Unstable Periodic Solutions Research Articles

The Sprott B system.

We will consider a thermostatic system, Sprott B, that is a generalization of the well-known one-parameter Sprott A system. Sprott B contains an explicit periodic solution for all positive values of the parameter a. As for Sprott A, we find dissipative KAM tori associated with time-reversal symmetry and canards in dissipative systems. The exact periodic solution is characterized by an infinite number of instability intervals of the parameter. The investigation of the dynamics in these intervals shows the presence of families of stable and unstable periodic solutions, tori, and strange attractors. For large values of the control parameter a, we find non-hyperbolic slow manifolds producing violent vibrations. We discuss a generalization of the Sprott B system with related dynamics.

A general and efficient harmonic balance method for nonlinear dynamic simulation

This paper presents a general and efficient harmonic balance (GE-HB) method to expeditiously evaluate the periodic responses of nonlinear systems. By employing the fast Fourier transform (FFT) technique, it rapidly computes harmonic expansion coefficients of the nonlinear terms and their corresponding constant tensor matrices. Notably, the nonlinear part of the Jacobian matrix is just the product of the coefficients and the constant tensor matrices, loop computations are no longer requisite for calculating the Jacobian matrix. This, facilitates accelerated determination of the nonlinear portion of the Jacobian matrix, thus obtaining the periodic responses of the system in an efficient manner. Furthermore, the constant tensor matrices are exclusively contingent upon the harmonic bases of the system's periodic response, devoid of reliance on the equation's specific formulation, facilitates obtaining the Jacobian matrix in a general manner. Simulation results on a 20-degree-of-freedom nonlinear oscillator chain system demonstrate that the efficiency of the GE-HB method is about tenfold that of the harmonic balance-alternating frequency/time domain (HB-AFT) method and twenty-fivefold that of the fourth-order Runge-Kutta (RK4) method. Further, the simulation of a 284-degree-of-freedom nonlinear rotor system exhibits the GE-HB method's computational efficiency surpassing HB-AFT method by over 500 times and Newmark method by over 3000 times. Moreover, in these two examples, the GE-HB method captures all solution branches, including unstable periodic solutions, providing comprehensive insights into periodic responses of the systems. The GE-HB method proposed here has the potential to explore dynamic characteristics for high-dimensional complex nonlinear systems in engineering.

Supercritical Hopf bifurcations in the stage-structured model of housefly populations

Insect populations, which are diverse and widespread, provide a principal area of utilization of the stage-structured modeling approach. In this paper, housefly populations incorporating a stage-structured model are investigated theoretically and graphically. First, stability charts and rightmost characteristic roots of the positive equilibrium are elucidated analytically and numerically. Furthermore, the Hopf bifurcation at the positive equilibrium is derived employing geometric stability switch criterion. Second, the properties of Hopf bifurcation are determined using the center manifold theorem and by reducing the equation to the Poincaré normal form. Finally, the correctness of the theoretical derivation is confirmed using a numerical simulation based on specific parameter values. Our results show that with an increase in delay [Formula: see text], the unique positive equilibrium may undergo two stability switches: from stable to unstable, and from unstable to stable. Interestingly, the characteristic equation has pure imaginary roots at the second pair and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also simulate the unstable periodic solutions at the second pair of critical values through a bifurcation diagram. Therefore, a pair of supercritical Hopf bifurcations appear around the positive equilibrium of the housefly population stage-structured model.

Double Hopf bifurcation analysis of time delay coupled active control system

In this paper, the stability analysis of double Hopf bifurcation is carried out based on the time delay coupled active control system by using the related theory of bifurcation analysis. We got the stability switching region of the system with respect to time delay. The time delay and coupling strength are selected as the bifurcation parameters, then the normal form of the coupled time delay active control system is obtained by using the multi-scale method, and the dynamic behavior near the bifurcation point is obtained. So the parameter region of the stable periodic solution and the stable quasi-periodic solution are obtained. The analytical solution of the system is then solved by using the multi-frequency homotopy analysis method (MFHAM) to verify the existence of the unstable periodic solution. Finally, numerical simulation is executed to verify the theoretical results.

