Sacker Bifurcation Research Articles

Nonlinear normal modes of multi-walled nanoshells with consideration of surface effect and nonlocal elasticity

System of nonlinear partial differential equations with respect to displacements is derived to describe free nonlinear vibrations of multi-walled nanoshell. The geometrically nonlinear vibrations of nanoshell, which is described by Sanders-Koiter nonlinear shell theory, are considered with account of nonlinear van der Waals forces, surface effect and nonlocal elasticity. Using the weighted residual method, the system of the partial differential equations is transformed into the system of nonlinear ordinary differential equations. The nonlinear boundary condition for the longitudinal stresses resultant is accounted in these differential equations.The essentially curvilinear Kauderer-Rosenberg nonlinear normal modes are studied numerically by combination of shooting technique and continuation algorithm. Such NNMs in double-walled nanoshell are undergone the Naimark- Sacker bifurcation. As a result, the curvilinear nonlinear modes are transformed into chaotic motions.

A Neimark--Sacker bifurcation in a discrete SIS model

A Neimark--Sacker bifurcation in a discrete SIS model

Further Investigations on the Dynamics and Multistability Coexisted in a Memory-Based Cobweb Model

Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.

Allee effect and Holling type - II response in a discrete fractional order prey - predator model

A gauss type prey - predator model is considered with Allee effect and Holling type II response. Fractional order two species system is discretized. The discretized system exhibits much richer and complex dynamics than its corresponding continuous version.Bifurcation types like flip, neimark sacker and chaos exist in the discretized system.Existence of the positive fixed points is established and local stability of discrete fractional order system is discussed with the variational matrix. It is also shown that the system allows a flip bifurcation and a neimark - sacker bifurcation. Rich dynamical nature of the system is established through time plots with phase trajectory, exciting invariant circles and periodic doubling bifurcation which leads to chaos.

A Housing Price Dynamics Model Using Heterogeneous Interacting Agents

The regulatory restraints on residential mortgage loans have a major impact on the housing market dynamics. Limits on the loan-to-value and debt-to-income ratio can be used to control credit availability, changing how agents behave and, therefore, how housing market price evolves. In this work, we propose a pricing model based on the heterogeneous agents theory, in which owners are influenced by historical housing demand, whereas buyers' behavior depends on credit availability, i.e., on regulatory restraints on residential mortgage loans. We show that the market price can be described by a Markov process that converges to a deterministic dynamical system as the number of agents increases. We establish the conditions under which the emerging system has a nontrivial equilibrium price, which depends on the regulatory restraints on mortgage loans, on buyers' reserve price, and on owners' memory and sensitivity to changes in the housing demand. Finally, we also show that changes in owners' memory and sensitivity to housing demand can lead to price oscillations or a Neimark--Sacker bifurcation, which are typically observed in bubble formation processes.

An Analysis of Discrete Stage-Structured Prey and Prey-Predator Population Models

Discrete stage-structured prey and prey-predator models are considered. Regarding the former, we prove that the models at hand are permanent (i.e., the population will neither go extinct nor exhibit explosive oscillations) and, moreover, that the transfer from stability to nonstationary behaviour always goes through a supercritical Neimark−Sacker bifurcation. The prey model covers species that possess a wide range of different life histories. Predation pressure may both stabilize and destabilize the prey dynamics but the strength of impact is closely related to life history. Indeed, if the prey possesses a precocious semelparous life history and exhibits chaotic oscillations, it is shown that increased predation may stabilize the dynamics and also, in case of large predation pressure, transfer the population to another chaotic regime.

An Analysis of a Semelparous Population Model with Density-Dependent Fecundity and Density-Dependent Survival Probabilities

A discrete age-structured semelparous Leslie matrix model where density dependence is included both in the fecundity and in the survival probabilities is analysed. Depending on strength of density dependence, we show in the precocious semelparous case that the nonstationary dynamics may indeed be rich, ranging from SYC (a dynamical state where the whole population is in one age class only) dynamics to cycles of low period where all age classes are populated. Quasiperiodic and chaotic dynamics have also been identified. Moreover, outside parameter regions where SYC dynamics dominates, we prove that the transfer from stability to instability goes through a supercritical Neimark−Sacker bifurcation, and it is further shown that when the population switches from possessing a precocious to a delayed semelparous life history both stability properties and the possibility of periodic dynamics become weaker.

The dynamics of a fold-and-twist map

Inspired by a discrete-time predator–prey model we introduce a planar, noninvertible map composed of a rigid rotation over an angle φ and a quadratic map depending on a parameter a. We study the dynamics of this map with a particular emphasis on the transitions from orderly to chaotic dynamics. For a=3 a stable fixed point bifurcates through a Hopf–Neĭmark–Sacker bifurcation which gives rise to the alternation of periodic and quasi-periodic dynamics organized by Arnold tongues in the (φ,a)-plane. Inside a tongue a periodic attractor typically either undergoes a period doubling cascade, which leads to chaotic dynamics, or a Hopf–Neĭmark–Sacker bifurcation, which leads in turn to a new family of Arnold tongues. Numerical evidence suggests the existence of strange attractors with both one and two positive Lyapunov exponents. The former attractors are conjectured to be Hénon-like, i.e., they are formed by the closure of the unstable manifold of a periodic point of saddle type. The folded nature of such attractors is the novel feature of this paper.

