Traditionally, applied mathematics and nonlinear analysis have focused on creating models of real-world systems using first principles. However, in many contemporary scientific domains, first-principles approaches face increasing limitations, including but not limited to the challenges posed by complex, high-dimensional, or poorly characterized systems. In response, data science leverages the availability of large datasets and computational power to study problems where empirical understanding is incomplete or the underlying mechanisms are only partially known. A combination of both paradigms is crucial when some data is available, but our understanding of the phenomenon remains limited. In this context, methods such as sparse identification of nonlinear dynamics (SINDy), a data-driven technique designed to discover nonlinear dynamical systems from empirical data using regularization methods, have proved to be successful in the study of multivariate time series and nonlinear dynamics. Sparse identification capitalizes on the observation that many natural phenomena can be described by systems with only a few nonlinear terms, yielding interpretable models. In this paper, we will discuss the effectiveness of sparse identification in accurately determining the nonlinear dynamics of systems such as the van der Pol equation and a coupled system of van der Pol oscillators, two systems often used as a major benchmark examples for testing new data-driven methods on systems with rich dynamics, also when chaotic behavior of solutions and synchronization are possible.

A gauge function for the van der Pol oscillator is constructed and used to perform stability analysis of this system as well as to determine its dynamical states without solving the corresponding equation of motion. Despite previous claims that the oscillator is a non-Lagrangian system, it is shown that the gauge function allows deriving its first Lagrangian. The presented results demonstrate that the gauge function is a new and powerful tool to study this system and that it gives deeper physical insights into the system’s nature than any tool previously used.

Flow-induced oscillations (FIO) are energy-rich hydrodynamic phenomena that can be exploited to harvest renewable energy from ocean and river currents. The hydrodynamics of tandem cylinders have recently gained attention in the literature, and this study investigates a hybrid energy harvesting system based on tandem cylinder configurations. The upstream-downstream wake interference is modeled through coupled van der Pol and wake oscillators, with particular emphasis on accurately capturing both vortex-induced vibration and galloping mechanisms. Three configurations, including piezoelectric (PZT-H), electromagnetic (EMT-H), and a new proposed hybrid piezoelectric-electromagnetic coupled with tandem cylinders (HEPT-H), are analyzed under varying spacing ratios and reduced velocities. Findings highlight that galloping is the dominant instability driving large-amplitude responses, and its proper modeling is critical for predicting and maximizing harvested energy. The proposed HEPT-H system takes advantage of this mechanism, nearly doubling the harvested power and improving efficiency by about 20% compared with single-harvester systems. A multi-criteria decision-making method (TOPSIS) was employed to rank the harvesters under different cylinder spacing configurations according to their relative closeness to the ideal solution. The HEPT-H system with a center-to-center cylinder spacing of four diameters indicated the best performance, achieving a maximum output of 0.071W and a peak efficiency of 69.78%. This research emphasizes the significant potential of HEPT-H systems in FIO and demonstrates that tandem configurations outperform isolated cylinders, underscoring their effectiveness for advancing sustainable hydrokinetic energy applications.

Mixed-mode Oscillations (MMOs), which are characterized by alternating patterns of comparatively small- and large-amplitude oscillations, can be observed in many nonlinear systems. Understanding MMOs is essential to research in the dynamics of chemical reactions, electronic circuits, and biological systems. In many studies, a certain type of MMO bifurcations, referred to as MMO-incrementing bifurcations, has been observed. These bifurcations exhibit a repeated motif, where two basic oscillations switch in a regular way. However, recent studies have shown that novel, more complex MMO structures exist. In particular, a new class of MMO patterns has been discovered in periodically forced systems, such as the Bonhoeffer–van der Pol (BVP) and van der Pol oscillators; these oscillations have been denoted as nested MMOs. Nested MMOs are distinct from previously known MMOs because they show a hierarchical or nested structure. In this nested structure, a sequence of MMOs appears inside another repeating sequence, forming a two-level pattern. The motivation of this study is to understand how these nested MMO structures form and how they are related to the more well-known unnested MMOs; and for this purpose, we consider a nonautonomous BVP oscillator. The second-order nested MMOs observed in this work are confirmed to occur more than 20 times in succession in the nonautonomous BVP dynamics, and it is observed that the value of the scaling constants that describe the dynamics of the system, which are defined in a similar manner to Feigenbaum’s constant, may converge to a value in the vicinity of unity. This discovery adds further depth to the theory of oscillatory dynamics and opens new directions for research in nonlinear science.

