Bifurcation Curves Research Articles

Steady-state dynamics and discontinuity-induced sliding bifurcation of a multi-module piecewise-smooth vibration-driven system with dry friction

The vibration-driven locomotion system subject to the Coulomb dry friction is a typical multi-degree-of-freedom piecewise-smooth dynamical system with rigid-body displacement. Rather than ignoring the sticking behaviors caused by dry friction in previous studies, this paper focuses on the steady-state dynamics and the discontinuity-induced sliding bifurcations of a non-smooth three-module vibration-driven system. It is shown that the presence of sticking partially invalidates the averaging method and prevents an accurate prediction of the locomotion performance. As a result, numerical methods are employed in this research, which not only gives the distribution pattern of the average steady-state velocity with respect to the actuation phase differences but also elucidates its variation trend with respect to the actuation frequency and friction anisotropism, revealing the significance of the resonance effect and phase coordination for locomotion performance enhancement. Subsequently, this paper focuses on the stick–slip dynamics associated with the non-smooth boundaries induced by dry friction. Through comprehensive numerical calculations, we identify four different types of stick–slip motions and their distributed zones, whose boundaries are essentially the sliding bifurcation curves. Analytical expressions of these bifurcation curves are derived by means of the least-square regression, and they together constitute a bifurcation diagram for a given actuation and environment conditions. Each time a sliding bifurcation curve is traversed, the stick–slip trajectories experience qualitative changes. We also systematically reveal the evolution of the bifurcation diagram with the actuation frequency and friction anisotropism.

Global bifurcation curve for fourth-order MEMS/NEMS models

Global solution curve and exact multiplicity of positive solutions for a class of semilinear systems are obtained. The results extend some known global bifurcation results of P. Korman (2002) by allowing the nonlinearity to possess a singularity. Applications to fourth-order problems arising from MEMS/NEMS models give new global bifurcation results.

Existence of S-shaped type bifurcation curve with dual cusp catastrophe via variational methods

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form −Δpu=f(u), with p>1. We deal with relatively unexplored cases when f is non-Lipschitz at 0, f(0)=0 and f(u)<0, u∈(0,r), for some r<+∞. Using the nonlinear generalized Rayleigh quotients method we find a range of parameters where the equation may have distinct branches of solutions. As a consequence, applying the variational methods, we prove that the equation has at least three positive solutions with two of them linearly unstable and one linearly stable. The results evidence that the bifurcation curve is S-shaped and exhibits the so-called dual cusp catastrophe. Our results are new even in the one-dimensional case and p=2.

Control-Based Continuation: A New Approach to Prototype Synthetic Gene Networks.

Control-Based Continuation (CBC) is a general and systematic method to carry out the bifurcation analysis of physical experiments. CBC does not rely on a mathematical model and thus overcomes the uncertainty introduced when identifying bifurcation curves indirectly through modeling and parameter estimation. We demonstrate, in silico, CBC applicability to biochemical processes by tracking the equilibrium curve of a toggle switch, which includes additive process noise and exhibits bistability. We compare the results obtained when CBC uses a model-free and model-based control strategy and show that both can track stable and unstable solutions, revealing bistability. We then demonstrate CBC in conditions more representative of an in vivo experiment using an agent-based simulator describing cell growth and division, cell-to-cell variability, spatial distribution, and diffusion of chemicals. We further show how the identified curves can be used for parameter estimation and discuss how CBC can significantly accelerate the prototyping of synthetic gene regulatory networks.

Assessing the impact of parametric uncertainty on tipping points of the Atlantic meridional overturning circulation

Various elements of the Earth system have the potential to undergo critical transitions to a radically different state, under sustained changes to climate forcing. The Atlantic meridional overturning circulation (AMOC) is of particular importance for North Atlantic heat transport and is thought to be potentially at risk of passing such a tipping point (TP). In climate models, the location and likelihood of such TPs depends on model parameters that may be poorly known. Reducing this parametric uncertainty is important to understand the likelihood of tipping behaviour. In this letter, we develop estimates for parametric uncertainty in a simple model of AMOC tipping, using a Bayesian inversion technique. When applied using synthetic (‘perfect model’) salinity timeseries data, the technique drastically reduces the uncertainty in model parameters, compared to prior estimates derived from previous literature, resulting in tighter constraints on the AMOC TPs. To visualise the impact of parametric uncertainty on TPs, we extend classical tipping diagrams by showing probabilistic bifurcation curves according to the inferred distribution of the model parameter, allowing the uncertain locations of TPs along the probabilistic bifurcation curves to be highlighted. Our results show that suitable palaeo-proxy timeseries may contain enough information to assess the likely position of AMOC (and potentially other Earth system) TPs, even in cases where no tipping occurred during the period of the proxy data.

