Branches Of Periodic Solutions Research Articles

On the detection of nonlinear normal mode-related isolated branches of periodic solutions for high-dimensional nonlinear mechanical systems with frictionless contact interfaces

This contribution introduces a methodology for the detection of isolated branches of periodic solutions to the nonlinear mechanical equation of motion for systems featuring frictionless contact interfaces. This methodology relies on a harmonic balance method-based solving procedure combined with the application of the Melnikov energy principle. It is able to predict the location of isolated branches of solutions in the vicinity of families of autonomous periodic solutions, namely nonlinear normal modes. The methodology is first applied on a two-degree-of-freedom phenomenological system in order to illustrate its relevance and accuracy. In particular, for this academic application, the proposed methodology yields an understanding of the discontinuous evolution of the first nonlinear resonance frequency as the forcing amplitude increases. Isolated branches of solutions featuring high amplitudes of vibration are detected far beyond nonlinear resonance frequencies obtained with a typical application of the harmonic balance method coupled with an arc-length continuation algorithm. Demonstration of the applicability of the proposed methodology to high-dimensional industrial finite element models is made with the nonlinear vibration analysis of a transsonic compressor blade, NASA rotor 37, subjected to an harmonic loading and blade-tip/casing structural contacts. The proposed methodology yields numerous isolated branches of solutions, which relevance is assessed by means of time integration simulations. In the end, the presented results underline that typical continuation algorithms may yield a significant underestimation of nonlinear resonance frequencies.

Discontinuity Structures and Solitary Waves in Electromagnetic Hydrodynamics Associated with Linear and Nonlinear Alfvén Wave Resonances

Nondissipative and weakly dissipative discontinuity structures are considered. A special numerical method for studying periodic waves is used. The location of branches of periodic solutions is investigated. Solitary waves and nondissipative discontinuity structures are sought as limiting solutions. It is found that, in addition to the resonance of long Alfvén waves with short fast and slow magnetosonic waves, there is also resonance with long waves, which leads to the occurrence of hybrid-type solitary waves and hybrid-type discontinuity structures. Partial differential equations are solved to find out if the found structures are actually observed.

Understanding oscillons: Standing waves in a ball

Oscillons are localized long-lived pulsating states in the three-dimensional ${\ensuremath{\phi}}^{4}$ theory. We gain insight into the spatiotemporal structure and bifurcation of the oscillons by studying time-periodic solutions in a ball of a finite radius. A sequence of weakly localized Bessel waves---nonlinear standing waves with the Bessel-like $r$-dependence---is shown to extend from eigenfunctions of the linearized operator. The lowest-frequency Bessel wave serves as a starting point of a branch of periodic solutions with exponentially localized cores and small-amplitude tails decaying slowly toward the surface of the ball. A numerical continuation of this branch gives rise to the energy-frequency diagram featuring a series of resonant spikes. We show that the standing waves associated with the resonances are born in the period-multiplication bifurcations of the Bessel waves with higher frequencies. The energy-frequency diagram for a sufficiently large ball displays sizeable intervals of stability against spherically symmetric perturbations.

Modelling and nonlinear analysis of the wave-induced vibrations of a single hybrid riser conveying two-phase flow

This paper analyses the nonlinear dynamic response of a single hybrid riser pipe to wave excitations. Detailed nonlinear equations of motion are derived for the riser pipe as it conveys two-phase flow. Hence its dynamics are described by two coupled nonlinear equations, relating to both longitudinal and transverse displacements. Wave excitations forcing the system are imposed as oscillating inline flow. In addition, considering that the riser is a structure that is vibrating in a moving fluid, the inline force is modelled with the extended Morison equation. These equations are solved by the method of multiple scale perturbation. Theoretical results show that longitudinal and transverse responses are coupled at some specified natural frequencies. Similarly, the riser pipe is also prone to primary and super-harmonic resonance excitations at some wave frequencies. This presents a complex nonlinear multi-frequency excitation problem identical to the numerical studies conducted on a typical single hybrid riser pipe based on the frequency response curves. The uncoupled primary resonance behaviour of the transverse displacements for all the void fractions studied exhibited a hardening nonlinear behaviour of non-unique responses and jump phenomenon. However, a distinctive feature of the harmonics was observed where the curves separate into two to form an isolated branch of periodic solutions. These are known as Isola. Also, the presence of internal resonance results in the transfer of energy between the planar axis resulting in undesirable axial resonance peaks which are destabilized by the presence of the nonlinear anti-resonance effect.

