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Showing papers on "Bifurcation diagram published in 2014"


Journal ArticleDOI
TL;DR: In this paper, the authors considered a predator-prey system with generalized Holling type III functional response and showed that the model exhibits subcritical Hopf and Bogdanov-Takens bifurcation simultaneously in corresponding small neighborhoods of the two degenerate equilibria.

167 citations


Journal ArticleDOI
TL;DR: A new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium is introduced.
Abstract: This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincare map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

167 citations


Journal ArticleDOI
TL;DR: In this paper, the global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis, where the threshold values of the control parameter for the occurrence of homocallinic Bifurcation and onset of chaos are predicted.
Abstract: The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.

107 citations


Book
09 Oct 2014
TL;DR: This paper presents a model for dynamic bifurcation and linearization of the Fitzhugh-Nagumo Model and some examples of the models used in this study showed good agreement on both the static and the dynamic aspects of the model.
Abstract: Introduction. 1. Models and Dynamics. 2. Static Bifurcation and Linearization of the Fitzhugh-Nagumo Model. 3. Dynamic Bifurcation for the Fitzhugh-Nagumo Model. 4. Models of Asymptotic Approximation for the Fitzhugh-Nagumo System as c --> ? 5. Global Bifurcation Diagram and Phase Dynamics for the Fitzhugh-Nagumo Model. References. Index.

97 citations


Journal ArticleDOI
TL;DR: In this article, an SIR model with a standard incidence rate and a nonlinear recovery rate was established to consider the impact of available resource of the public health system especially the number of hospital beds.

94 citations


Journal ArticleDOI
TL;DR: A modified mathematical model of Kuznetsov et al. (1994) representing tumor-immune interaction with discrete time delay with bifurcation parameter is proposed, based on the normal form theory and center manifold theorem, to establish the sufficient condition for local stability of interior steady state.

71 citations


Journal ArticleDOI
TL;DR: In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented, and the chaotic behaviors are validated by the positive Lyapunov exponents.
Abstract: In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented. The chaotic behaviors are validated by the positive Lyapunov exponents. Furthermore, the fractional Hopf bifurcation is investigated. It is found that the system admits Hopf bifurcations with varying fractional order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the Hopf bifurcations are proposed. Numerical simulations are given to illustrate and verify the results.

69 citations


Journal ArticleDOI
TL;DR: In this paper, a novel method is proposed for prediction of the chaos in the micro- and nano-electro-mechanical resonators based on the proposed method, first an accurate analytical solution for the dynamics behavior of the nano-resonators is derived using the multiple scales method up to the second order.

65 citations


Journal ArticleDOI
TL;DR: Normal forms associated with codimension-two Hopf–Turing bifurcation are derived, which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the Hopf-Turing pitting point.
Abstract: Spatiotemporal dynamics in a ratio-dependent predator–prey model with diffusion is studied by analytical methods. Normal forms associated with codimension-two Hopf–Turing bifurcation are derived, which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the Hopf–Turing bifurcation point. In the vicinity of this degenerate point, a wealth of complex spatiotemporal dynamics are observed. Our theoretical results are confirmed by numerical simulations.

64 citations


Journal ArticleDOI
TL;DR: The paper discusses the dynamical behaviors of a discrete-time SIR epidemic model and it is shown that the model undergoes flip bifurcation and Hopf bIfurcation by using center manifold theorem and b ifurcation theory.

62 citations


Journal ArticleDOI
TL;DR: In this paper, a nonlinear time-varying dynamic model for a multistage planetary gear train, considering time varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated.
Abstract: A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincare map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method for computing modal coupling coefficients for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. But their method is not suitable for the case of a single oscillator.
Abstract: We propose a direct method for computing modal coupling coefficients--due to geometrically nonlinear effects--for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the FE formulation, with specificities pertaining to the shell elements considered, namely, here elements of the "Mixed Interpolation of Tensorial Components" family. Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing--with a continuation procedure --the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes, and we find that the most important part of the dynamics can be predicted with a single oscillator equation.

Journal ArticleDOI
TL;DR: A new modification to the Chua oscillator is proposed, changing the nonlinear term of the original oscillator to a smooth and bounded nonlinear function and an application to secure communications is proposed in which two channels are used.

