Showing papers on "Bifurcation diagram published in 2015"

The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems

Abstract: The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed. A new procedure for the tracking of Neimark–Sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed. The HB method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device.

Hidden attractors in a chaotic system with an exponential nonlinear term

Abstract: Studying systems with hidden attractors is new attractive research direction because of its practical and threoretical importance. A novel system with an exponential nonlinear term, which can exhibit hidden attractors, is proposed in this work. Although new system possesses no equilibrium points, it displays rich dynamical behaviors, like chaos. By calculating Lyapunov exponents and bifurcation diagram, the dynamical behaviors of such system are discovered. Moreover, two important features of a chaotic system, the possibility of synchronization and the feasibility of the theoretical model, are also presented by introducing an adaptive synchronization scheme and designing a digital hardware platform-based emulator.

Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Delay

Abstract: In the present paper, we investigate the impact of fear in a tri-trophic food chain model. We propose a three-species food chain model, where the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is suppressed due to the cost of fear of middle predator. Mathematical properties such as equilibrium analysis, stability analysis, bifurcation analysis and persistence have been investigated. We also describe the global stability analysis of the equilibrium points. Our numerical simulations reveal that cost of fear in basal prey may exhibit bistability by producing unstable limit cycles, however, fear in middle predator can replace unstable limit cycles by a stable limit cycle or a stable interior equilibrium. We observe that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon. We conclude that chaotic dynamics can be controlled by the fear factors. We apply basic tools of nonlinear dynamics such as Poincare s...

A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium

Abstract: A new hyper-chaotic system is presented in this paper by adding a smooth flux-controlled memristor and a cross-product item into a three-dimensional autonomous chaotic system. It is exciting that this new memristive system can show a four-wing hyper-chaotic attractor with a line equilibrium. The dynamical behaviors of the proposed system are analyzed by Lyapunov exponents, bifurcation diagram and Poincare maps. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of hyperchaos in the system is verified theoretically. Finally, an electronic circuit is designed to implement the hyper-chaotic memristive system.

A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model

Abstract: The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as coupled phase oscillators. In this paper, a bifurcation structure of the infinite-dimensional Kuramoto model is investigated. A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength , a non-trivial stable solution, which corresponds to the synchronization, bifurcates from the de-synchronous state. One of the difficulties in proving the conjecture is that a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, has the continuous spectrum on the imaginary axis. Hence, the standard spectral theory is not applicable to prove a bifurcation as well as the asymptotic stability of the steady state. In this paper, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid the continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator will be estimated with the aid of the spectral theory on a rigged Hilbert space to prove the linear stability of the steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite-dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimension because of the continuous spectrum on the imaginary axis. These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto.

Chaos in the fractionally damped broadband piezoelectric energy generator

Abstract: Piezoelectric materials play a significant role in harvesting ambient vibration energy. Due to their inherent characteristics and electromechanical interaction, the system damping for piezoelectric energy harvesting can be adequately characterized by fractional calculus. This paper introduces the fractional model for magnetically coupling broadband energy harvesters under low-frequency excitation and investigates their nonlinear dynamic characteristics. The effects of fractional-order damping, excitation amplitude, and frequency on dynamic behaviors are proposed using the phase trajectory, power spectrum, Poincare map, and bifurcation diagram. The numerical analysis shows that the fractionally damped energy harvesting system exhibits chaos, periodic motion, chaos and periodic motion in turn when the fractional order changes from 0.2 to 1.5. The period doubling route to chaos and the inverse period doubling route from chaos to periodic motion can be clearly observed. It is also demonstrated numerically and experimentally that the magnetically coupling piezoelectric energy harvester possesses the usable frequency bandwidth over a wide range of low-frequency excitation. Both high-energy chaotic attractors and large-amplitude periodic response with inter-well oscillators dominate these broadband energy harvesting.

Bifurcation analysis and Turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality

Abstract: In this paper, we consider a predator-prey model with herd behavior and hyperbolic mortality subject to the homogeneous Neumann boundary condition. Firstly, we prove the existence and uniqueness of positive equilibrium for this model by analytical skills. Then we analyze the stability of the positive equilibrium, Turing instability, and the existence of Hopf, steady state bifurcations. Finally, by calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Meanwhile, for the steady state bifurcation, the possibility of pitchfork bifurcation can be concluded by the normal form, which does also determine the stability of spatially inhomogeneous steady states. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out and expand our theoretical results.

Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems

Abstract: Dynamic mode decomposition (DMD) belongs to a class of data-driven decomposition techniques, which extracts spatial modes of a constant frequency from a given set of numerical or experimental data. Although the modal shapes and frequencies are a direct product of the decomposition technique, the determination of the respective modal amplitudes is non-unique. In this study, we introduce a new algorithm for defining these amplitudes, which is capable of capturing physical growth/decay rates of the modes within a transient signal and is otherwise not straightforward using the standard DMD algorithm. In addition, a parametric DMD algorithm is introduced for studying dynamical systems going through a bifurcation. The parametric DMD alleviates multiple applications of the DMD decomposition to the system with fixed parametric values by including the bifurcation parameter in the decomposition process. The parametric DMD with amplitude correction is applied to a numerical and experimental data sequence taken from thermo-acoustically unstable systems. Using DMD with amplitude correction, we are able to identify the dominant modes of the transient regime and their respective growth/decay rates leading to the final limit-cycle. In addition, by applying parametrized DMD to images of an oscillating flame, we are able to identify the dominant modes of the bifurcation diagram.

