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Numerical continuation tools are nowadays standard to analyse nonlinear dynamical systems by numerical means. These powerful methods are unfortunately not available in real experiments without having access to an accurate mathematical model. Implementing such a concept in real world experiments using control and data processing to track unstable states and their bifurcations, requires robust control techniques with large basins and good global properties. Here we propose design principles for control techniques for periodic states which lead to large basins and which are robust, without the need to have access to a detailed mathematical model. Our analytic considerations for the control design will be based on weakly nonlinear analysis of periodically driven oscillator systems. We then demonstrate by numerical means that in strong nonlinear regimes successful control with large basins of attraction can be achieved when only plain time series data are available. [ABSTRACT FROM AUTHOR]

We provide multiplicity results for the periodic problem associated with Hamiltonian systems coupling a system having a Poincaré–Birkhoff twist-type structure with a system presenting some asymmetric nonlinearities, with possible one-sided superlinear growth. We investigate nonresonance, simple resonance and double resonance situations, by implementing some kind of Landesman–Lazer conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a problem of periodic solution for a second order nonautonomous differential equation with a discontinuity line. Under two types of assumptions, we analyze the geometric properties of solutions respectively. Then by using a fixed point theorem, we obtain several existence criteria of periodic solutions, where the periodic solutions are crossing harmonic and subharmonic solutions. [ABSTRACT FROM AUTHOR]

We deal with the following system of coupled asymmetric oscillators x ¨ 1 + a 1 x 1 + - b 1 x 1 - + ϕ 1 (x 2) = p 1 (t) x ¨ 2 + a 2 x 2 + - b 2 x 2 - + ϕ 2 (x 1) = p 2 (t) , where ϕ i : R → R is locally Lipschitz continuous and bounded, p i : R → R is continuous and 2 π -periodic and the positive real numbers a i , b i satisfy 1 a i + 1 b i = 2 n , for some n ∈ N. We define a suitable function L : T 2 → R 2 , appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincaré map, in action-angle coordinates. [ABSTRACT FROM AUTHOR]

The mechanical horizontal platform (MHP) system exhibits a rich chaotic behavior. The chaotic MHP system has applications in the earthquake and offshore industries. This article proposes a robust adaptive continuous control (RACC) algorithm. It investigates the control and synchronization of chaos in the uncertain MHP system with time-delay in the presence of unknown state-dependent and time-dependent disturbances. The closed-loop system contains most of the nonlinear terms that enhance the complexity of the dynamical system; it improves the efficiency of the closed-loop. The proposed RACC approach (a) accomplishes faster convergence of the perturbed state variables (synchronization errors) to the desired steady-state, (b) eradicates the effect of unknown state-dependent and time-dependent disturbances, and (c) suppresses undesirable chattering in the feedback control inputs. This paper describes a detailed closed-loop stability analysis based on the Lyapunov-Krasovskii functional theory and Lyapunov stability technique. It provides parameter adaptation laws that confirm the convergence of the uncertain parameters to some constant values. The computer simulation results endorse the theoretical findings and provide a comparative performance. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations x ˙ = y , y ˙ = - x - ε · (f (x) · y + sgn (y) · g (x)) . Using the averaging method, we were able to generalize a previous result for Liénard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of | ε | , the number h m , n = n 2 + m 2 + 1 serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center x ˙ = y , y ˙ = - x . Furthermore, we demonstrate that it is indeed possible to obtain a system with h m , n limit cycles. [ABSTRACT FROM AUTHOR]

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component. [ABSTRACT FROM AUTHOR]

This paper studies an impact of media epidemic system with diffusion and linear source. We first derive the uniform bounds of solutions to impact on media reaction diffusion system. Then, the basic reproduction number is calculated and the threshold dynamics of impact media reaction diffusion system is also given and the Kuratowski measure κ of non-compactness is also considered. In addition, assume the spatial environment is homogeneous, it is shown that the unique endemic equilibrium of the system is global stability by constructing suitable Lyapunov function. Finally, we discuss the asymptotic profile of the system when the diffusion rate of the susceptible (infected) individuals for the system tends to zero or infinity. The main results show that the activities of infected individuals can only be at low risk, and then the virus eventually will be extinct, that is, to control the entry of viruses from abroad and increase the detection of domestic viruses. Finally, some numerical simulations are worked out to confirm the results obtained in this paper. [ABSTRACT FROM AUTHOR]