Journal ArticleDOI
TL;DR: The conditions for existence and stability of the fixed points of the discrete reduced Lorenz system are derived and the complex dynamics, bifurcations and chaos are displayed by numerical simulations.

Journal ArticleDOI
TL;DR: In this paper, the authors adopt the time-delayed feedback control, and convert chaos control to the Hopf bifurcation of the delayed feedback system, showing that the excitable neuron can emit spikes via the subcritical Hopf Bifurcation, and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current.
Abstract: This paper is concerned with bifurcations and chaos control of the Hindmarsh-Rose (HR) neuronal model with the time-delayed feedback control. By stability and bifurcation analysis, we find that the excitable neuron can emit spikes via the subcritical Hopf bifurcation, and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current. For the purpose of control of chaos, we adopt the time-delayed feedback control, and convert chaos control to the Hopf bifurcation of the delayed feedback system. Then the analytical conditions under which the Hopf bifurcation occurs are given with an explicit formula. Based on this, we show the Hopf bifurcation curves in the two-parameter plane. Finally, some numerical simulations are carried out to support the theoretical results. It is shown that by appropriate choice of feedback gain and time delay, the chaotic orbit can be controlled to be stable. The adopted method in this paper is general and can be applied to other neuronal models. It may help us better understand the bifurcation mechanisms of neural behaviors.

Journal ArticleDOI
09 Jun 2014-Chaos
TL;DR: It is demonstrated how the organizing centers-points corresponding to codimension-two homoclinic bifurcations-along with fold and period-doubling bIfurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotics due to the hysteresis.
Abstract: We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines—exact bifurcation curves. We demonstrate how the organizing centers—points corresponding to codimension-two homoclinic bifurcations—along with fold and period-doubling bifurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis.

Journal ArticleDOI
TL;DR: In this paper, a family of graphs that develop turning singularities (i.e., their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function is analyzed.
Abstract: We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.

Journal ArticleDOI
TL;DR: In this article, the stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied in a ratio-dependent predator-prey model with diffusion.
Abstract: In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.


Journal ArticleDOI
TL;DR: In this article, complex bifurcation scenarios occurring in the dynamic response of a piecewise-linear impact oscillator with drift are investigated, which is able to describe qualitatively the behaviour of impact drilling systems.
Abstract: We investigate the complex bifurcation scenarios occurring in the dynamic response of a piecewise-linear impact oscillator with drift, which is able to describe qualitatively the behaviour of impact drilling systems. This system has been extensively studied by numerical and analytical methods in the past, but its intricate bifurcation structure has largely remained unknown. For the bifurcation analysis, we use the computational package TC-HAT, a toolbox of AUTO 97 for numerical continuation and bifurcation detection of periodic orbits of non-smooth dynamical systems (Thota and Dankowicz, SIAM J Appl Dyn Syst 7(4):1283–322, 2008) The study reveals the presence of co-dimension-1 and -2 bifurcations, including fold, period-doubling, grazing, flip-grazing, fold-grazing and double grazing bifurcations of limit cycles, as well as hysteretic effects and chaotic behaviour. Special attention is given to the study of the rate of drift, and how it is affected by the control parameters.

Journal ArticleDOI
TL;DR: A ratio-dependent predator-prey system with Holling type II functional response, two time delays and stage structure for the predator is investigated and the sufficient conditions for the local stability and the existence of Hopf bifurcation with respect to both delays are established.

Journal ArticleDOI
Hongbin Fang1, Jian Xu1
TL;DR: In this article, a two-parameter bifurcation problem is theoretically analyzed and the corresponding bifurburcation diagram is presented, where branches of the BIFurcation are derived in view of classical mechanics.
Abstract: Vibration-driven systems can move progressively in resistive media owing to periodic motions of internal masses. In consideration of the external dry friction forces, the system is piecewise smooth and has been shown to exhibit different types of stick-slip motions. In this paper, a vibration-driven system with Coulomb dry friction is investigated in terms of sliding bifurcation. A two-parameter bifurcation problem is theoretically analyzed and the corresponding bifurcation diagram is presented, where branches of the bifurcation are derived in view of classical mechanics. The results show that these sliding bifurcations organize different types of transitions between slip and sticking motions in this system. The bifurcation diagram and the predicted stick-slip transitions are verified through numerical simulations. Considering the effects of physical parameters on average steadystate velocity and utilizing the sticking feature of the system, optimization of the system is performed. Better performance of the system with no backward motion and higher average steady-state velocity can be achieved, based on the proposed optimization procedures. [DOI: 10.1115/1.4025747]

Journal ArticleDOI
TL;DR: Numerical evidence shows that the presence of harvesting can impact the existence of species and over harvesting can result in the extinction of the prey or the predator which is in line with reality.