Bifurcation structure and stability of steady gravity water waves with constant vorticity

This paper studies the local bifurcation direction, stability properties and global structure for a nonlinear pseudodifferential equation, which describes the periodic travelling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation points. In particular, their signs can be strictly judged when constant vorticity vanishes. Furthermore, we present the stability analysis for the travelling water waves that have small vorticity and amplitude. We also show that the global bifurcation curves can't form a loop. Moreover, if the total head is bounded, the existence of waves of all amplitudes from zero up to that of Stokes' highest wave has been established.

On the non-linear dynamics of vehicle shimmy coupled with handling motions: modelling, bifurcation analysis, and validations

A novel vehicle dynamic model considering the coupling mechanism between the front wheel shimmy and the vehicle lateral handling motions is established to investigate the interacting effect between the steering shimmy instability and the general motion instability, that focuses especially on the influence of the tyre force saturation induced by lateral-longitudinal coupling when different degrees and allocation strategy of braking force are applied. The typical 3-degree-of-freedom front wheel shimmy structure of dependent suspension is dynamically modelled by using the Lagrange method, alongside the equations for plane vehicle handling motions introduced. The stretched-string-based tyre kinematic equations, that describe the transient tyre-road contact status, are proposed by analytically deriving the coupling relations between the angular shimmy variables and vehicle motion status. A theoretically approximated linear tyre model, based on the coupling between vertical, longitudinal, and lateral directional forces, is analytically derived for stability analysis. Meanwhile, a complete PAC-2002 tyre model, with experimentally identified parameters, is employed for simulation verification. The stability chart of linear stability analysis shows the variation of the system bifurcation curves and stable parameter region under different parametric configurations, which gives that the degrees and allocations of braking force largely affect the stable parameter domain and the global stability. The variations of the system characteristic roots indicate that both Hopf and S-N bifurcation could happen due to shimmy oscillations and destabilization of motion, respectively, and their co-existence at certain parameter domains make the dynamic behaviour further unpredictable and complex. Numerical simulation verifies the dynamical characteristics of shimmy, and further analysis shows that three types of vehicle motion instability which coexisted with shimmy instability could be triggered by the coupling effect. In addition, the variation law of limit-cycle amplitude under varying vehicle driving inputs is summarized, and finally, a preliminary experiment is conducted for validations.

Nine equilibrium points of four point charges on the plane

We find a specific configuration for four point charges on the plane and show, with charge values in a small region, there are nine equilibrium points on the same plane, which reaches the claimed upper bound of Maxwell’s conjecture. A procedure of computing bifurcation curves is presented for assisting in locating the region in the parameter space yielding the nine equilibrium points.

Nonlinear dynamical modelling of high frequency electrostatic drift waves using fluid theoretical approach in magnetized plasma

A novel third order nonlinear evolution equation governing the dynamics of the recently observed high frequency electrostatic drift waves has been derived in the framework of a plasma fluid model in an inhomogeneous magnetized plasma. The corresponding linear dispersion relation arising out of the fluid equations has been studied in detail for the conventional low frequency as well as the recently observed high frequency electrostatic drift waves along with the electrostatic ion cyclotron waves. The derived nonlinear evolution equation of third order is then decomposed into two second order equations as the order of the equation becomes reduced after this kind of decomposition under certain conditions, and one equation becomes linear whereas the other equation becomes nonlinear. The detailed analysis of fixed points leading to bifurcations has been performed using the qualitative theory of differential equations and the bifurcation theory of planar dynamical systems for the reduced second order nonlinear equation. The bifurcation curves for the individual fixed points are also shown for the variations of different parameters in the parameter space of this reduced nonlinear equation. Then some exact as well as approximate travelling wave solutions of this reduced nonlinear equation have been derived. As the other second order reduced equation is linear in nature, its exact oscillatory and exponential solutions are obtained. The intersection of the solutions of these two reduced second order equations provides the solutions of the original third order nonlinear evolution equation; it is verified that the solutions of the reduced nonlinear second order equation are subsets of the oscillatory solution of the reduced linear second order equation in most cases whereas the intersection is different if the exponential solution of the reduced linear second order equation is considered. This further implies that the solutions of the reduced second order nonlinear equation can directly represent the solutions of the original third order nonlinear evolution equation in most cases representing the dynamics of the nonlinear high frequency electrostatic drift waves. These solutions of the novel third order nonlinear evolution equation represent certain vortex-like structures as our work is restricted to (1 + 1) dimensions only. The possible applications of our novel results on the dynamics of the high frequency electrostatic drift waves are discussed along with future directions for research in this field.