We consider homogeneous coupled cell networks with asymmetric inputs. We obtain general results concerning codimension-one steady-state bifurcations for networks with any number of cells and any number of asymmetric inputs. These results rely solely on the network adjacency matrices eigenvalue structure and the existence, or not, of network synchrony subspaces. For networks with three cells, we describe the possible lattices of synchrony subspaces annotated with the eigenvalues on each synchrony subspace. Applying the previous results, we classify the synchrony-breaking steady-state bifurcations that can occur for three-cell minimal networks with one, two or six asymmetric inputs. [ABSTRACT FROM AUTHOR]

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators. [ABSTRACT FROM AUTHOR]

An easily stackable multi-layer quasi-zero-stiffness (ML-QZS) meta-structure is proposed to achieve highly efficient vibration isolation performance at low frequency. First, the distributed shape optimization method is used to design the unit cel, i.e., the single-layer QZS (SL-QZS) meta-structure. Second, the stiffness feature of the unit cell is investigated and verified through static experiments. Third, the unit cells are stacked one by one along the direction of vibration isolation, and thus the ML-QZS meta-structure is constructed. Fourth, the dynamic modeling of the ML-QZS vibration isolation meta-structure is conducted, and the dynamic responses are obtained from the equations of motion, and verified by finite element (FE) simulations. Finally, a prototype of the ML-QZS vibration isolation meta-structure is fabricated by additive manufacturing, and the vibration isolation performance is evaluated experimentally. The results show that the vibration isolation performance substantially enhances when the number of unit cells increases. More importantly, the ML-QZS meta-structure can be easily extended in the direction of vibration isolation when the unit cells are properly stacked. Hence, the ML-FQZS vibration isolation meta-structure should be a fascinating solution for highly efficient vibration isolation performance at low frequency. [ABSTRACT FROM AUTHOR]

Modal analysis is a widely applied method to study the vibration phenomenon of continuum structures, but there is no clear method to solve the modal truncation problem at present. To determine the contribution of different modes to the whole system, a new mode truncation method based on perturbation theory is proposed in this paper. The modes are subjected to perturbation parameters during discretization, and using norm error analysis on the stiffness matrix in different degrees of freedom (DOFs) systems confirms the model number of the continuum structure system. The results show that the DOF identified by the modal perturbation method is related to the perturbation parameter, and the smaller the perturbation parameter is, the fewer modes need to be considered. When the perturbation parameter is large enough, the response of the system can only be accurately explained by truncation to higher-order modes. Finally, the perturbation parameter is fixed to 1, and the traditional Galerkin method is connected to the modal perturbation, making traditional discretization a unique case for the modal perturbation method. This method can significantly reduce the modal truncation error, which is of great significance to the dynamic analysis of engineering applications. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order α ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits α to 0 or α to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation. [ABSTRACT FROM AUTHOR]

The Riccati equation method is used to establish some global solvability criteria for a classes of Lane–Emdem–Fowler and Van der Pol type equations. Two oscillation theorems are proved. The results obtained are applied to the Emden–Fowler equation and to the Van der Pol type equation. [ABSTRACT FROM AUTHOR]

We study an n-dimensional system of ordinary differential equations with a constant matrix, a relay-type nonlinearity, and an external disturbance in the right-hand side. We consider a nonideal relay characteristic. The external disturbance is described by the product of an exponential function and a sine function with an initial phase as a parameter. We assume the matrix of the linear part and the vector at the relay characteristic such that, by a nonsingular transformation, the system is reduced to the form with the diagonal matrix and the vector being opposite to the unit vector. We establish a necessary and sufficient condition for the existence of two-point oscillatory solutions, i.e., the solutions with two fixed points on the hyperplanes of the relay switching in phase space. Also, we give the sufficient conditions under which such solutions do not exist. We provide a supporting example, which demonstrates how to apply the obtained results. [ABSTRACT FROM AUTHOR]