Stability and Hopf bifurcation of a complex-valued neural network with two time delays

Abstract: In this paper, a class of complex-valued neural networks with two time delays is considered. By considering that the activation function can be expressed by separating into its real and imaginary part and regarding the sum of time delays as a bifurcating parameter, the dynamical behaviors that include local asymptotical stability and local Hopf bifurcation are investigated. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. The linearized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

The Bogdanov-Takens bifurcation study of 2 m coupled neurons system with 2 m + 1 $2m+1$ delays

Abstract: In this paper the Bogdanov-Takens (BT) bifurcation of an 2m coupled neurons network model with multiple delays is studied, where one neuron is excitatory and the next is inhibitory. When the origin of the model has a double zero eigenvalue, by using center manifold reduction of delay differential equations (DDEs), the second-order and third-order universal unfoldings of the normal forms are deduced, respectively. Some bifurcation diagrams and numerical simulations are presented to verify our main results.

Cusp bifurcation in the eigenvalue spectrum of PT -symmetric Bose-Einstein condensates

Abstract: A Bose-Einstein condensate in a double-well potential features stationary solutions even for attractive contact interaction as long as the particle number and therefore the interaction strength do not exceed a certain limit. Introducing balanced gain and loss into such a system drastically changes the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at which the symmetric and antisymmetric states emerge, one tangent bifurcation between two formerly independent branches arises [D. Haag et al., Phys. Rev. A 89, 023601 (2014)]. We study this transition in detail using a bicomplex formulation of the time-dependent variational principle and find that in fact there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp bifurcation.

Novel Grid Multiwing Butterfly Chaotic Attractors and Their Circuit Design

Abstract: A new method is presented generating grid multiwing butterfly chaotic attractors in this brief. By designing piecewise hysteresis functions to take the place of the state variables of the Lorenz system directly, a novel grid multiwing butterfly chaotic system is constructed. The Lyapunov exponent and the bifurcation diagram are studied. Furthermore, an electronic circuit is designed to implement the system. The experimental results are in agreement with numerical simulation results, which verify the availability and feasibility of this method.

Higher dimensional solitary waves generated by second-harmonic generation in quadratic media

Abstract: Schrodinger type soliton waves generated by second-harmonic generation in higher dimensional quadratic optical media are considered The existence of ground state solutions for spatial dimension from two to five is proved, and the continuous dependence on the parameter and asymptotic behavior of ground state solutions are established Multi-pulse solutions with certain symmetry are also obtained In a bounded domain setting, global bifurcation diagram of multi-pulse solutions are shown by using new technique of double saddle-node bifurcation

Pattern formation in the FitzHugh-Nagumo model

Abstract: In this paper, we investigate the effect of diffusion on pattern formation in FitzHugh-Nagumo model. Through the linear stability analysis of local equilibrium we obtain the condition how the Turing bifurcation, Hopf bifurcation and the oscillatory instability boundaries arise. By using the method of the weak nonlinear multiple scales analysis and Taylor series expansion, we derive the amplitude equations of the stationary patterns. The analysis of amplitude equations shows the occurrence of different complex phenomena, including Turing instability Eckhaus instability and zigzag instability. In addition, we apply this analysis to FitzHugh-Nagumo model and find that this model has very rich dynamical behaviors, such as spotted, stripe and hexagon patterns. Finally, the numerical simulation shows that the analytical results agree with numerical simulation.

Bifurcation and chaos analysis of spur gear pair in two-parameter plane

Abstract: A developed algorithm is designed based on the simple cell mapping method and escape time algorithm to examine every state cell and the dynamic characteristics of the multi-parameter coupling in torsion-vibration gear system. Two different types of bifurcation caused by the intersection of the period-doubling bifurcation curves are researched by analyzing the distribution map and the bifurcation diagram of system’s dynamic characteristic in the parameter plane, $$\omega -F$$ . The occurrence processes of periodic bubbles and saltatory periodic bifurcation are studied. The stationary solution and its phase trajectory of the fractal of the periodic motion attractor boundary are researched too. The sufficient condition of the fractal structure of the periodic attractor domain is achieved. Homoclinic or heteroclinic trajectory in phase space is found caused by the intersection of the different periodic motion trajectories in non-smooth system.

Stability and Bifurcation in a Predator-Prey Model with Age Structure and Delays

Abstract: This paper investigates a predator–prey model with age structure and two delays. By formulating the age-structured model with delays as a non-densely defined Cauchy problem and applying the theory of integrated semigroup and recently established Hopf bifurcation theory for abstract Cauchy problems with non-dense domain, we show that Hopf bifurcation occurs in the model. This also shows the sensitivity of the model dynamics on the threshold $$\tau $$ which might be taken as a measure of a biological maturation period and a time lag between conception and birth. Numerical simulations are performed to illustrate the obtained results and a summary is given.

Nonlinear dynamic characteristics of variable inclination magnetically coupled piezoelectric energy harvesters

Abstract: This paper investigates the nonlinear dynamic characteristics of a magnetically coupled piezoelectric energy harvester under low frequency excitation where the angle of the external magnetic field is adjustable. The nonlinear dynamic equation with the identified nonlinear magnetic force is derived to describe the electromechanical interaction of variable inclination angle harvesters. The effect of excitation amplitude and frequency on dynamic behavior is proposed by using the phase trajectory, power spectrum, and bifurcation diagram. The numerical analysis shows that a rotating magnetically coupled energy harvesting system exhibits rich nonlinear characteristics with the change of external magnet inclination angle. The nonlinear route to and from large amplitude high-energy motion can be clearly observed. It is demonstrated numerically and experimentally that lumped parameters equations with an identified polynomials for magnetic force could adequately describe the characteristics of nonlinear energy harvester. The rotating magnetically coupled energy harvester possesses the usable frequency bandwidth over a wide range of low frequency excitation by adjusting the angular orientation.