Patterns, Bifurcations, Multistability and Hysteresis in an Inhomogeneous Coupled Map Lattice

We investigate local interaction between logistic maps in an inhomogeneous lattice numerically with respect to an inhomogeneity parameter [Formula: see text] and the coupling constant [Formula: see text]. In our model, the inhomogeneity appears in the form of different values of the map parameter at different sites. The phase diagram of the model in the [Formula: see text]–[Formula: see text] plane gives seven qualitatively different patterns. These are: synchronized patterns, steady (fixed in time) patterns with spatial period-two, spatially chaotic together with temporally periodic patterns, spatially coherent accompanied with temporally quasi-periodic patterns, spatial intermittency with temporal period-two patterns, spatiotemporal intermittency patterns or spatiotemporal chaotic patterns. Our system exhibits, tangent bifurcations in the transition from synchronized patterns to steady patterns, period-doubling bifurcation from steady patterns to temporal periodic patterns and Neimark–Sacker (Hopf) bifurcation from steady patterns to temporal quasi-periodic patterns. The system also shows the possibility of multistable attractors and the phenomena of hysteresis for some parameter values. We identify our results using techniques such as time series, space-time plot, Fourier transform, bifurcation diagram, stability analysis.

A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

In this paper a degenerate case of a 2:3 resonant Hopf--Hopf bifurcation is studied. This codimension-four bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two- and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and Neimark--Sacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (FF) bubble interacting with curves of fold-Neimark--Sacker ($FNS$), generalized period-doubling (GPD), 1:2 strong resonances ($R_{1:2}$), 1:1 strong resonances of period-two cycles ($R_{1:1}^{(2)}$), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1,-1,-1), where the bifurcation curves $FF$, $FNS$, $R_{1:2}$, and CH interact, are detected. A second pair of codimension-three points appears when FF interacts with $GPD$ and $R_{1:1}^{(2)}$ (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf--Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.

Stability and Bifurcation of a Class of Discrete-Time Cohen–Grossberg Neural Networks with Discrete Delays

A class of discrete-time Cohen---Grossberg neural networks with discrete delays and ring-architecture are investigated in this paper. By analyzing the corresponding characteristic equations, the existence of Neimark---Sacker bifurcations at the origin are obtained. By applying the normal form theory and the center manifold theorem, the direction of the Neimark---Sacker bifurcation and the stability of bifurcating periodic solutions are obtained. Sufficient conditions to guarantee the global stability of the null solution of such networks are established by using suitable Lyapunov function and the properties of M-matrix. Numerical simulations are given to illustrate the obtained results.

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

NONLINEAR VIBRATION ANALYSIS FOR AN AIRFLOW-EXCITED TRANSLATING STRING

The nonlinear vibration of transverse motion of a translating string excited by steady wind force is investigated in this paper. The stability of the equilibrium configuration is analyzed and the generation of limit cycles via multiple Hopf bifurcations is presented. Single-, double-, and quadruple-Hopf bifurcations are determined in the parametric space. The limit-cycle response is solved through the method of Incremental Harmonic Balance, with its stability determined by Floquet multipliers. For the forced vibration, the coexistence of periodic and quasiperiodic motions is found with varying excitation frequency and amplitude. The Neimark–Sacker (NS) bifurcation and the flip bifurcation are demonstrated in an example. The continuation software MATCONT is adopted to identify the fold and NS bifurcations of periodic motions, as well as other codim-2 bifurcations of NS–NS, the Chenciner and the 1:1, 1:3, and 1:4 resonances. These bifurcations present the complexity of the string dynamics induced by steady wind excitations.

Dynamics of a discrete-time host–parasitoid system with cannibalism

A discrete-time, two-stage host population model in which adults may consume their offspring is proposed and analysed. It is shown that the population is stabilized at the unique interior equilibrium if the basic reproductive number of the population is larger than one and the population is not cannibalistic. If the mechanism of cannibalism is incorporated, then the model undergoes a discrete Hopf (Neimark–Sacker) bifurcation when the interior equilibrium loses its stability. We also study a system of host–parasitoid interaction based on the two-stage host population model. Numerical simulations suggest that the introduction of parasitoid may stabilize the system when the host population is oscillating in the absence of parasitoid.