We study pairs of symmetrically coupled, identical Lengyel-Epstein oscillators, where the coupling can be through both the fast and slow variables. We find a plethora of strong symmetry breaking rhythms, in which the two oscillators exhibit qualitatively different oscillations, and their amplitudes differ by as much as an order of magnitude. Analysis of the folded singularities in the coupled system shows that a key folded node, located off the symmetry axis, is the primary mechanism responsible for the strong symmetry breaking. Passage through the neighborhood of this folded node can result in splitting between the amplitudes of the oscillators, in which one is constrained to remain of small amplitude, while the other makes a large-amplitude oscillation or a mixed-mode oscillation. The analysis also reveals an organizing center in parameter space, where the system undergoes an asymmetric canard explosion, in which one oscillator exhibits a sequence of limit cycle canards, over an interval of parameter values centered at the explosion point, while the other oscillator executes small amplitude oscillations. Other folded singularities can also impact properties of the strong symmetry breaking rhythms. We contrast these strong symmetry breaking rhythms with asymmetric rhythms that are close to symmetric states, such as in-phase or anti-phase oscillations. In addition to the symmetry breaking rhythms, we also find an explosion of anti-phase limit cycle canards, which mediates the transition from small-amplitude, anti-phase oscillations to large-amplitude, anti-phase oscillations. [ABSTRACT FROM AUTHOR]

In this paper we study the periodic orbits of a class of 2n-dimensional control dynamical systems with a perturbation of continuous piecewise smooth quadratic polynomial and a perturbation of discontinuous piecewise smooth polynomial of an arbitrarily given degree, respectively. By applying the averaging theory for continuous and discontinuous piecewise smooth systems, the number of isolated zeros of the averaging function can be determined, which provides a lower bound of the maximum number of isolated periodic orbits. At last, we give examples to show that the lower bound of the maximum number is accessible by the properties of algebraic closure and transcendental extension. [ABSTRACT FROM AUTHOR]

This study proposes a module designed for the automatic parameter control of peristaltic locomotion systems. The module comprises two populations of oscillators, each responsible for controlling the phase of periodic elongation–contraction motion and the friction force exerted by the ground, respectively. The parameter control algorithm operates independently in each module, with parameters updated through a selection algorithm applied to the elements in each module. Peristaltic locomotion systems, equipped with the proposed modules in the body segments, exhibit autonomous parameter controls, resulting in stable unidirectional locomotion. Consequently, the system can adapt to various environmental changes. Because the modules do not interact with each other, the system acts as a decentralized distributed system; thus, it is scalable to several body segments. [ABSTRACT FROM AUTHOR]

We show that a recently-derived model for the propagation of nonlinear waves in the atmosphere admits undular bores as travelling-wave solutions. These solutions represent waves consisting of a damped oscillation behind a front that is preceded by a uniform breeze-type flow. The generation of such wave profiles requires a jump in the heat source across the leading front of the wave, a feature that is consistent with observations. [ABSTRACT FROM AUTHOR]

A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix with a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian network on a ring with some extra couplings. [ABSTRACT FROM AUTHOR]

In this paper, some resonance behaviors and stability for the solutions of the Duffing-van der Pol system under multi-frequency excitation are studied. First, the first-order asymptotic solution of the system under resonance circumstance is obtained by the method of multiple scales, the approximation sketch is compared with the numerical solution of the system, and the result indicates the validity of the asymptotic approximate solution. Then, the amplitude-frequency equation of steady-state response is derived. The amplitude-frequency equation of the system shows phenomenon of multiple solutions. Through numerical simulation, the influences of the Duffing parameter, van der Pol parameter and external force amplitude on the dynamic behavior and stability of the system were studied. Compared to ones of the Duffing system or van der Pol system, the amplitude-frequency relationship of the Duffing-van der Pol system is more complex. Finally, based on Lyapunov stability theory, the stability condition for a steady-state solution involving system parameters in case of the combination resonance is obtained. The research achievements provide a theoretical basis for the analysis and control design of such systems in engineering. [ABSTRACT FROM AUTHOR]