Outlet boundary condition and mean temperature gradient effects on the minimum acoustics disturbances energy in triggering nonlinear thermoacoustic instability

In this study, we theoretically investigate the impact of outlet boundary conditions and mean temperature gradients on the maximum transient growth rate of acoustical energy and the critical energy required for triggering. Our analysis encompasses open–open and open–closed thermoacoustic systems. The theoretical models developed focus on horizontal ducts with a mean temperature jump over the heat source, employing the modified King's law. By linearizing the unsteady heat release, the nonlinear thermoacoustic equations transform into linearized-delay ones. This approach enables us to predict optimal initial perturbations for linearized-delay and nonlinear systems, corresponding to maximum transient growth rates of acoustic energy over short and long periods, respectively, thus providing insights into critical energy for triggering. We find that a closed outlet leads to higher transient energy growth and a lower critical energy for triggering compared to an open outlet. The increased mean temperature gradient has a “destructive” impact on triggering in open–open systems but a “constructive” effect in open–closed systems. Raising the mean temperature ratio generally increases the critical energy for triggering in the open–open system, whereas it decreases the critical energy in the open–closed system. The critical energy for nonlinear optimal initial perturbations is notably affected by the minimum energy of critical unstable periodic solutions, while the critical energy for linearized-delay optimal initial perturbations is closely tied to the energy level of stable periodic solutions. Due to the transient energy growth rate, the critical energy for nonlinear optimal initial perturbations is significantly lower than that for linearized-delay optimal initial perturbations.

Stabilization of Laminars in Chaos Intermittency

Chaos intermittency is composed of a laminar regime, which exhibits almost periodic motion, and a burst regime, which exhibits chaotic motion; it is known that in chaos intermittency, switching between these regimes occurs irregularly. In the laminar regime of chaos intermittency, the periodic solution before the saddle node bifurcation is closely related to its generation, and its behavior becomes periodic in a short time; the laminar is not, however, a periodic solution, and there are no unstable periodic solutions nearby. Most chaos control methods cannot be applied to the problem of stabilizing a laminar response to a periodic solution since they refer to information about unstable periodic orbits. In this paper, we demonstrate a control method that can be applied to the control target with laminar phase of a dynamical system exhibiting chaos intermittency. This method records the time series of a periodic solution prior to the saddle node bifurcation as a pseudo-periodic orbit and feeds it back to the control target. We report that when this control method is applied to a circuit model, laminar motion can be stabilized to a periodic solution via control inputs of very small magnitude, for which robust control can be obtained.

Periodic solutions and frequency lock-in of vortex-induced vibration energy harvesters with nonlinear stiffness

The phenomenon of amplitude jump usually means the presence of multi-solution ranges and unstable periodic solutions, which has not been fully explored in the field of vortex-induced vibration energy harvesting. In this study, semi-analytical periodic solutions of vortex-induced vibration energy harvesters (VIVEHs) are obtained using the incremental harmonic balance method, and their stabilities are determined via the Floquet theory. It is found that the VIVEH with linear stiffness and those with three-order nonlinear stiffness have four saddle-node bifurcation points and two multi-solution ranges, which leads to the occurrence of amplitude jumps. The hardening effect can shift frequency lock-in regions to higher wind speeds, thus increasing the aerodynamic forces, and realizing wide wind speed bandwidths and high voltage outputs. The wind tunnel experiments verify the accuracy of theoretical calculations, the occurrence of amplitude jump phenomena, and the existence of multi-solution ranges. The VIVEH with nonlinear stiffness experimentally reached a maximum voltage of 9.87 V, and a frequency lock-in region of 2.27–5.36 m/s.

Quantum scars in quantum field theory

We develop the theory of quantum scars for quantum fields. By generalizing the formalisms of Heller and Bogomolny from few-body quantum mechanics to quantum fields, we find that unstable periodic classical solutions of the field equations imprint themselves in a precise manner on bands of energy eigenfunctions. This indicates a breakdown of thermalization at certain energy scales, in a manner that can be characterized via semiclassics. As an explicit example, we consider time-periodic non-topological solitons in complex scalar field theories. We find that an unstable variant of Q-balls, called Q-clouds, induce quantum scars. Some technical contributions of our work include methods for characterizing moduli spaces of periodic orbits in field theories, which are essential for formulating our quantum scar formula. We further discuss potential connections with quantum many-body scars in Rydberg atom arrays.

Nonlinear analysis of the delayed tyre model with control-based continuation

In this study, the numerical bifurcation analysis of a shimmying wheel is performed with a non-smooth, time-delayed model of the tyre-ground contact. This model is capable of reproducing the bistable behaviour often observed in experiments: a stable equilibrium and a stable periodic orbit coexisting for the same set of system parameters, that the simpler quasi-steady tyre models fail to capture. In the bistable parameter domain, there also exists an unstable periodic orbit within the separatrix between the domains of attractions of the two stable steady-state solutions. Although this solution never appears in a real-life system, one may still gain valuable information from tracing it as it gives an indication about the level of perturbation that would drive the system from one stable solution to the other. However, the complexity of the laws governing partial sticking and sliding in the tyre-ground contact makes the numerical bifurcation analysis with the traditional, collocation-based techniques infeasible. Instead, this study is based on numerical simulations and the technique of control-based continuation (CBC) to track the stable and unstable periodic solutions of the system allowing for the assessment of the accuracy of the non-smooth, delayed tyre model in replicating the dynamics observed in experiments. In the meantime, the physics-based model provides an insight into the relationship between the sticking and sliding regions appearing in the tyre-ground contact and the global dynamics of the system.

Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force

The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored. Unlike the existing work in detecting early warning signals, this paper presents the most probable transition paths to characterize the regime shifts. The most probable transition paths are obtained by minimizing the Freidlin–Wentzell (FW) action functional and Onsager–Machlup (OM) action functional, respectively. The most probable path shows the movement trend of the lake eutrophication system under noise excitation, and describes the global transition behavior of the system. Under the excitation of Gaussian noise, the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and the most probable transition paths. Under the excitation of Gaussian white noise and periodic force, we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.

Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts

The circumferential vibration of a gear pair is a parametric excitation caused by nonlinear tooth stiffness, which fluctuates with meshing. In addition, the vibration characteristics of the gear pair become complicated owing to the tooth profile error and backlash. It is considered that the circumferential vibration of the gear pair is affected by the torsional vibration of the shafts. It is important to understand quantitatively the vibration characteristics of the gear system considering the shafts. Therefore, the purpose of this research was to clarify the nonlinear vibration characteristics of a gear pair considering the influence of the shafts using theoretical methods. To achieve this objective, calculations were performed using equations of motion in which the circumferential vibration of the gear pair and the torsional vibration of the shafts were coupled. The nonlinear tooth stiffness was represented by a sine wave. The influence of tooth separation was considered by defining a nonlinear function using backlash and the tooth profile error. For the numerical calculations, both stable and unstable periodic solutions were obtained by using the shooting method. The effect of the shafts on the gear system vibration were clarified by comparing the results in the cases in which the shaft was not considered, one shaft was considered, and both shafts were considered.

Bifurcation analysis of an ecological model with nonlinear state–dependent feedback control by Poincaré map defined in phase set

Ecosystems often need to integrate instantaneous interventions such as biological resource management or integrated pest control. The implementation of interventions, in turn, depends on the state and its change rate of the system, which brings great challenges for the stability and bifurcation analyses. To develop qualitative analysis techniques, we propose a predator–prey model with state-dependent pulse interventions, including spraying insecticides and releasing natural enemies, in which the implementation of control measures depends on whether the weighted value of pest density and its change rate reaches the action threshold. We first address the threshold condition for the existence and stability of the boundary periodic solutions and then define a one-parameter family of discrete (Poincaré) maps. By analyzing the properties of those discrete maps, we conclude that if there is no interior equilibrium for the system without control measures and the releasing amount of natural enemies is greater than zero, then there is at least one positive periodic solution. While if the system without control measures has an internal equilibrium and only the chemical tactic is applied, then there could be an unstable positive periodic solution near the boundary periodic solution. Consequently, backward bifurcation and bi-stability occur. The analytical techniques developed here could be applied to analyze more generalized models and other fields, including infectious disease control.

Robustness of nonlinear parameter identification in the presence of process noise using control-based continuation

In this study, we consider the experimentally obtained, periodically forced response of a nonlinear structure in the presence of process noise. Control-based continuation is used to measure both the stable and unstable periodic solutions, while different levels of noise are injected into the system. Using these data, the robustness of the control-based continuation algorithm and its ability to capture the noise-free system response are assessed by identifying the parameters of an associated Duffing-like model. We demonstrate that control-based continuation extracts system information more robustly, in the presence of a high level of noise, than open-loop parameter sweeps and so is a valuable tool for investigating nonlinear structures.

Unbounded solutions of models for glycolysis

The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This is done with the help of a Poincaré compactification of the system and a shooting argument. This system was originally derived from another system with Michaelis–Menten kinetics. A Poincaré compactification of the latter system is carried out and this is used to show that the Michaelis–Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models for the same biological system in the literature.

Broadband energy harvesting from time-delayed nonlinear oscillations of magnetic levitation

This article investigates quasiperiodic energy harvesting in a nonlinear vibration-based harvester, consisting of delayed nonlinear vibrations from magnetic levitation subjected to harmonic base acceleration, in which time-delay is introduced. For energy harvesting, a coil made up of seven layers of 36 gauges wound around the outer casing is utilized. The first- and second-order perturbation approximations are used to obtain the frequency responses of the power and vibration amplitudes in the vicinity of primary resonance. Numerical results are presented to verify the correctness of the analytical solutions. The results show that there is a good agreement between the second approximation and the numerical method; thus, we used the second-order perturbation to present the results. The impact of several parameters on the power and vibration amplitudes in the absence and presence of delay is studied. It was observed that using the nonlinear vibrations due to magnetic levitation in the absence and presence of the delay enables energy harvesting over broadband frequencies distant from primary resonance, which benefits preventing the instability and hysteresis regions near resonance. Dynamic behaviors of the motion are shown in the form of time histories and phase portraits. The results indicate periodic and quasiperiodic motions for the different values of the delay parameters. Also, the effects of the separation distance between magnets [Formula: see text], without and with time-delay, on the vibration and power amplitudes were examined. It was observed that the [Formula: see text], for both points chosen from stable periodic solution and unstable periodic solution, has a significant influence on the system behavior. Thus, choosing proper values for these parameters enables the device to extract more power in lower and broadband excitation frequencies. Besides, we illustrate that for specific values of [Formula: see text] and delay time parameters, the maximum system power output is not necessarily accompanied by maximum vibration amplitude.

