Nonlinear Dynamics最新文献

Noise limits the information that can be experimentally extracted from dynamical systems. In this study, we review the Control-based Continuation (CBC) approach, which is commonly used for experimental characterisation of nonlinear systems with coexisting stable and unstable steady states. The CBC technique, however, uses a deterministic framework, whereas in practice, almost all measurements are subject to some level of random perturbation, and the underlying dynamical system is inherently noisy. In order to discover what the CBC is capable of extracting from inherently noisy experiments, we study the Hopf normal form with quintic terms with additive noise. The bifurcation diagram of the deterministic core of this system is well-known, therefore the discrepancies introduced by noise can be easily assessed. First, we utilise the Step-Matrix Multiplication based Path Integral (SMM-PI) method to approximate the system's steady state probability density function (PDF) for different intensity noise perturbations. We associate the local extrema of the resulting PDFs with limit cycles, and compare the resulting bifurcation diagram to those captured by CBC. We show that CBC estimates the bifurcation diagram of the noisy system well for noise intensities varying from small to moderate, and in practice, the amplitudes provided by CBC may be accepted as a 'best guess' proxy for the vibration amplitudes characteristic to the near periodic solutions in a wide range of experiments.

The dynamical systems view of a turbulent fluid flow provides a tantalizing connection between the self-sustaining nonlinear mechanics of turbulence and its more well-known statistical properties, and promises to open up new avenues in our ability to understand, predict and control complex fluid motion. However, successful application of these ideas to a high Reynolds number (Re) problem requires the discovery and convergence of an expansive library of simple invariant solutions (e.g. equilibria, periodic orbits). The key challenge for the field has been that algorithms to compute dynamically relevant structures struggle for a variety of reasons outside of the weakly turbulent regime. It is here that ideas from deep learning have started to show promise, and this review describes how various techniques from the machine learning community have accelerated progress. First, the use of autoencoders - neural networks which perform a nonlinear analogue to PCA - will be described. There is compelling evidence that the low-order representations of the flow learned by these models are closely connected to the unstable simple invariant solutions embedded in the turbulent attractor. As such, these representations can be used to measure shadowing of periodic solutions, to parameterize reduced order models and to estimate manifold dimension. The other key technique adapted from deep learning reviewed here is the advance in high-dimensional, gradient-based optimization that has been driven by the requirements of neural network training. To exploit these tools, the search for simple invariant solutions is converted to a hunt for minima of a scalar loss function, and gradient computation is performed efficiently within a fully differentiable flow solver. Using forced, two-dimensional turbulence as a test case, these new methods reveal an order of magnitude more solutions than has been possible using earlier approaches and converge periodic orbits where previous methods have been ineffective. An assessment will be made as to what the large set of new exact solutions says about the 'dynamical systems' exercise in general and the prospects for application at high Re.

This paper presents a novel investigation into the dynamics of a micro ring structure subjected to harmonic base excitation, designed as a highly sensitive MEMS mass sensor or bifurcation-based switch. Leveraging the in-plane nature of the motion, the system exhibits an exceptionally low damping ratio, making it ideal for detecting subtle changes in dynamic behaviour. The governing nonlinear differential equations, incorporating the geometric nonlinearities of the support beams, were derived and simplified into a reduced-order model consisting of coupled nonlinear Duffing-type equations. A key innovation of this study lies in the tunability of the system's frequency ratios, enabling the activation of a 1:3 internal resonance. By varying the length of the support beams while keeping the central ring geometry fixed, the first two natural frequencies were carefully examined, revealing a significant influence on the dynamic response. Frequency response curves confirmed the presence of 1:3 internal resonance near the primary resonance of the first mode, highlighting the potential for efficient energy transfer between modes. Furthermore, a detailed bifurcation analysis uncovered a range of complex nonlinear phenomena, including nonlinear modal interactions, torus bifurcations, quasi-periodic motion, and cyclic fold bifurcations. These bifurcations not only provide deeper insight into the system's dynamics but also offer additional operational mechanisms for switching applications. The findings demonstrate the system's capability to exploit nonlinear dynamics for enhanced sensitivity and robustness, paving the way for the development of next-generation MEMS sensors and bifurcation-based devices.