Since Littlewood works in the 1960s, the boundedness of solutions of Duffing-type equations x ¨ + g (x) = p (t) has been extensively investigated. More recently, some researches have focused on the family of non-smooth forced oscillators x ¨ + sign (x) = p (t) , mainly because it represents a simple limit scenario of Duffing-type equations for when g is bounded. Here, we provide a simple proof for the boundedness of solutions of the non-smooth forced oscillator in the case that the forcing term p(t) is a T-periodic Lebesgue-integrable function with vanishing average. We reach this result by constructing a sequence of invariant tori whose union of their interiors covers all the (t , x , x ˙) -space, (t , x , x ˙) ∈ S 1 × R 2 . [ABSTRACT FROM AUTHOR]

The presence or absence of synaptic plasticity can dramatically influence the collective behavior of populations of coupled neurons. In this work, we consider spike-timing dependent plasticity (STDP) and its resulting influence on phase cohesion in computational models of heterogeneous populations of conductance-based neurons. STDP allows for the influence of individual synapses to change over time, strengthening or weakening depending on the relative timing of the relevant action potentials. Using phase reduction techniques, we derive an upper bound on the critical coupling strength required to retain phase cohesion for a network of synaptically coupled, heterogeneous neurons with STDP. We find that including STDP can significantly alter phase cohesion as compared to a network with static synaptic connections. Analytical results are validated numerically. Our analysis highlights the importance of the relative ordering of action potentials emitted in a population of tonically firing neurons and demonstrates that order switching can degrade the synchronizing influence of coupling when STDP is considered. [ABSTRACT FROM AUTHOR]

In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes. [ABSTRACT FROM AUTHOR]

Numerical continuation tools are nowadays standard methods for the bifurcation analysis of dynamical systems. Unfortunately, the full power of these methods is still unavailable in experiments, in particular, if no underlying mathematical model is at hand. We here aim to narrow this gap by providing control based continuation of periodic states which can be ultimately implemented in real-world experimental set-ups. Taking inspiration from atomic force microscopy, we develop experimentally relevant control and tracking tools for time periodic solutions in driven nonlinear oscillator systems based on stroboscopic maps. [ABSTRACT FROM AUTHOR]

In recent papers by the authors (Motonaga and Yagasaki, Arch. Ration. Mech. Anal. 247:44 (2023), and Yagasaki, J. Nonlinear Sci. 32:43 (2022)), two different techniques which allow us to prove the real-analytic or complex-meromorphic nonintegrability of forced nonlinear oscillators having the form of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems were provided. Here the concept of nonintegrability in the Bogoyavlenskij sense is adopted and the first integrals and commutative vector fields are also required to depend real-analytically or complex-meromorphically on the small parameter. In this paper we review the theories and continue to demonstrate their usefulness. In particular, we consider the periodically forced damped pendulum, which provides an especially important differential equation not only in dynamical systems and mechanics but also in other fields such as mechanical and electrical engineering and robotics, and prove its nonintegrability in the above meaning. [ABSTRACT FROM AUTHOR]

The paper focuses on the problem on monochromatic electromagnetic waves propagation in a shielded planar dielectric waveguide. The waveguide is filled with nonhomogeneous nonlinear medium. The nonlinearity is expressed by nonnegative unbounded monotonically increasing function with power growth. Such nonlinearity is a generalization of the well-known Kerr nonlinear law. The existence of propagation constants and eigenwaves is proved. Besides, it is proved that the studied problem has nonlinearized solutions as well as linearized ones. [ABSTRACT FROM AUTHOR]

In this work, we present a system composed of three identical Duffing oscillators coupled bidirectionally. Starting from a Lagrangian that describes the system, an integral of motion is obtained by means of Noether's theorem. The dynamics of the model is studied using bifurcation diagrams, Lyapunov exponents, time-series analysis, phase spaces, Poincaré sections, spatiotemporal and integral of motion planes. The analysis focuses on the monostable and bistable cases of the Duffing oscillator potential, in which a confined movement is guaranteed. In particular, it is observed that the system shows a chaotic behavior for small values of the coupling parameter for the bistable case. This is one of the first articles in the literature in which non-trivial integrals of motion are obtained for a system of three Duffing oscillators coupled bidirectionally. It is worth pointing out that there are some reports in the literature on integrals of motion for unidirectionally coupled nonlinear Duffing oscillators, but the study carried out in this work for bidirectionally coupled systems with more than two nonlinear Duffing oscillators is certainly one of the first. [ABSTRACT FROM AUTHOR]