Nonlinear dynamics analysis of a dual-rotor-bearing-casing system based on a modified HB-AFT method

This paper presents a modified harmonic balance-alternating frequency/time domain (HB-AFT) method to analyze the nonlinear dynamics of a dual-rotor-bearing-casing system. The motion equations of a dual-rotor-bearing-casing system subjected to the unbalanced excitations of the two rotors are formulated with 284 degrees of freedom, in which, the complex nonlinear factors of the inter-shaft bearing such as exponential nonlinear restoring force, radial clearance and varying stiffness excitation are taken into consideration. The HB-AFT method is modified by applying the AFT procedure to obtain the harmonic expending coefficients of the harmonic balance residuals rather than that of the nonlinear terms, hence, the Jacobian matrix is calculated programmatically and efficiently, as a result the performance of the HB-AFT method is improved greatly. The primary resonance of the system is analyzed by employing the three-dimensional diagrams of amplitude-frequency responses of each node of the system, including the inner casing and the outer casing. In addition, the amplitude-frequency response curves with separated frequencies and orbits of the two rotors are also discussed. Furthermore, the pseudo-arclength continuation procedure is introduced into the modified HB method, then all periodic solution branches of the system including the unstable solutions are obtained. The results show that there are two resonant peaks in the amplitude-frequency response curves of the system and the vibration of each part of the system, i.e., LPC, LPT, HPC, HPT, inner casing and outer casing, is synchronous. Besides, the nonlinear dynamic behaviors such as vibration jumping, bi-stable and resonance hysteresis are revealed. Accordingly, the dynamic load of the inter-shaft bearing is calculated. The results show that the evolution of the inter-shaft bearing’s dynamic load is associated with the vibration responses of the two rotors. The nonlinear characteristics of the force-frequency responses of the inter-shaft bearing are consistent with that of the two rotors and casings. In addition, with the increase of the inter-shaft bearing clearance, the nonlinear characteristics of the force-frequency responses of the inter-shaft bearing are more significant. In comparison with the Newmark method, the modified HB-AFT method proposed in this paper has a great advantage in computing efficiency, and it can also grasp all solution branches of the system including the unstable solutions. Consequently, it has great potential to deal with high-dimensional systems with complex nonlinearities and multifrequency excitations in practical engineering.

Harmonic balance analysis of magnetically coupled two-degree-of-freedom bistable energy harvesters

Because a magnetically coupled two-degree-of-freedom bistable energy harvester (2-DOF MCBEH) shows the rich, complicated nonlinear behaviors caused by its coupled cubic nonlinearities, understanding the dynamics remains challenging. This paper reports and investigates the important nonlinear dynamical phenomena of the 2-DOF MCBEHs by performing the harmonic balance analysis (HBA). All periodic solution branches are identified in order to study and comprehend the complicated dynamics of the 2-DOF MCBEHs. This end requires care when truncating the harmonic balance solution. For a 1-DOF MCBEH, which is the conventional type, the fundamental harmonic is able to approximately describe the steady-state periodic response. However, high-order harmonics are significant for the 2-DOF MCBEH. This paper demonstrates that the harmonic balance solution should involve the high-order terms instead of using the oversimplified single-harmonic solution. By performing the proposed HBA, important solution branches are reported, and their dynamical behaviors are studied. Moreover, the complete architecture of the frequency response of the 2-DOF MCBEH is disclosed across the entire frequency range. The HBA also reveals the underlying physics of building a bridge between the first and second primary resonant areas under a strong excitation. In the future, the findings in the present report can be utilized in the design process of the 2-DOF MCBEHs.

Investigation on multiple periodic solution branches of a system with piecewise linear nonlinear energy sink

The multiple solution problem of nonlinear energy sinks (NESs) is of great interest in academic research and engineering applications; however, there exists only limited work on the multiple solution branches of piecewise linear NESs. Piecewise linear NESs can give rise to the appearance of isolated solution branches (ISBs), which are harmful to the performance of dynamical systems. It is difficult to detect the ISBs of a dynamical system with a piecewise linear NES. Hence, in this study, the multiple periodic solution branch distribution of a four-degree-of-freedom dynamical system with a piecewise linear NES is investigated. A method based on polynomial homotopy that can capture ISBs is developed to search multiple solution branches. Spurious periodic solutions can exist owing to the use of the implicit polynomial approach; therefore, the harmonic balance method with high harmonic orders and alternating frequency/time-domain technique are employed to distinguish true periodic solutions. Numerical experiments demonstrate that the proposed method can search for multiple periodic solution branches. A parameter analysis is conducted on the multiple solution branches, and the inner and outer ISBs are observed.

