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1.
The dynamical behaviour of the generalized Korteweg-de
Vries (KdV) equation under a periodic perturbation is investigated
numerically. The bifurcation and chaos in the system are observed by
applying bifurcation diagrams, phase portraits and Poincaré maps.
To characterise the chaotic behaviour of this system, the spectra of
the Lyapunov exponent and Lyapunov dimension of the attractor are also
employed. 相似文献
2.
Looking for new light sources, especially short wavelength laser light sources has attracted widespread attention. This paper analytically describes the radiation of a crystalline undulator field by the sine-squared potential. In the classical mechanics and the dipole approximation, the motion equation of a particle is reduced to a generalized pendulum equation with a damping term and a forcing term. The bifurcation behavior of periodic orbits is analyzed by using the Melnikov method and the numerical method, and the stability of the system is discussed. The results show that, in principle, the stability of the system relates to its parameters, and only by adjusting these parameters appropriately can the occurrence of bifurcation be avoided or suppressed. 相似文献
3.
An improved hyper-chaotic system based on the hyper-chaos generated
from Chen's system is presented, and some basic dynamical properties
of the system are investigated by means of Lyapunov exponent
spectrum, bifurcation diagrams and characteristic equation roots.
Simulations show that the new improved system evolves into
hyper-chaotic, chaotic, various quasi-periodic or periodic orbits
when one parameter of the system is fixed to be a certain value
while the other one is variable. Some computer simulations and
bifurcation analyses are given to testify the findings. 相似文献
4.
Phase synchronization and its bursting neurons: theoretical transition in two coupled and numerical analysis 下载免费PDF全文
It is crucially important to study different synchronous regimes in coupled neurons because different regimes may correspond to different cognitive and pathological states. In this paper, phase synchronization and its transitions are discussed by means of theoretical and numerical analyses. In two coupled modified Morris-Lecar neurons with a gap junction, we show that the occurrence of phase synchronization can be investigated from the dynamics of phase equation, and the analytical synchronization condition is derived. By defining the phase of spike and burst, the transitions from burst synchronization to spike synchronization and then toward nearly complete synchronization can be identified by bifurcation diagrams, the mean frequency difference and time series of neurons. The simulation results suggest that the synchronization of bursting activity is a multi-time-scale phenomenon and the phase synchronization deduced by the phase equation is actually spike synchronization. 相似文献
5.
Direct time delay feedback can make non-chaotic Chen
circuit chaotic. The chaotic Chen circuit with direct time delay
feedback possesses rich and complex dynamical behaviours. To reach a
deep and clear understanding of the dynamics of such circuits
described by delay differential equations, Hopf bifurcation in the
circuit is analysed using the Hopf bifurcation theory and the
central manifold theorem in this paper. Bifurcation points and
bifurcation directions are derived in detail, which prove to be
consistent with the previous bifurcation diagram. Numerical
simulations and experimental results are given to verify the
theoretical analysis. Hopf bifurcation analysis can explain and
predict the periodical orbit (oscillation) in Chen circuit with
direct time delay feedback. Bifurcation boundaries are derived using
the Hopf bifurcation analysis, which will be helpful for determining
the parameters in the stabilisation of the originally chaotic
circuit. 相似文献
6.
Phase synchronization and its transition in two coupled bursting neurons: theoretical and numerical analysis 下载免费PDF全文
It is crucially important to study different synchronous
regimes in coupled neurons because different regimes may correspond
to different cognitive and pathological states. In this paper, phase
synchronization and its transitions are discussed by means of
theoretical and numerical analyses. In two coupled modified
Morris--Lecar neurons with a gap junction, we show that the occurrence
of phase synchronization can be investigated from the dynamics of
phase equation, and the analytical synchronization condition is
derived. By defining the phase of spike and burst, the transitions
from burst synchronization to spike synchronization and then toward
nearly complete synchronization can be identified by bifurcation
diagrams, the mean frequency difference and time series of neurons.
The simulation results suggest that the synchronization of bursting
activity is a multi-time-scale phenomenon and the phase
synchronization deduced by the phase equation is actually spike
synchronization. 相似文献
7.
Bifurcation diagram globally underpinning neuronal firing behaviors modified by SK conductance 下载免费PDF全文
Neurons in the brain utilize various firing trains to encode the input signals they have received.Firing behavior of one single neuron is thoroughly explained by using a bifurcation diagram from polarized resting to firing,and then to depolarized resting.This explanation provides an important theoretical principle for understanding neuronal biophysical behaviors.This paper reports the novel experimental and modeling results of the modification of such a bifurcation diagram by adjusting small conductance potassium(SK)channel.In experiments,changes in excitability and depolarization block in nucleus accumbens shell and medium-spiny projection neurons are explored by increasing the intensity of injected current and blocking the SK channels by apamin.A shift of bifurcation points is observed.Then,a Hodgkin–Huxley type model including the main electrophysiological processes of such neurons is developed to reproduce the experimental results.The reduction of SK channel conductance also shifts the bifurcations,which is in consistence with experiment.A global bifurcation paradigm of this shift is obtained by adjusting two parameters,intensity of injected current and SK channel conductance.This work reveals the dynamics underpinning modulation of neuronal firing behaviors by biologically important ionic conductance.The results indicate that small ionic conductance other than that responsible for spike generation can modify bifurcation points and shift the bifurcation diagram and,thus,change neuronal excitability and adaptation. 相似文献
8.