Journal ArticleDOI
TL;DR: In this article, a wing-like plate in supersonic flow cantilevered at its root is studied, and the Rayleigh-Ritz approach is adopted to discretize (and reduce the order of) the partial differential equations of the plate and the resulting ODEs are solved numerically by the fourth-order Runge-Kutta (RK4) method.

Journal ArticleDOI
TL;DR: In this paper, the authors investigate an inertial two-neural coupling system with multiple delays and find that multiple delays can induce the system to exhibit stable switching between the resting state and periodic motion.
Abstract: In this paper, we investigate an inertial two-neural coupling system with multiple delays. We analyze the number of equilibrium points and demonstrate the corresponding pitchfork bifurcation. Results show that the system has a unique equilibrium as well as three equilibria for different values of coupling weights. The local asymptotic stability of the equilibrium point is studied using the corresponding characteristic equation. We find that multiple delays can induce the system to exhibit stable switching between the resting state and periodic motion. Stability regions with delay-dependence are exhibited in the parameter plane of the time delays employing the Hopf bifurcation curves. To obtain the global perspective of the system dynamics, stability and periodic activity involving multiple equilibria are investigated by analyzing the intersection points of the pitchfork and Hopf bifurcation curves, called the Bogdanov-Takens (BT) bifurcation. The homoclinic bifurcation and the fold bifurcation of limit cycle are obtained using the BT theoretical results of the third-order normal form. Finally, numerical simulations are provided to support the theoretical analyses.

Journal ArticleDOI
TL;DR: In this article, a four-dimensional four-parameter Chua model with cubic nonlinearity was studied applying numerical continuation and numerical solutions methods, and the bifurcation curves of the model were obtained with the possibility to describe the shrimp-shaped domains and their endoskeletons.

Journal ArticleDOI
TL;DR: In this article, the dynamics of a two-dimensional discrete Hindmarsh-Rose model is discussed, and it is shown that the system undergoes flip bifurcation, Neimark-Sacker bifurbcation, and 1:1 resonance by using a center manifold theorem and bifurlcation theory.
Abstract: In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.

Journal ArticleDOI
TL;DR: The alteration of the membrane properties of the Morris–Lecar neurons is discussed and different membrane excitability is obtained by bifurcation analysis and frequency-current curves.
Abstract: In this paper, we investigate the dynamical behaviors of a Morris---Lecar neuron model. By using bifurcation methods and numerical simulations, we examine the global structure of bifurcations of the model. Results are summarized in various two-parameter bifurcation diagrams with the stimulating current as the abscissa and the other parameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay much attention to the emergence of periodic solutions and bistability. Different membrane excitability is obtained by bifurcation analysis and frequency-current curves. The alteration of the membrane properties of the Morris---Lecar neurons is discussed.

Journal ArticleDOI
TL;DR: A two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point is considered, and the bifurcation structures in a parameter plane of the map are investigated.
Abstract: We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to co...

Journal ArticleDOI
TL;DR: In this article, the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of fractional order in this study, and the discharge frequency of the neuron is greater than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic or chaotic firing.
Abstract: We find that the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of the fractional order in this study. There exists a clear difference in the bifurcation diagram between the fractional-order Hindmarsh-Rose model and the corresponding integer-order model even though the neuron undergoes a Hopf bifurcation to oscillation and then starts a period-doubling cascade to chaos with the decrease of the externally applied current. Interestingly, the discharge frequency of the fractional-order Hindmarsh-Rose model neuron is greater than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic or chaotic firing. Then we demonstrate that the firing behavior of the fractional-order Hindmarsh-Rose model neuron has a higher complexity than that of the integer-order counterpart. Also, the synchronization phenomenon is investigated in the network of two electrically coupled fractional-order model neurons. We show that the synchronization rate increases as the fractional order decreases.