Influence of a Non-Ideal External Force on the Complexity of the Basins of Attraction of Coupled Van der Pol Oscillators

Phase resetting with temporal template explains complexity matching in finger tapping to fractal rhythms.

Abstract This paper presents a novel approach for estimating the Koopman operator defined on a reproducing kernel Hilbert space (RKHS) and its spectra. We propose an estimation method, what we call Jet Extended Dynamic Mode Decomposition (JetEDMD) , leveraging the intrinsic structure of RKHS and the geometric notion known as jets to enhance the estimation of the Koopman operator. This method refines the traditional Extended Dynamic Mode Decomposition (EDMD) in accuracy, especially in the numerical estimation of eigenvalues. This paper proves JetEDMD’s superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also investigate the spectral analysis of the Koopman operator, proposing the notion of an extended Koopman operator within a framework of a rigged Hilbert space. This notion leads to a deeper understanding of estimated Koopman eigenfunctions and capturing them outside the original function space. Through the theory of rigged Hilbert space, our study provides a principled methodology to analyse the estimated spectrum and eigenfunctions of Koopman operators, and enables eigendecomposition within a rigged RKHS. We also propose a new effective method for reconstructing the dynamical system from temporally-sampled trajectory data of the dynamical system with solid theoretical guarantee. We conduct several numerical simulations using the van der Pol oscillator, the Duffing oscillator, the Hénon map, and the Lorenz attractor, and illustrate the performance of JetEDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.

Incorporating force bounds is crucial for realistic control implementations in physical systems. Here, we investigate the fastest possible synchronization of a Liénard system to its limit cycle using a bounded external force. To tackle this challenging nonlinear optimal control problem, our approach involves applying Pontryagin’s maximum principle with a combination of analytical and numerical tools. We show that the optimal control develops a remarkably complex structure in phase space as the force bound is lowered. Trajectories rewound from the limit cycle’s extreme points turn out to play a key role in determining the maximum number of control bangs for optimal connection. We illustrate these intricate features using the paradigmatic van der Pol oscillator model.

Within the scope of the qualitative study of dynamical systems in the language of graph theory, we proceed to a generalization of the notion of structuralgraph to dynamical systems on monoids. This approach considers the actionof an ordered monoid on a metric space, which encompasses a broader rangeof dynamical systems than those written in terms of ordinary differential ordifference equations. We examine some essential properties of the structuralgraph, such as connectedness and degeneracy, and it is shown, for example,that in the finite case, the structural graph associated to a C1-vector field ona closed differential manifold is always connected. The paper also introducesthe new concept of DNA of a dynamical system, that is, roughly, the simplestgraph allowing the recovery of the corresponding phase portrait up to a continuous deformation. Topological equivalence and conjugacy have been set ina more general context and a quite simple example of calculation of the DNAInternational Journal of Applied MathematicsVolume 38 No. 9s 2025ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version)Received: August 08, 2025 895is given by the van der Pol oscillator. As a perspective, in addition to thenotion of 'physical' or 'hyperstructural' graphs, the last section suggests a newform of linearization of dynamical systems with a possible use in the study ofglobal qualitative properties such as structural stability

We introduce a quantum spin van der Pol (vdP) oscillator as a prototypical model of quantum spin-based limit-cycle oscillators, which coincides with the quantum optical vdP oscillator in the high-spin limit. The system is described as a noisy limit-cycle oscillator in the semiclassical regime at large spin numbers, exhibiting frequency entrainment to a periodic drive. Even in the smallest spin-1 case, mutual synchronization, Arnold tongues, and entanglement tongues in two dissipatively coupled oscillators, and collective synchronization in all-to-all coupled oscillators are clearly observed. The proposed quantum spin vdP oscillator will provide a useful platform for analyzing quantum spin synchronization.