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov–Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity. [ABSTRACT FROM AUTHOR]

A novel vibration absorber is designed to suppress vibrations in fluid-conveying pipes subject to varying fluid speeds. The proposed absorber combines the fundamental principles of nonlinear energy sinks (NESs) and nonlinear energy harvesters (NEHs). The governing equation is derived, and a second-order discrete system is used to assess the performance of the developed device. The results demonstrate that the proposed absorber achieves significantly enhanced energy dissipation efficiency, reaching up to 95%, over a wider frequency range. Additionally, it successfully harvests additional electric energy. This research establishes a promising avenue for the development of new nonlinear devices aimed at suppressing fluid-conveying pipe vibrations across a broad frequency spectrum. [ABSTRACT FROM AUTHOR]

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems. [ABSTRACT FROM AUTHOR]

The continuum limit provides a useful tool for analyzing coupled oscillator networks. Recently, Medvedev (Commun Math Sci 17(4):883–898, 2019) gave a mathematical foundation for such an approach when the networks are defined on single graphs which may be dense or sparse, directed or undirected, and deterministic or random. In this paper, we consider coupled oscillator networks depending on multiple graphs, and extend his results to show that the continuum limit is also valid in this situation. Specifically, we prove that the initial value problem (IVP) of the corresponding continuum limit has a unique solution under general conditions and that the solution becomes the limit of those to the IVP of the networks in some adequate meaning. Moreover, we show that if solutions to the networks are stable or asymptotically stable when the node number is sufficiently large, then so are the corresponding solutions to the continuum limit, and that if solutions to the continuum limit are asymptotically stable, then so are the corresponding solutions to the networks in some weak meaning as the node number tends to infinity. These results can also be applied to coupled oscillator networks with multiple frequencies by regarding the frequencies as a weight matrix of another graph. We illustrate the theory for three variants of the Kuramoto model along with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study bifurcations of double grazing periodic orbits, namely periodic orbits with two grazing points, in a one-degree-of-freedom impact oscillator constrained by two rigid blocks. By applying the method of Poincaré discontinuity mapping, we approximate the normal Poincaré map near the double grazing orbit to the leading order, which is a two-dimensional piecewise smooth map defined in four regions that has square root singularities. Based on the Poincaré map, the stability and co-dimension-two bifurcation conditions of the double grazing orbit are presented. Furthermore, the results of computing Monte Carlo bifurcation diagrams of the map are compared with those from direct simulations of the original system, and we find that they are in excellent agreement. Through numerical simulations, we find that the system undergoes very complicated bifurcations near the double grazing periodic orbit, such as the period-adding and period-incrementing cascades. Hysteresis effects are also discussed. [ABSTRACT FROM AUTHOR]

The Kuramoto model was developed to describe the coupling of oscillators, motivated by the natural synchronization phenomena. We are interested in modeling an epileptic seizure considering it as the synchronization of action potentials using and modifying this model. In this article, we propose to modify this model, changing the constant coupling force for a function with logistic growth to simulate the onset and epileptic seizure level in an adult male rat caused by the administration of lithium–pilocarpine. Later, we select some frequencies and their respective amplitude values using an algorithm based on the fast Fourier transform (FFT) from an electroencephalography signal when the rat is in basal conditions. Then, we take these values as the natural frequencies of the oscillators in the modified Kuramoto model, considering every oscillator as a single neuron to simulate the emergence of an epileptic seizure numerically by increasing the synchronization value in the coupling function. Finally, using Dynamic Time Warping algorithm, we compare the simulated signal by the Kuramoto model with an FFT approximation of the epileptic seizure. [ABSTRACT FROM AUTHOR]

This article is devoted to studying a class of second order nonlinear dynamic equations on time scales with unbounded neutral coefficients. We give some sufficient conditions for the oscillations of solutions for the investigated dynamic equations. The main tool to prove our main results is a generalized Riccati transformation. Our results are provided with examples. [ABSTRACT FROM AUTHOR]