Recognition of resonance type in periodically forced oscillators

This paper deals with families of periodically forced oscillators undergoing a Hopf–Neĭmarck–Sacker bifurcation. The interest is in the corresponding resonance sets, regions in parameter space for which subharmonics occur. It is a classical result that the local geometry of these sets in the non-degenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our main contribution is providing corresponding recognition conditions, that determine to which of these cases a given family of periodically forced oscillators corresponds. The conditions are constructed from known results for families of diffeomorphisms, which in the current context are given by Poincaré maps. Our approach also provides a skeleton for the local resonant Hopf–Neĭmarck–Sacker dynamics in the form of planar Poincaré–Takens vector fields. To illustrate our methods two case studies are included: A periodically forced generalized Duffing–Van der Pol oscillator and a parametrically forced generalized Volterra–Lotka system.

EFFECTS OF CONTRARIAN INVESTOR TYPE IN ASSET PRICE DYNAMICS

We develop an asset pricing model based on the interaction of heterogeneous trading groups. In addition to the two main trader groups, fundamentalists and trend-chasing chartists, we include a third significant group known as contrarian chartists. We model the case of opportunistic contrarian behavior, where the contrarian group disagrees with the trend-chasing chartists only when the return differential is high. We also consider absolute contrarian behavior, in which the contrarians consistently disagree with trend-chasers. The models are nonlinear planar maps, exhibiting period doubling, Neimark–Sacker and global bifurcations leading to local chaotic behavior. Absolute contrarian behavior is found to have a moderating effect on price change, while opportunistic contrarian behavior is found to further complicate the price cycles present in other models.

Geometry and dynamics of mildly degenerate Hopf–Neĭmarck–Sacker families near resonance

This paper deals with families of planar diffeomorphisms undergoing a Hopf–Neĭmarck–Sacker bifurcation and focuses on bifurcation diagrams of the periodic dynamics. In our universal study the corresponding geometry is classified using Lyapunov–Schmidt reduction and contact-equivalence singularity theory, equivariant under an appropriate cyclic group. This approach recovers the non-degenerate standard Arnol'd resonance tongues. Our main concern is a mildly degenerate case, for which we analyse a 4-parameter universal model in detail. The corresponding resonance set has a Whitney stratification, which we explain by giving its incidence diagram and by giving both two- and three-dimensional cross-sections of parameter space. We investigate the further complexity of the dynamics of the universal model by performing a bifurcation study of an approximating family of Takens normal form vector fields. In particular, this study demonstrates how the bifurcation set extends an approximation of the resonance set of the universal model.

Arnol$'$d Tongues Arising from a Grazing-Sliding Bifurcation

The Nei˘mark–Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At this bifurcation a periodic orbit loses stability, and, except at certain “strong” resonances, an invariant torus is born. The dynamics on the torus is organized by Arnol$'$d tongues in parameter space; inside the Arnol$'$d tongues phase-locked periodic orbits exist that disappear in saddle-node bifurcations on the tongue boundaries, and outside the tongues the dynamics on the torus is quasi-periodic. In this paper we investigate whether a piecewise-smooth system with sliding regions may exhibit an equivalent of the Nei˘mark–Sacker bifurcation. The vector field defining such a system changes from one region in phase space to the next, and the dividing (or switching) surface contains a sliding region if the vector fields on both sides point toward the switching surface. We consider the grazing-sliding bifurcation at which a periodic orbit becomes tangent to the sliding region and prov...

Recognition of the bifurcation type of resonance in mildly degenerate Hopf–Neĭmark–Sacker families

This paper deals with families of diffeomorphisms, a fixed point of which undergoes a Hopf–Neĭmark–Sacker bifurcation with its characteristic array of resonance tongues organizing the alteration of periodic and quasi-periodic dynamics. Our interest is with the periodic dynamics as this corresponds to subharmonic periodic solutions in the case of flows. We zoom in on the shape of one such tongue, as a subset of the resonance bifurcation diagram, briefly reviewing the classical non-degenerate case, but then turning to a next case of degeneracy. It has already been established that the generic tongue geometry involves both tongues and flames. A description of this can be given in terms of contact-equivalence singularity theory, equivariant under an appropriate cyclic group given by the resonance at hand. At an intermediate stage of the theory a Lyapunov–Schmidt reduction is applied. This gives a finite classification of such bifurcation diagrams. This paper solves the ensuing recognition problem, which aims to classify any given generic family of diffeomorphisms with respect to this.

Generic Hopf–Neĭmark–Sacker bifurcations in feed-forward systems

We show that generic Hopf–Neĭmark–Sacker bifurcations occur in the dynamics of a large class of feed-forward coupled cell networks. To this end we present a framework for studying such bifurcations in parametrized families of perturbed forced oscillators near weak resonance points. Our approach is based on fine-tuning existing normal form techniques. We then apply this framework to show that certain cells in the feed-forward networks exhibit this forced oscillator dynamics and, hence, undergo generic Hopf–Neĭmark–Sacker bifurcations. These bifurcations correspond to the occurrence of resonance tongues in parameter space with a ‘standard’ geometry.