Partial differential equations, and their chaotic solutions, are pervasive in the modelling of complex systems in engineering, science, and beyond. Data-driven methods can find solutions to partial differential equations with a divide-and-conquer strategy: The solution is sought in a latent space, on which the temporal dynamics are inferred ("latent-space" approach). This is achieved by, first, compressing the data with an autoencoder, and, second, inferring the temporal dynamics with recurrent neural networks. The overarching goal of this paper is to show that a latent-space approach can not only infer the solution of a chaotic partial differential equation, but it can also predict the stability properties of the physical system. First, we employ the convolutional autoencoder echo state network (CAE-ESN) on the chaotic Kuramoto-Sivashinsky equation for various chaotic regimes. We show that the CAE-ESN (i) finds a low-dimensional latent-space representation of the observations and (ii) accurately infers the Lyapunov exponents and covariant Lyapunov vectors (CLVs) in this low-dimensional manifold for different attractors. Second, we extend the CAE-ESN to a turbulent flow, comparing the Lyapunov spectrum to estimates obtained from Jacobian-free methods. A latent-space approach based on the CAE-ESN effectively produces a latent space that preserves the key properties of the chaotic system, such as Lyapunov exponents and CLVs, thus retaining the geometric structure of the attractor. The latent-space approach based on the CAE-ESN is a reduced-order model that accurately predicts the dynamics of the chaotic system, or, alternatively, it can be used to infer stability properties of chaotic systems from data.

This research develops a novel geometrically nonlinear aeroelastic model of a cantilevered flexible wing capable of capturing unsteady aerodynamics with finite span effects. The structural model is developed using a Chebyshev-Ritz approach with the ONERA unsteady aerodynamic formulation coupled to a three-dimensional wing analysis based on the lifting line approach. The resulting system describing the evolution of the generalised structural coordinates and the aerodynamic states is formulated as a continuous state space model. The model is validated against experimental results from the wind tunnel testing of a highly flexible wing demonstrator at the University of Bristol. The numerical results are computed by applying a numerical continuation procedure, exercising the benefits of the state space formulation. The extensive numerical-experimental comparative analysis includes the bifurcation characteristics of the system and the behaviour of the aeroelastic modes. The developed model reflected the experimentally identified bifurcation behaviour. The predicted variations of the flutter onset speed with the wing root pitch angles were found to be within 5% of the experimental values. The model also correctly captured the post-flutter Limit Cycle Oscillation (LCO). This work further showed that the predicted flutter speeds are sensitive to the semi-empirical coefficients present in the ONERA unsteady aerodynamics formulation. Consequently, further calibration of the model was based on the analysis of the experimental flutter speeds and airspeed-driven eigenvalue variations. This approach was necessitated by the influence of application-specific conditions (e.g., the Reynolds number) on the unsteady aerodynamic characteristics. Despite this limitation, the tuned model agreed with the experimental observations across a wide range of test conditions. The approaches presented in this work can benefit multiple applications during research on geometrically nonlinear wings, such as the analysis of root causes influencing critical aeroelastic phenomena or design of aeroelastic control laws.

Functionalised nonlinear structures employing structural instabilities for rapid response shape-shifting are emerging technologies with a wide range of potential applications. The von Mises truss is a widely employed model for such functionalised nonlinear structures; however, few studies have delved into its functionality when integrated with a complex dynamical system. This paper investigates its efficacy on enhancing the progression speed of a vibration-driven robot, known as the vibro-impact capsule robot, which is a piecewise-smooth dynamical system having abundant coexisting attractors. Bifurcation analysis of the capsule robot integrated with a von Mises truss is conducted for this purpose. Our numerical studies focus on the influence of the frequency and amplitude of the robot's driving force on its progression. Specifically, we compare the periodic responses of both the capsule robots with and without the von Mises truss, utilising the numerical continuation techniques for piecewise-smooth dynamical systems. Our studies confirm the advantage of using the von Mises truss when the driving force of the robot is significantly small. Additionally, we identify an optimal operational regime on the amplitude-frequency control plane, where the maximum robot speed is achieved for a given amount of power consumption. The numerical studies presented in this work provide a promising indication of the advantages offered by the von Mises truss for vibration-driven robots. This research underscores the significant potential of functionalised nonlinear structures to enhance the efficiency of small-scale robots operating under power limitations.