This paper examines the application of self-sustained oscillator systems, particularly the van der Pol-Duffing oscillator, to understand the complex dynamics of Pleistocene glacial cycles. We investigate how asymmetric forcing (β ̸= 0) influences frequency locking and bifurcation behavior through geometric singular perturbation theory (GSPT) analysis and Poincar ́e return map construction. Our numerical results demonstrate that the van der Pol-Duffing oscillator possesses substantially larger regions of stable periodic behavior in parameter space compared to standard van der Pol oscillators. As asymmetry increases from β = 0.25 to β = 1.2, we observed progressive narrowing of Arnold tongue structures with most frequency locking regions requiring stronger forcing amplitudes (a ≥ 1.5) to initiate synchronization. However, remarkably resilient 2:1 frequency locking regions persist across all asymmetry levels. This provides a mathematical framework for explaining dominant frequency transitions observed in paleoclimate records, particularly the Mid-Pleistocene Transition from 41 kyr to 100 kyr glacial cycles (4-significance)

The analysis of complex engineering systems governed by nonlinear fractional differential equations (NFDEs) remains a significant challenge due to memory-dependent behaviour and strong nonlinearity. Traditional analytical and numerical techniques often fail to ensure convergence or stability in the fractional domain. This paper proposes a novel Hybrid Analytical Framework (HAF) that synergistically integrates the Homotopy Perturbation Method (HPM) and Adomian Decomposition Method (ADM) with a fractional residual correction mechanism to efficiently solve NFDEs arising in engineering dynamics. The hybridization enhances solution accuracy, accelerates convergence, and extends applicability to stiff and chaotic systems. A new Hybrid Analytical Convergence Theorem is established, guaranteeing the existence, uniqueness, and uniform convergence of the derived fractional series solution under Lipschitz continuity conditions. The theorem is rigorously validated using contraction mapping principles in a Banach space framework. The proposed technique is applied to benchmark fractional models, including the Duffing and Van der Pol oscillators, to demonstrate robustness and computational efficiency. Comparative error analysis and phase space evaluations confirm the superior precision of the HAF over existing fractional analytical methods. This framework contributes a mathematically consistent and computationally tractable tool for modeling nonlinear dynamics in modern engineering systems.

This study introduces a novel control strategy aimed at achieving projective synchronization in incommensurate fractional-order chaotic systems (IFOCS). The approach integrates the mathematical framework of fractional calculus with the recursive structure of the backstepping control technique. A key feature of the proposed method is the systematic use of the Mittag–Leffler function to verify stability at every step of the control design. By carefully constructing the error dynamics and proving their asymptotic convergence, the method guarantees the overall stability of the coupled system. In particular, stabilization of the error signals around the origin ensures perfect projective synchronization between the master and slave systems, even when these systems exhibit fundamentally different fractional-order chaotic behaviors. To illustrate the applicability of the method, the proposed fractional order backstepping control (FOBC) is implemented for the synchronization of two representative systems: the fractional-order Van der Pol oscillator and the fractional-order Rayleigh oscillator. These examples were deliberately chosen due to their structural differences, highlighting the robustness and versatility of the proposed approach. Extensive simulations are carried out under diverse initial conditions, confirming that the synchronization errors converge rapidly and remain stable in the presence of parameter variations and external disturbances. The results clearly demonstrate that the proposed FOBC strategy not only ensures precise synchronization but also provides resilience against uncertainties that typically challenge nonlinear chaotic systems. Overall, the work validates the effectiveness of FOBC as a powerful tool for managing complex dynamical behaviors in chaotic systems, opening the way for broader applications in engineering and science.

Classical self-sustained oscillators, which generate periodic motion without periodic external forcing, are ubiquitous in science and technology. The realization of nonclassical self-oscillators is an important goal of quantum physics. We here present the experimental implementation of a quantum van der Pol oscillator, a paradigmatic autonomous quantum driven-dissipative system with nonlinear damping, using a single trapped atom. We demonstrate the existence of a quantum limit cycle in phase space in the absence of a drive and the occurrence of quantum synchronization when the nonlinear oscillator is externally driven. We additionally show that synchronization can be enhanced with the help of squeezing perpendicular to the direction of the drive and, counterintuitively, linear dissipation. We also observe the bifurcation to a bistable phase-space distribution for large squeezing. Our results pave the way for the exploration of self-sustained quantum oscillators and their application to quantum technology.

