the highest classification accuracy compared with three existing approaches: sample entropy, fuzzy entropy, and permutation entropy.

An Image Matching Method for SAR Orthophotos from Adjacent Orbits in Large Area Based on SAR-Moravec

In producing orthophoto mosaic in a large area from spaceborne synthetic aperture radar (SAR) images, SAR image matching from adjacent orbits is a technical difficulty due to the speckle noise and different imaging mechanism between azimuth and range direction. In this paper, an area-based method, SAR-Moravec, is proposed for SAR orthophoto matching from adjacent orbits in a large area. Compared with the classical area-based Moravec, the template of SAR-Moravec is characterized by more directions for speckle noise restraint and a main direction consistent with the azimuth. In order to get evenly distributed matching points with high accuracy, the grid control mechanism and Gaussian pyramid from coarse to fine are introduced in matching. The whole process contains three steps. First, the pyramid images are constructed by the down-sampling process. Second, under grid control, the feature points are evenly extracted by the modified template. Third, the transformation model is iteratively calculated from the first to the last layer of the pyramid. After the matching process layer-by-layer, the final matching points and transformation model can be obtained. In the experiments, we compare the SAR-Moravec with three widely used methods, including the Moravec, the SAR-scale invariant feature transform (SAR-SIFT), and the SAR-features from an accelerated segment test (SAR-FAST). The results indicate that the proposed method has the best global matching accuracy among these methods and the matching efficiency is better than SAR-SIFT and SAR-FAST methods in large area.

PLC-C: An Integrated Method for Sentinel-2 Topographic and Angular Normalization

Topographic and angular corrections on Sentinel-2 imagery are crucial for the generation of consistent surface reflectance. We propose a novel topographic-angular integrated normalization approach based on the combination of the path length correction (PLC) and C-factor approaches. The PLC-C normalization approach is a semiphysical method with limited use of auxiliary data: only a digital elevation model and a fixed set of kernel coefficients, ensuring its transferability for operational implementation. For the validation, we used two Sentinel-2A images over a mountainous area observed in backward (BS) and forward scattering (FS) directions from laterally adjacent orbit swaths. PLC-C significantly reduced both the topographic and directional anisotropy effects: the overlapping ratio between BS and FS observations was increased from 84.1% to 92.8% for the near-infrared band, and from 81.0% to 93.1% for the red band; the coefficient of variation of the reflectances across different aspects, which was used as a criterion of topographic effects, was reduced from 9.8%/12.2% to 3.6%/5.7% in BS/FS direction for the near-infrared band, and from 8.1%/9.7% to 4.5%/4.2% for the red band. PLC-C will contribute to the generation of analysis ready data from Sentinel-2 top of canopy reflectance.

Object-based landfast sea ice detection over West Antarctica using time series ALOS PALSAR data

Landfast sea ice (fast ice) is an important feature prevalent around the Antarctic coast, which is affected by climate change and energy exchanges with the atmosphere and ocean. This study proposed a method for detection of the West Antarctic fast ice using the Advanced Land Observing Satellite Phased Array L-band SAR (ALOS PALSAR) images. The algorithm has combined image segmentation, image correlation analysis, and machine learning techniques (i.e., random forest (RF), extremely randomized trees (ERT), and logistic regression (LR)). We used SAR images with a baseline of 5 days that are not in the same orbit but overlap each other as overlaps between swaths in adjacent orbits are often available in the polar regions. The underlying assumption for the proposed fast ice detection algorithm is that fast ice regions in SAR images with a time interval of 5 days are highly correlated. The object-based approach proposed in this study was well suited to high-resolution SAR images in deriving spatially homogeneous fast ice regions. The image segmentation results using the optimized parameters showed a distinct difference in the backscatter temporal evolution between fast ice and pack ice regions. Correlation and STD of backscattering coefficients were found to be the most significant variables for the object-based fast ice detection from two temporally separated images. In overall, the quantitative and qualitative evaluation demonstrated that the algorithm was an effective approach to detect fast ice with high accuracies. The models well detected various fast ice regions in the West Antarctica but misclassified some objects. The misclassifications occurred toward the edge of fast ice regions with relatively rapid changes in backscattering between both data acquisitions. On the other hand, few fast ice objects were misclassified as uniform backscattering over time occurred by chance on very small objects far from the coast. Very old multi-year fast ice regions with high backscattered signals were also a source for some misclassifications. This may be due to the sensitivity of L-band to snow structure to some extent and a thinner ice over the region with either ice growth (no deformation) or closing (slight deformation) between both images. Heavy snow load on the ice could be another error source for some misclassification as well. The approach allowed for the reliable detection of fast ice regions by using L-band SAR images with a small local incidence angle difference.

Next-Generation Sequencing in the Diagnosis of Rare Pediatric Sinonasal Tumors

The diagnosis of desmoid fibromatosis or other spindle cell tumors in the sinonasal region is very rare in children and needs to be thoroughly confirmed with immunohistochemical and/or molecular tests. We report 2 patients with such rare tumors and describe the use of next-generation sequencing in their evaluation. A 3-year-old female had a 4.4-cm midline nasal cavity mass involving the bony septum and extending into the base of the skull bilaterally. The moderate cellular fibroblastic proliferation revealed areas of thick keloid-like collagen bands and other areas with myxoid edematous stroma. Deep targeted sequencing identified a novel G34V mutation in the CTNNB1 gene consistent with desmoid fibromatosis. An 11-month-old male infant presented with a right nasal mass that extended through the cribriform plate into the anterior cranial fossa and involved the right ethmoid sinus and adjacent right orbit. Histology revealed an infiltrative atypical fibrous proliferation with focal calcifications that was negative for CTNNB1 and GNAS mutations. A novel RET E511K variant was identified in the tumor and later was also found in the germline and hence rendered of unknown significance. Both cases highlight the utility of next-generation sequencing in the evaluation of pediatric sinonasal spindle cell tumors that may have overlapping pathologic features. Reporting of rare or novel variants in tumor-only sequencing should be cautiously evaluated in children and pairing with germline sequencing may be needed to avoid the pitfall of assigning uncommon variants.

Early Fault Diagnosis of Rolling Bearing Based on Lyapunov Exponent

Lyapunov exponent is an important quantitative index to measure the dynamic characteristics of the system. It represents the average exponential rate of convergence or divergence between adjacent orbits in phase space. Whether there is dynamic chaos in the system can be judged intuitively from whether the maximum Lyapunov exponent is greater than zero. The Lyapunov exponent obtained by the small data method has the characteristics of simple calculation process and accurate calculation results, and it is applied to the fault diagnosis of rolling bearings. By calculating the Lyapunov exponents of bearing rollers, bearing outer rings, bearing inner rings and normal state bearings, and comparing the calculation results under different working conditions, it is concluded that Lyapunov exponents are of great significance in judging the early failure of rolling bearings.