In order to analyze the motion characteristics of the spring pendulum under the action of magnetic field force, the motion of the spring pendulum will be studied by applying a uniform magnetic field in the vertical direction. Firstly, a first-order approximate solution is given by studying the micro-vibration around its equilibrium point. And an approximate solution similar to the Foucault pendulum is also presented in the case of a soft spring with strong ductility. Then, according to the resonance conditions of mechanical vibration, the internal resonance phenomenon of magnetic spring pendulum is discovered, and then the conclusion that the energy of the system is cyclically transmitted between the three modes of breathing, oscillating and deflection is presented subsequently. Finally, the influence of magnetic field strength on the motion stability of the spring pendulum is explored, and not only the bifurcation phenomenon at its equilibrium point is found, but also the complex dynamic behavior including chaotic motion occurs. [ABSTRACT FROM AUTHOR]

We study the oscillator x ¨ + n 2 x + h (x) = p (t) , where h is a piecewise linear saturation function and p is a continuous 2 π -periodic forcing. It is shown that there is recurrence if and only if p satisfies the Lazer–Leach condition. This condition relates the n-th Fourier coefficient of p(t) with the maximum of h and was first introduced to characterize the existence of periodic solutions. [ABSTRACT FROM AUTHOR]

We consider the temporal asymptotic behavior of the all-to-all coupled Kuramoto model with inertia and time-periodic natural frequencies. Due to the inertial effect, there are three cases of the dynamical ensemble with respect to the coupling strength; large coupling, near boundary, and small coupling. For each case, we present the asymptotic behavior of the solution to the inertial Kuramoto model with periodic natural frequencies: the solutions commonly consist of a macroscopic phase, a mean-centered-periodic solution, and an exponential decay term. The macroscopic phase is a drift-type term determined by initial data and natural frequencies, and the mean-centered-periodic solution is a standing wave independent of initial data. We provide sufficient conditions for the existence of a mean-centered-periodic solution with a time-periodic phase difference between nodes for each case and its exponential stability. We also provide several simulations to confirm our mathematical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods. [ABSTRACT FROM AUTHOR]

The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management (PHM). However, most data-driven models are still black-box models that cannot be interpreted. In this study, we use the neural ordinary differential equations (ODEs), especially the inherent computational relationships of a system added to the loss function calculation, to approximate the governing equations. In addition, a new strategy for identifying the local parameters of the system is investigated, which can be utilized for system parameter identification and damage detection. The numerical and experimental examples presented in the paper demonstrate that the strategy has high accuracy and good local parameter identification. Moreover, the proposed method has the advantage of being interpretable. It can directly approximate the underlying governing dynamics and be a worthwhile strategy for system identification and PHM. [ABSTRACT FROM AUTHOR]

In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality's weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero. [ABSTRACT FROM AUTHOR]

This paper focuses on the vector fields on the quasi-periodically forced circle flow φ ˙ = ρ + f (φ , θ) , θ ˙ = α where the forcing term f is real analytic in its arguments and small enough, the frequency vector α ∈ R d (d ≥ 2) is beyond Brjuno frequency. We prove that the flow above is reducible analytically provided fibered rotation number ρ f : = ρ (ρ + f (φ , θ)) is Diophantine with respect to the base frequency α. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite-dimensional systems with multi-dimensional frequency weaker than Brjuno frequency. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the oscillation of second-order half-linear differential equations whose coefficients are given by the products of (typically bounded) functions and power functions. Applying the generalized Riccati technique, we find a very general oscillation criterion for the studied equations. This criterion covers several half-linear equations with unbounded non-power parts of coefficients, but it is new also for linear equations with bounded non-power parts. This fact is appropriately documented by new corollaries. [ABSTRACT FROM AUTHOR]

For the 1+5-body problem, we study the relationship between the masses of 5 satellites and given symmetric configurations, where the symmetry axis contains one satellite. Under certain assumptions, we find analytically some central configurations for suitable positive masses. Also, we prove that for certain symmetric configuration of these satellites, there exists a one-parameter family of mass vectors for which such configuration is a central configuration. Furthermore, we present some numerical results for configurations and derive the positive masses for these satellites, such that these configurations are central configurations. [ABSTRACT FROM AUTHOR]