This study develops a dynamics model of a microrobot vibrating in a blood vessel aiming to detect potential cancer metastasis. We derive an analytical solution for microrobot's motion, considering interactions with the vessel walls modelled by a linear spring-dashpot and a constant damping value for blood viscosity. The model facilitates instantaneous state transitions of the microrobot, such as contact with the vessel wall and free motion within the fluid. Amplitudes and phase angles from the transient solutions of dynamics model of the microrobot are solved at arbitrary moments, providing insights into its transient dynamics. The analytical solution of the proposed system is validated by experimental data, serving as a benchmark to examine the influence of pertinent parameters on microrobot's dynamic response. It is found that the contact force transmitted to the vessel wall, assessed by system's transmissibility function dependent on damping and frequency ratios, decreases with increasing damping ratio and intensifies when the frequency ratio is below $\sqrt{2}$ . At the frequency ratio is equal to 1, resonance phenomenon is dominated by the magnification factor linked to the damping ratio, increasing the amplitude of resonance as damping decreases. Finally, different sets of system parameters, including excitation frequency and magnitude, fluid damping, vessel wall's stiffness and damping, reveal multi-periodic motions and fake collision of the microrobot with the vessel wall. Simulation results imply that these phenomena are minimally affected by vessel wall's stiffness but are significantly influenced by other parameters, such as fluid damping coefficient and damping coefficient of the blood vessel wall. This research provides a robust theoretical foundation for developing control strategies for microrobots aimed at detecting cancer metastasis.

In confined environments, the propulsion speed of torque-driven helical untethered magnetic robots (UMRs) depends on vessel diameter and helical pitch. This phenomenon likely affects the swimming speed of these robots when navigating in vivo within blood vessels. Achieving a consistent swimming speed across a range of vessel diameters is therefore essential for precise control of UMRs. With this goal, we investigate how vessel diameter and helical pitch influence UMR propulsion to identify optimal robot designs. UMRs from two groups-fixed length (FL) and fixed wave (FW)-were analyzed. In the FW group, longer robots achieved higher speeds in larger vessels due to better vessel guidance, whereas shorter robots performed best in medium-sized vessels. The FL group showed peak speeds at intermediate normalized wavenumbers ( $\nu \approx 1$ to 1.8), significantly influenced by robot geometry and vessel confinement. Normalized wavenumber affected swimming speed and stability, with maximum efficiency at $\nu \approx 1$ and stability increasing with higher wavenumbers. Swimming performance was modeled using Stokeslets method and additional effects were considered by correction factors to improve agreement of experimental and modeling results. Ex vivo tests under physiological conditions demonstrated clear performance differences between robust (FL-9 ( $\nu =2.3$ ), FW-5 ( $\nu =2$ )) and unstable swimmers, with robust designs showing consistent correlation between phantom and ex vivo trials. Unstable designs exhibited unpredictable behavior, emphasizing the importance of specific design parameters for stability and maneuverability. These findings highlight critical design factors-such as the normalized wavenumber, body length, and degree of vessel confinement-that are essential for enhancing UMR propulsion efficiency and control consistency, which are fundamental for successful navigation in targeted vascular biomedical applications.

Supplementary information: The online version contains supplementary material available at 10.1007/s11071-025-11646-7.

Time series is a data structure prevalent in a wide range of fields such as healthcare, finance and meteorology. It goes without saying that analyzing time series data holds the key to gaining insight into our day-to-day observations. Among the vast spectrum of time series analysis, time series classification offers the unique opportunity to classify the sequences into their respective categories for the sake of automated detection. To this end, two types of mainstream approaches, recurrent neural networks and distance-based methods, have been commonly employed to address this specific problem. Despite their enormous success, methods like Long Short-Term Memory networks typically require high computational resources. It is largely as a consequence of the nature of backpropagation, driving the search for some backpropagation-free alternatives. Reservoir computing is an instance of recurrent neural networks that is known for its efficiency in processing time series sequences. Therefore, in this article, we will develop two reservoir computing based methods that can effectively deal with regular and irregular time series with minimal computational cost, both while achieving a desirable level of classification accuracy.

This paper introduces a multi-step, off-policy adaptive dynamic programming approach, in both model-free and model-based variants, intending to solve optimal control problems under disturbances and safety constraints. To provide a more accurate estimation of the performance function in the policy evaluation step, we employ an interleaved training method in the model-free scheme and utilize a prior model in the model-based version to mitigate the underestimation issue of the accumulated utility function. To further counteract the underestimation of the terminal performance function, dual critic neural networks are utilized. Additionally, to ensure a well-balanced trade-off between safety and performance requirements, the original unconstrained policy improvement process is transformed into a constrained optimization task with a far-sighted safety function. Furthermore, an actor-critic-disturbance framework is designed to handle safety constraints during the zero-sum game process, in which the disturbance policy and the performance function are alternately updated during the PEV step. Based on this, a rigorous theoretical analysis is conducted to evaluate the convergence property of the proposed method. Finally, simulation results and practical experiments demonstrate the effectiveness and safety of the proposed method.