On the existence and stabilization of an upper unstable limit cycle of the damped forced pendulum

By the use of interval methods it is proven that there exists an unstable periodic solution to the damped and periodically forced pendulum around the upper equilibrium. It is also proved that this solution can be stabilized by a control which does not need the knowledge of values of the state variables but of the unstable periodic solution.

An independent period-3 motion to chaos in a nonlinear flexible rotor system

In this paper, a bifurcation tree of an independent period-3 motion to chaos in a flexible nonlinear rotor system is developed semi-analytically. The period-3 motion was traditionally called the subharmonic periodic motion of order-1/3, but one did not achieve the corresponding solutions yet. Herein, stable and unstable periodic solutions on the bifurcation tree in the flexible rotor system are achieved and the corresponding stability is analyzed by eigenvalue analysis. Harmonic frequency-amplitude characteristics for periodic motions on the bifurcation tree are presented. For comparison of analytical and numerical results, numerical simulation of periodic motions is completed. Phase trajectories, displacement orbits and velocity planes are illustrated, and harmonic amplitude spectrums of period-3 and period-6 motions are presented to show harmonic terms effects. Such studies presented in this paper help one better understand the subharmonic periodic oscillation of order-1/3 in nonlinear rotor systems.

The extended periodic motion concept for fast limit cycle detection of self-excited systems

Limit cycle solutions of self-excited dynamic systems can be determined by continuation of solutions along a system parameter variation or by brute-force testing. While the brute-force search for basins of attraction is computationally intractable, continuation methods compute only those branches that are connected to others, thus neglecting a-priori unknown solutions and detached branches, such as isolas. In this work, a method is proposed for finding limit cycles of self-excited dynamic systems. The method is based on the continuation of nonlinear modes for non-conservative systems, for which the Extended Periodic Motion Concept (E-PMC) is applied. The E-PMC allows for finding stable and unstable periodic solutions along the nonlinear mode and is especially helpful for determining solutions that are detached from other solution branches. Hence, the a-priori selection of proper initial conditions for the limit cycle computation is no longer necessary. A self-excited frictional oscillator with cubic stiffness terms is studied. The proposed technique proves to be robust and finds all isolated periodic solutions that were published previously by other authors. In an extended model configuration, the E-PMC finds co-existing stable limit cycles and unstable periodic orbits, one of which gives rise to hyper-chaotic motion with multiple positive Lyapunov exponents.

Stable and unstableperiodic solutions for quadratic contact Hamiltonians with a small parameter

In this paper, we study the existence of periodic solutions for a class of quadratic contact Hamiltonian system with a small parameter by averaging theory.We have proved that the quadratic contact Hamiltonian system has a stable periodic solution and an unstable periodic solution when the corresponding parameter is small enough.

Routes to long‐term atmospheric predictability in reduced‐order coupled ocean–atmosphere systems: Impact of the ocean basin boundary conditions

The predictability of the atmosphere at short and long time‐scales associated with coupling to the ocean is explored in a new version of the Modular Arbitrary‐Order Ocean–Atmosphere Model (MAOOAM), based on a two‐layer quasigeostrophic atmosphere and a one‐layer reduced‐gravity quasigeostrophic ocean. This version features a new ocean basin geometry with periodic boundary conditions in the zonal direction. The analysis presented in this article considers a low‐order version of the model with 40 dynamical variables. First, the increase of surface friction (and the associated heat flux) due to the ocean can either induce chaos when the aspect ratio between the meridional and zonal directions of the domain of integration is small or suppress chaos when it is large. This reflects the potentially counterintuitive role that the ocean can play in the coupled dynamics. Second, and perhaps more importantly, the emergence of long‐term predictability within the atmosphere for specific values of the friction coefficient occurs through intermittent excursions in the vicinity of a (long‐period) unstable periodic solution. Once close to this solution, the system is predictable for long times, i.e. a few years. The intermittent transition close to this orbit is, however, erratic and probably hard to predict. This new route to long‐term predictability contrasts with the one found in the closed ocean‐basin low‐order version of MAOOAM, in which the chaotic solution is permanently wandering in the vicinity of an unstable periodic orbit for specific values of the friction coefficient. The model solution is thus influenced by the unstable periodic orbit at any time, and inherits from its long‐term predictability.