It is shown that the coexistence of synchronous and asynchronous states, being typical for chimera states in large networks of coupled oscillators, can be formed in a minimal chain of two coupled van der Pol oscillators with dissipative delay coupling. This means that a stable limit cycle corresponding to synchronization and an attractive two-dimensional torus (quasiperiodicity) related to an incoherence regime of interacting oscillators coexist in the phase space at the same parameter values. Bistability is observed within the main synchronization region in the control parameter plane at the boundary between the locking and suppression regions. This phenomenon emerges as the delay time in the communication channel increases. The bifurcation mechanism of bistability formation is revealed and studied for a system of delay differential equations, its finite-dimensional model of ordinary differential equations, and by using the amplitude and phase approach.

Dynamics of a discontinuous electromechanical system excited by the Van der Pol oscillator and its microcontroller implementation

Purpose This research aims to enhance the modelling of cardiac rhythm dynamics through the integration of fractional derivatives into Van der Pol oscillators. By employing the Lucas wavelet method (LWM), the study effectively addresses the inherent nonlinearity of cardiac models, providing a more accurate representation of heartbeats, including relaxation, chaos, and bifurcations. The model’s adaptability with two adjustable parameters increases its practical applicability. The performance of LWM is validated through comparisons with the Runge-Kutta fourth-order method, highlighting its reliability. This approach offers potential advancements in computational techniques for understanding and diagnosing cardiovascular conditions. Design/methodology/approach This study employs the LWM to solve Van der Pol-like nonlinear second-order differential equations modelling cardiac rhythm dynamics. The approach incorporates fractional derivatives to capture memory effects and complex temporal dynamics of heartbeats. Operational matrices of Lucas wavelets are used to transform the differential equations into algebraic systems, which are solved numerically. The model’s flexibility is ensured through two adjustable parameters, enhancing its adaptability. The accuracy and reliability of LWM are validated by comparing its results with the Runge-Kutta fourth-order method, demonstrating superior performance in handling nonlinearity and fractional dynamics. Findings The study demonstrates that the LWM effectively approximates solutions for cardiac rhythm models governed by Van der Pol-like nonlinear differential equations. Incorporating fractional derivatives enhances the model’s ability to capture complex cardiac dynamics, including relaxation, chaos, and bifurcations. LWM shows high accuracy and reliability, with error metrics such as RMSE and MAE indicating strong consistency with the Runge-Kutta fourth-order method. Sensitivity analysis reveals that parameters like the pulse shape modification factor and asymmetric damping significantly influence cardiac behaviour. The results highlight LWM’s potential for advancing computational techniques in cardiovascular modelling. Research limitations/implications While the LWM effectively models cardiac rhythm dynamics, the study has limitations. The Van der Pol oscillator, being a phenomenological model, simplifies the heart’s complex physiological processes and does not account for spatial variations in electrical activity. Its ability to simulate pathological conditions like arrhythmias is limited, requiring more complex models for detailed clinical insights. Additionally, the absence of experimental validation restricts the model’s applicability in real-world scenarios. Despite these limitations, the research provides a strong foundation for future studies to enhance cardiac modelling using advanced wavelet-based and fractional-order techniques. Practical implications The study’s findings have significant practical implications for computational cardiology and biomedical engineering. The LWM offers an efficient and accurate approach to modelling cardiac rhythm dynamics, aiding in the analysis of heart conditions such as arrhythmias. Its ability to handle nonlinearities and fractional dynamics makes it suitable for simulating complex cardiac behaviours. The model’s simplicity, with adjustable parameters, enhances its adaptability for various clinical scenarios. This approach can support the development of diagnostic tools, improve the understanding of cardiac disorders, and assist in designing effective treatments and control strategies for cardiovascular diseases. Social implications This research contributes to improving public health by enhancing the understanding of cardiac rhythm disorders through advanced mathematical modelling. The LWM offers insights into heart dynamics, potentially aiding in early diagnosis and more effective management of cardiovascular diseases, which are leading causes of mortality worldwide. By supporting the development of reliable, non-invasive diagnostic tools, the study can help reduce healthcare costs and improve patient outcomes. Additionally, its applications in medical research and education can raise awareness about heart health, contributing to preventive measures and better cardiovascular care in society. Originality/value This research presents a novel approach by integrating fractional derivatives into Van der Pol-like cardiac rhythm models, offering a more comprehensive representation of heart dynamics, including memory effects and complex oscillatory behaviour. The application of the LWM is a key innovation, providing an efficient, accurate, and effective technique for solving nonlinear and fractional differential equations. Unlike traditional methods, LWM effectively handles system nonlinearity and enhances computational performance. The study’s originality lies in its combination of fractional calculus with wavelet-based spectral analysis, offering valuable insights for advancing mathematical modelling in cardiovascular research.