Nonlinear and chaotic vibrations of FG double curved sandwich shallow shells resting on visco-elastic nonlinear Hetenyi foundation under combined resonances

In this study, the nonlinear and chaotic instability of functionally graded (FG) double curved shallow sandwich shells resting on a viscoelastic Hetenyi foundation, under simultaneous effect of in-plane and transverse excitations is studied. Employing the third-order Reddy theory and von-Karman relations and the Hamilton principle, the partial differential equations of motion under movable SS boundary conditions are derived. Introducing the trigonometric Airy stress function and applying the Galerkin’s method, the equations are reduced to a set of nonlinear ODEs with time. Then, under forcing resonance conditions and using the perturbation method, the modulation equations at the stationary conditions are derived and solved numerically. Then stability of non-trivial solutions for the resonance amplitude corresponding to the presence of limit cycle oscillation is investigated. Then by extracting the characteristic resonance amplitude curves, the effect of various parameters, including: frequency detuning parameter, in-plane excitation amplitude, linear, shear and nonlinear stiffness and damping parameters of the foundation on nonlinear response are analyzed. Finally, by extracting two degrees of freedom system time response, bifurcation and chaotic characteristic curves of the problem, the conditions for occurrence the periodic, double periodic, multi-periodic and chaotic behaviors under the simultaneous effect of parametric and internal resonances are studied comprehensively.

Bifurcations in the symmetric viability model with weak selection

In this paper we discuss bifurcations of equilibria in the symmetric viability model with weak selection. Overcoming the difficulties from irrational fractions of the 4 parameters in determining the signs of determinant and trace, we find that the system has at most 5 interior equilibria, 4 edge equilibria and 4 corner equilibria and give a complete qualitative analysis for all equilibria in both hyperbolic cases and non-hyperbolic cases, including the case of fully null degeneracy. On this basis we display all bifurcations of equilibria such as pitchfork bifurcation, transcritical bifurcation and bifurcations at corner equilibria of fully null degeneracy, which is of codimension 2 but none of known results of codimension 2 can be applied to. We further discuss the unfolding system of fully null degeneracy, displaying all its bifurcations and giving bifurcation curves analytically. Our results provide an intuitive basis for us to predict the genetic composition of a population and provide the basic mathematical theory for the protection and loss of alleles.

Dynamics and bifurcations of a discrete time neural network with self connection

This paper investigates the dynamical behavior of a discrete-time neural network system from both analytical and numerical points of view. The conditions as well as the critical coefficients for the pitchfork, flip (period-doubling), Neimark-Sacker, and strong resonances are computed analytically. Using critical coefficients, the bifurcation scenarios were determined for each bifurcation point. By changing one or two parameters, bifurcation curves of fixed points and cycles with periods up to four iterates, were obtained. Numerical analysis validates our analytical results and reveals more complex dynamical behaviors.

Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.

Kosambi–Cartan–Chern Analysis of the Nonequilibrium Singular Point in One-Dimensional Elementary Catastrophe

This paper analyzes the properties of the nonequilibrium singular point in one-dimensional elementary catastrophe. For this analysis, the Kosambi–Cartan–Chern (KCC) theory is applied to characterize the dynamical system based on differential geometrical quantities. When both the nonlinear connection and deviation curvature are zero, that is, when the geometric stability of the KCC theory is neutral, two bifurcation curves are obtained: one is the known curve with an equilibrium singular point, and the other is a new curve with a nonequilibrium singular point. The two singular points are distinguished based on the vanishing condition of the Berwald connection. Applied to the ecosystem described by the Hill function, the absolute value of the cuspidal curvature of the nonequilibrium singular point is larger than that of the equilibrium singular point. The ecological interpretation of this result is that the range of bistability of the ecosystem in the nonequilibrium state is greater than that in the equilibrium state. The type of singular points in equilibrium and nonequilibrium bifurcation curves are not necessarily the same. For instance, there is a combination in which even if the former has one cusp, the latter may show various types, depending on the parametric space. These results demonstrate that there are cases where simply shifting the system from the equilibrium to nonequilibrium state expands the range of bistability and changes the type of singularity. Although singularity analysis is often performed near the equilibrium point, nonequilibrium analysis, i.e. analysis based on the KCC theory, provides a useful perspective for analyzing singularity theory according to the bifurcation phenomenon.

Exploiting multiple-frequency bursting of a shape memory oscillator

In this paper we explored a novel multiple-frequency bursting of a shape memory oscillator under parametrically and externally excitation. Based on the fast-slow dynamics analysis method, the one-parameter bifurcation curves under different frequency ratios are proposed for some specific parameters. The corresponding jumping phenomenon are verified via transformed phase, and the multi-valued characteristic are observed by the domain of attraction. The results express that the multiple-frequency oscillation can induce complex bursting phenomenon. In addition, the jump multi-value phenomena of tristability are more abundant than bistability and monostability.