A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation

Harmonic balance method (HBM) incorporated with arc-length continuation and Floquet-Hill formulation now becomes an essential and strong tool for computing periodic solution branches and performing stability analysis, which can provide great insight into the responses of nonlinear dynamic systems. However, due to the redundancy of eigenvalues of Hill’s matrix over the actual Floquet exponents of periodic solutions and the distortion on the redundancy relation of the eigenvalues induced by order truncation of harmonics, it is not routine work to correctly select the actual Floquet exponents from the eigenvalues of Hill’s matrix in order to achieve correct stability and bifurcation analysis, especially the order of harmonics in HBM is low. In this work, a new robust strategy for Floquet exponents filtering is introduced, which is chosen based on the character of the real part (FEF-RP) of the Hill’s eigenvalues. Furthermore, an adaptive arc-length control for the arc-length continuation (ALC) is proposed, which is continuously governed by the minimum parameterized eigenvalue of the Jacobian matrix (ALC-MPE) and can provide smooth and efficient continuation of the turning points of periodic solutions. The study on a 4DOFs rotor–stator rubbing system and a 3DOFs damped Duffing oscillator system shows the feasibility of the proposed methods for arc-length control (ALC-MPE) and Floquet exponents filtering (FEF-RP).

Nonlinear oscillations of a dual-joint system involving simultaneous 1:1 and 1:2 internal resonances

Bolted flange joints are typical connections form used in engineering structures. However, nonlinear factors such as discontinuity of the structure and mechanical contact of the connecting interface significantly affect and complicate the responses of the connection system. In this paper, the near-resonant response of a dual-joint system driven by harmonic excitation is investigated. Considering the effect of the nonlinear connection stiffness on the response of the dual-joint system, a lump mass model is established by transforming the contact nonlinearity into piecewise-smooth springs. The model captures the interaction of three modes (two bending modes and a longitudinal mode) with eigenfrequencies ω1, ω2, and ω3 such that 2ω1≈ω3 and ω2≈ω3, displaying 1:1, 1:2, and 1:2:2 internal resonances under specific excitation conditions. Harmonic Balance Method combined with Asymptotic Numerical Method (HBM–ANM method) is utilized to trace the branch of periodic solutions. Instability limits are derived numerically to show a more direct bifurcation scenario, determining the existence and stability of periodic solutions of the system. Four classes of periodic solutions are found for the longitudinal mode of the system driven at its resonance. In addition, three classes of periodic solutions are observed for the second-order bending mode driven at its resonance. Remarkable coupling oscillations are detected induced by the internal resonances such that ignoring internal resonances in the system will lead to notably different dynamic behavior and possibly misleading results.

Hopf bifurcations for an oversteer vehicle – the influence of wheel load changes

AbstractIn [1] the occurence of a Hopf bifurcation at steady‐state cornering for a two‐wheel oversteering vehicle with a brush model for the tyre forces was found and following the bifurcating branch of periodic solutions we observed a Canard phenomenon and relaxation oscillations, which are typical for singularly perturbed systems. In that article we assumed for simplicity, that the car's center of mass lies at street level. In this presentation we investigate the influence of an elevated height on the stability of the car's steady motion and the postcritical behaviour.

Spatial Relative Equilibria and Periodic Solutions of the Coulomb $$(n+1)$$-Body Problem

We study a classical model for the atom that considers the movement of n charged particles of charge \(-1\) (electrons) interacting with a fixed nucleus of charge \(\mu >0\). We show that two global branches of spatial relative equilibria bifurcate from the n-polygonal relative equilibrium for each critical value \(\mu =s_{k}\) for \(k\in [2,\ldots ,n/2]\). In these solutions, the n charges form n/h-groups of regular h-polygons in space, where h is the greatest common divisor of k and n. Furthermore, each spatial relative equilibrium has a global branch of relative periodic solutions for each normal frequency satisfying some nonresonant condition. We obtain computer-assisted proofs of existence of several spatial relative equilibria on global branches away from the n-polygonal relative equilibrium. Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs.

Rigorous continuation of periodic solutions for impulsive delay differential equations

We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitly computable error bounds. As an application, we prove the existence of a global branch of periodic solutions in the pulse-harvested Hutchinson equation, connecting the state at carrying capacity in the absence of harvesting to the transcritical bifurcation at the extinction steady state.

Computer assisted proof of branches of stationary and periodic solutions, and Hopf bifurcations, for dissipative PDEs

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.

A mathematical analysis of Hopf-bifurcation in a prey-predator model with nonlinear functional response

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings.

Singular Limit in Hopf Bifurcation for Doubly Diffusive Convection Equations II: Bifurcation and Stability

A singular perturbation problem from the artificial compressible system to the incompressible system is considered for a doubly diffusive convection when a Hopf bifurcation from the motionless state occurs in the incompressible system. It is proved that the Hopf bifurcation also occurs in the artificial compressible system for small singular perturbation parameter, called the artificial Mach number. The time periodic solution branch of the artificial compressible system is shown to converge to the corresponding bifurcating branch of the incompressible system in the singular limit of vanishing artificial Mach number.

Stability and Hopf bifurcations in a competitive tumour-immune system with intrinsic recruitment delay and chemotherapy.