A time-domain semi-empirical simulation model based on the wake oscillator approach is developed to investigate the coupled in-line (IL) and cross-flow (CF) vortex-induced vibration (VIV) of a flexible riser in uniform oscillatory flow. A novel nondimensionalization method is introduced by utilizing the dimensionless parameter StKC, which effectively replicates the fundamental lift frequency caused by the complex vortex motion around the riser. The structural responses of the riser are described using the Euler–Bernoulli beam theory, and the van der Pol equations are used to calculate the fluid forces acting on the riser, which can replicate the nonlinear vortex dynamics. The coupled equations are discretized in both time and space with a finite difference method (FDM), enabling iterative computations of the VIV responses of the riser. A total of six cases are examined with four different Keulegan–Carpenter (KC) numbers (i.e., KC=31, 56, 121, and 178) to investigate the VIV characteristics of small-scale and large-scale risers in uniform oscillatory flow. Key features such as intermittent VIV, amplitude modulation, and hysteresis, as well as the VIV development process, are analyzed in detail. The simulation results show good agreement with the experimental data, indicating that the proposed numerical model is able to reliably reproduce the riser VIV in uniform oscillatory flow. Overall, the VIV characteristics of the large-scale riser resemble those of the small-scale riser but exhibit higher vibration modes, stronger traveling wave features, and more complex energy transfer mechanisms.

ABSTRACTThe equivalence problem in differential geometry investigates whether two geometric structures can be transformed into one another through an appropriate change of variables. This issue is central to understanding the invariants associated with dynamical systems, which remain unchanged under such transformations. The Cartan equivalence method provides a powerful and systematic framework for addressing these problems by reducing complex equivalence questions to more tractable ones involving coframes and structure equations. In this study, we apply Cartan's method to second‐order autonomous ordinary differential equations (ODEs), offering insights into the intrinsic geometric structure of these systems under coordinate transformations. Specifically, we examine a class of second‐order ODEs that model nonlinear dynamical behavior, employing the pseudogroup of web transformations to identify differential invariants and symmetry classes. The study introduces a step‐by‐step construction of an invariant coframe, elucidates the structure equations governing the system, and provides a classification of the equations based on their differential invariants. Two representative physical systems are investigated in detail: the motion of a damped spring‐mass system and the van der Pol oscillator. In addition to these applications, the paper explores the interplay between equivalence, symmetry reduction, and integrability. By leveraging Cartan's method, we aim to provide a deeper understanding of the equivalence problem in the context of higher order differential equations, highlighting both the theoretical elegance and practical utility of the approach in uncovering hidden geometric features, symmetries, and conserved quantities.

In this integrative review paper, we explore the parallels between the physical models of gyrotrons and some equations used in diverse and broad scientific fields. These include Adler’s famous equation, Van der Pol equation, the Lotka–Volterra equations and the Kuramoto model. The paper is written in the form of a pedagogical discourse and aims to provide additional insights into gyrotron physics through analogies and parallels to theoretical approaches used in other fields of research. For the first time, reachability analysis is used in the context of gyrotron physics as a modern tool for understanding the behavior of nonlinear dynamical systems.