Role of limiting dispersal on metacommunity stability and persistence

The role of dispersal on the stability and synchrony of a metacommunity is a topic of considerable interest in theoretical ecology. Dispersal is known to promote both synchrony, which enhances the likelihood of extinction, and spatial heterogeneity, which favors the persistence of the population. Several efforts have been made to understand the effect of diverse variants of dispersal in the spatially distributed ecological community. Despite that environmental change strongly affects the dispersal, the effects of controlled dispersal on the metacommunity stability and their persistence remain unknown. We study the influence of limiting the immigration using two-patch prey-predator metacommunity at both local and spatial scales. We find that the spread of the inhomogeneous stable steady states (asynchronous states) decreases monotonically upon limiting the predator dispersal. Nevertheless, at the local scale, the spread of the inhomogeneous steady states increases up to a critical value of the limiting factor, favoring the metacommunity persistence, and then starts decreasing for a further decrease in the limiting factor with varying local interaction. Interestingly, limiting the prey dispersal promotes inhomogeneous steady states in a large region of the parameter space, thereby increasing the metacommunity persistence at both spatial and local scales. Further, we show similar qualitative dynamics in an entire class of complex networks consisting of a large number of patches. We also deduce various bifurcation curves and stability conditions for the inhomogeneous steady states, which we find to agree well with the simulation results. Thus, our findings on the effect of the limiting dispersal can help to develop conservation measures for ecological communities.

Influence of asymmetric parameters in higher-order coupling with bimodal frequency distribution.

We investigate the phase diagram of the Sakaguchi-Kuramoto model with a higher-order interaction along with the traditional pairwise interaction. We also introduce asymmetry parameters in both the interaction terms and investigate the collective dynamics and their transitions in the phase diagrams under both unimodal and bimodal frequency distributions. We deduce the evolution equationsfor the macroscopic order parameters and eventually derive pitchfork and Hopf bifurcation curves. Transition from the incoherent state to standing wave pattern is observed in the presence of the unimodal frequency distribution. In contrast, a rich variety of dynamical states such as the incoherent state, partially synchronized state-I, partially synchronized state-II, and standing wave patterns and transitions among them are observed in the phase diagram via various bifurcation scenarios, including saddle-node and homoclinic bifurcations, in the presence of bimodal frequency distribution. Higher-order coupling enhances the spread of the bistable regions in the phase diagrams and also leads to the manifestation of bistability between incoherent and partially synchronized states even with unimodal frequency distribution, which is otherwise not observed with the pairwise coupling. Further, the asymmetry parameters facilitate the onset of several bistable and multistable regions in the phase diagrams. Very large values of the asymmetry parameters allow the phase diagrams to admit only the monostable dynamical states.

Codimension-Two Bifurcations and Bifurcation Controls in a Discrete Biological System with Weak Allee Effect

In this paper, the codimension-two bifurcations and bifurcation controls of a discrete predatory system with weak Allee effect on prey are investigated. After analyzing the qualitative properties of all the fixed points, we explore the existences of 1:2 and 1:3 strong resonances at the unique interior fixed point. Combining the adjoint operator method with an approximate identity transformation, we obtain the normal forms of these strong resonances. Meanwhile, we calculate the approximate expressions of bifurcation curves around the resonance point, which include Neimark–Sacker and period doubling bifurcation curves. And we give the conditions under which these bifurcations are supercritical or subcritical. To study global bifurcations near the resonance point, we use an ordinary differential equation to approximate the original discrete system by the time-one approximation flow theory. In biology, bifurcations and unstable fluctuations may be harmful for the breeding of biological population. Then we extend the hybrid control strategy to control 1:2 and 1:3 strong resonances. Numerical simulations are in good agreement with our theoretical results. By using the MatContM package, we perform numerical bifurcation analyses to further support our theoretical results, in which the bifurcation points and the normal form coefficients are completely consistent with our theories. According to the results, we finally analyze the impact of weak Allee effect on the system. And a reasonable biological explanation is given for these conclusions.

On the classification and evolution of bifurcation curves for a quasilinear regularized MEMS model

The quasilinear equation −(u′(x)1+(u′(x))2)′=λ(1−u)2−λɛ2(1−u)4 with the boundary condition u(−L)=u(L)=0 governs the steady-state solutions from a regularized MEMS model. We prove that for any evolution parameters ɛ∈(0,1) and L>0, the global bifurcation curve of positive solutions is strictly increasing or ⊃-like shaped or S-like shaped in the (λ,‖u‖∞)-plane. The bifurcation curves present a variety of shapes and structures, significantly different from those in non-regularized case (i.e., ɛ=0) and in the simplified semilinear case. The main tools are some new time-map techniques, the total positivity theory, and Sturm’s Theorem.

Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.