About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique. [ABSTRACT FROM AUTHOR]

We deal with a weakly coupled system of ODEs of the type x j ′ ′ + n j 2 x j + h j (x 1 , ... , x d) = p j (t) , j = 1 , ... , d , with h j locally Lipschitz continuous and bounded, p j continuous and 2 π -periodic, n j ∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h 1 , ... , h d are assumed. [ABSTRACT FROM AUTHOR]

In this study, we proved that damped quadratic nonlinear oscillators similar to Duffing and Helmholtz–Duffing damped equations which emerge in bubble dynamics with time-periodic straining flows and solitary-like wave's dynamics may be derived from a new functional approach based on nonstandard Lagrangians and fractional frictions. The solutions of these equations are given in terms of the Jacobi elliptic functions. It was observed that the dynamical model constructed in this study is comparable to dynamical systems with natural Lagrangians for which the Riemann structure is conformally flat which has important implications in dynamical systems with position-dependent mass. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the oscillatory properties of two fourth order differential equations in dependence on boundary behavior of its coefficients at infinity. These properties are established based on two-sided estimates of the least constant of a certain weighted differential inequality. [ABSTRACT FROM AUTHOR]

The research presented in this paper compares the occurrence of limit cycles under different bifurcation mechanisms in a simple system of two-dimensional autonomous predator–prey ODEs. Surprisingly two unconventional approaches, for a singular system and for a system with a center, turn out to produce more limit cycles than the traditional Andronov–Hopf bifurcation. The system has a functional response function which is a monotonically increasing cubic function of x for 0 ≤ x ≤ 1 where x represents the prey density, and which is constant for x > 1 . It acts as a proxy for investigating more general systems. The following results are obtained. For the Andronov–Hopf bifurcation the highest order of the weak focus is 2 and at most 2 small-amplitude limit cycles can be created. In the center bifurcation cases are shown to exist with at least 3 limit cycles. In the singular perturbation cases are shown to exist with at least 4 limit cycles and in some cases an exact upper bound of 2 limit cycles is obtained. Finally we indicate how the conclusions can be extended to more general systems. We show how an arbitrary number of limit cycles can be created by choosing an appropriate functional response function and growth function for the prey. One special situation is the system with group defense: the three bifurcation mechanisms typically produce less limit cycles if a group defense element is included. [ABSTRACT FROM AUTHOR]

Mycobacterium tuberculosis infection features various disease outcomes: clearance, latency, active disease, and latent tuberculosis infection (LTBI) reactivation. Identifying the decisive factors for disease outcomes and progression is crucial to elucidate the macrophages-tuberculosis interaction and provide insights into therapeutic strategies. To achieve this goal, we first model the disease progression as a dynamical shift among different disease outcomes, which are characterized by various steady states of bacterial concentration. The causal mechanisms of steady-state transitions can be the occurrence of transcritical and saddle-node bifurcations, which are induced by slowly changing parameters. Transcritical bifurcation, occurring when the basic reproduction number equals to one, determines whether the infection clears or spreads. Saddle-node bifurcation is the key mechanism to create and destroy steady states. Based on these two steady-state transition mechanisms, we carry out two sample-based sensitivity analyses on transcritical bifurcation conditions and saddle-node bifurcation conditions. The sensitivity analysis results suggest that the macrophage apoptosis rate is the most significant factor affecting the transition in disease outcomes. This result agrees with the discovery that the programmed cell death (apoptosis) plays a unique role in the complex microorganism-host interplay. Sensitivity analysis narrows down the parameters of interest, but cannot answer how these parameters influence the model outcomes. To do this, we employ bifurcation analysis and numerical simulation to unfold various disease outcomes induced by the variation of macrophage apoptosis rate. Our findings support the hypothesis that the regulation mechanism of macrophage apoptosis affects the host immunity against tuberculosis infection and tuberculosis virulence. Moreover, our mathematical results suggest that new treatments and/or vaccines that regulate macrophage apoptosis in combination with weakening bacillary viability and/or promoting adaptive immunity could have therapeutic value. [ABSTRACT FROM AUTHOR]