In this paper, a three-dimensional nonlinear delay differential system including Tumour cells, cytotoxic-T lymphocytes, T-helper cells is constructed to investigate the effects of intrinsic recruitment delay and chemotherapy. It is found that the introduction of chemotherapy and time delay can generate richer dynamics in tumor-immune system. In particular, there exists bistable phenomenon and the tumour cells would be cleared if the effect of chemotherapy on depletion of the tumour cells is strong enough or the side effect of chemotherapy on the hunting predator cells is under a threshold. It is also shown that a branch of stable periodic solutions bifurcates from the coexistence equilibrium when the intrinsic recruitment delay of tumor crosses the threshold which is new mechanism, which can help understand the short-term oscillations in tumour sizes as well as long-term tumour relapse. Numerical simulations are presented to illustrate that larger intrinsic recruitment delay of tumor leads to larger amplitude and longer period of the bifurcated periodic solution, which indicates that there exists longer relapse time and then contributes to the control of tumour growth.

Global Nonlinear Normal Modes in the Fullerene Molecule $C_{60}$

In this paper we analyze nonlinear dynamics of the fullerene molecule. We prove, under isotypic nonresonant assumptions, the existence of global branches of periodic solutions emerging from an icos...

Nonlinear stochastic dynamics of a rub-impact rotor system with probabilistic uncertainties

This paper presents a stochastic model for performing the uncertainty and sensitivity analysis of a Jeffcott rotor system with fixed-point rub-impact and multiple uncertain parameters. A probabilistic nonlinear formulation is developed based on the combination of the harmonic balance method and an alternate frequency/time procedure (HB-AFT) in stochastic form. The non-intrusive generalized polynomial chaos expansion (gPCE) with unknown deterministic coefficients is employed to represent the propagation of uncertainties on rotor dynamics. In conjunction with the path continuation scheme and the Floquet theory, the developed model enables one to expediently evaluate the uncertainty bounds and probability density functions (PDFs) on periodic solution branches and associated stabilities. A global sensitivity analysis is then carried out by evaluating Sobol’s indices from gPCE to quantitatively ascertain the relative influence of different stochastic parameters on vibrational behaviors and conditions for the occurrence of rub-impact. The efficiency of the proposed algorithm for nonlinear stochastic dynamics of rub-impact rotors is validated with Monte Carlo simulation. Parametric studies are finally carried out to investigate the effects of multiple random parameters on the probabilistic variability in nonlinear responses of rub-impact rotors, which reveals the necessary to consider input uncertainties in analyses and designs to ensure the sustainable system performance.

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

<p style='text-indent:20px;'>In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us to rigorously validate numerically computed branches of periodic solutions. We derive the estimates in full generality and present sample continuation proofs obtained using an implementation in Matlab. The presented approach is applicable to any polynomial vector field of any order and dimension. A variety of examples is presented to illustrate the efficacy of the method.

Computational analysis of a 9D model for a small DRG neuron.

Small dorsal root ganglion (DRG) neurons are primary nociceptors which are responsible for sensing pain. Elucidation of their dynamics is essential for understanding and controlling pain. To this end, we present a numerical bifurcation analysis of a small DRG neuron model in this paper. The model is of Hodgkin-Huxley type and has 9 state variables. It consists of a Nav1.7 and a Nav1.8 sodium channel, a leak channel, a delayed rectifier potassium, and an A-type transient potassium channel. The dynamics of this model strongly depend on the maximal conductances of the voltage-gated ion channels and the external current, which can be adjusted experimentally. We show that the neuron dynamics are most sensitive to the Nav1.8 channel maximal conductance ([Formula: see text]). Numerical bifurcation analysis shows that depending on [Formula: see text] and the external current, different parameter regions can be identified with stable steady states, periodic firing of action potentials, mixed-mode oscillations (MMOs), and bistability between stable steady states and stable periodic firing of action potentials. We illustrate and discuss the transitions between these different regimes. We further analyze the behavior of MMOs. As the external current is decreased, we find that MMOs appear after a cyclic limit point. Within this region, bifurcation analysis shows a sequence of isolated periodic solution branches with one large action potential and a number of small amplitude peaks per period. For decreasing external current, the number of small amplitude peaks is increasing and the distance between the large amplitude action potentials is growing, finally tending to infinity and thereby leading to a stable steady state. A closer inspection reveals more complex concatenated MMOs in between these periodic MMO branches, forming Farey sequences. Lastly, we also find small solution windows with aperiodic oscillations which seem to be chaotic. The dynamical patterns found here-as consequences of bifurcation points regulated by different parameters-have potential translational significance as repetitive firing of action potentials imply pain of some form and intensity; manipulating these patterns by regulating the different parameters could aid in investigating pain dynamics.