Dynamic bifurcation of a modified Kuramotoben Sivashinsky equation with higher-order nonlinearity 下载免费PDF全文
Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto—Sivashinsky equation with a higher-order nonlinearity μ(ux)puxx are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained. 相似文献
9.
Bifurcation and chaos analysis of a nonlinear electromechanical coupling relative rotation system 下载免费PDF全文
Hopf bifurcation and chaos of a nonlinear electromechanical coupling relative rotation system are studied in this paper. Considering the energy in air-gap field of AC motor, the dynamical equation of nonlinear electromechanical coupling relative rotation system is deduced by using the dissipation Lagrange equation. Choosing the electromagnetic stiffness as a bifurcation parameter, the necessary and sufficient conditions of Hopf bifurcation are given, and the bifurcation characteristics are studied. The mechanism and conditions of system parameters for chaotic motions are investigated rigorously based on the Silnikov method, and the homoclinic orbit is found by using the undetermined coefficient method. Therefore, Smale horseshoe chaos occurs when electromagnetic stiffness changes. Numerical simulations are also given, which confirm the analytical results. 相似文献
10.
In this paper, new explicit and exact travelling wave solutions for a compound KdV-Burgers equation are obtained by using the hyperbola function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV-Burgers and mKdV equations can be solved by this method. The method can also solve other nonlinear partial differential equations. 相似文献
11.
Linear stability of solitary waves near transcritical bifurcations is analyzed for the generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions. Bifurcation of linear-stability eigenvalues associated with this transcritical bifurcation is analytically calculated. Based on this eigenvalue bifurcation, it is shown that both solution branches undergo stability switching at the transcritical bifurcation point. In addition, the two solution branches have opposite linear stability. These analytical results are compared with the numerical results, and good agreement is obtained. 相似文献
12.
自适应复杂网络是以节点状态与拓扑结构之间存在反馈回路为特征的网络. 针对自适应网络病毒传播模型, 利用非线性微分动力学系统研究病毒传播行为; 通过分析非线性系统对应雅可比矩阵的特征方程, 研究其平衡点的局部稳定性和分岔行为, 并推导出各种分岔点的计算公式. 研究表明, 当病毒传播阈值小于病毒存在阈值, 即R00c时, 网络中病毒逐渐消除, 系统的无病毒平衡点是局部渐近稳定的; R0c0<1时, 网络出现滞后分岔, 产生双稳态现象, 系统存在稳定的无病毒平衡点、较大稳定的地方病平衡点和较小不稳定的地方病平衡点; R0>1时, 网络中病毒持续存在, 系统唯一的地方病平衡点是局部渐近稳定的. 研究发现, 系统先后出现了鞍结分岔、跨临界分岔、霍普夫分岔等分岔行为. 最后通过数值仿真验证所得结论的正确性.
关键词:
自适应网络
稳定性
分岔
基本再生数 相似文献
13.
Haberman R 《Chaos (Woodbury, N.Y.)》2000,10(3):641-648
One-degree of freedom conservative slowly varying Hamiltonian systems are analyzed in the case in which a saddle-center pair undergo a transcritical bifurcation. We analyze the case in which the method of averaging predicts the solution crosses the unperturbed homoclinic orbit at the precise time at which the transcritical bifurcation occurs. For the slow passage through the nonhyperbolic homoclinic orbit associated with a transcritical bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits surrounded by autonomous nonlinear saddle approaches. The change in action is computed by matching these solutions to those obtained by averaging, valid before and after crossing the nonhyperbolic homoclinic orbit. For initial conditions near the stable manifold of the nonhyperbolic saddle point, one saddle approach has particularly small energy and instead satisfies a nonautonomous nonlinear equation, which provides a transition between nonhyperbolic homoclinic orbits, centers, and saddles. (c) 2000 American Institute of Physics. 相似文献
14.
Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons
The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV–Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior. 相似文献
15.
John David Crawford 《Journal of statistical physics》1994,74(5-6):1047-1084
We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens-Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes. 相似文献
16.
对含有两个时滞参数、受简谐激励作用下的van der Pol-Duffing方程进行了研究,着重研究了时滞参数对该类参数激励系统的主共振的分岔响应控制.首先采用摄动法从理论上推导出时滞动力系统的分岔响应方程,用奇异性理论得到了退化余维一分岔和余维二分岔的条件,以及Hopf分岔的存在性及发生该分岔的条件,最后用数值模拟的方法研究了时滞参数对系统分岔响应的影响.研究结果表明,适当选取时滞参数,不仅可以改变分岔响应曲线的拓扑形态, 还可以改变分岔点的位置.
关键词:
摄动法
分岔控制
时滞动力系统 相似文献
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19.
Bifurcation and solitary waves of the nonlinear wave equation with quartic polynomial potential 下载免费PDF全文
For the nonlinear wave equation with quartic polynomial potential, bifurcation and solitary waves are investigated. Based on the bifurcation and the energy integral of the two-dimensional dynamical system satisfied by the travelling waves, it is very interesting to find different sufficient and necessary conditions in terms of the bifurcation parameter for the existence and coexistence of bright, dark solitary waves and shock waves. The method of direct integration is developed to give all types of solitary wave solutions. Our method is simpler than other newly developed ones. Some results are similar to those obtained recently for the combined KdV-mKdV equation. 相似文献
20.
We investigate the role of multistable states on the occurrence of vibrational resonance in a periodic potential system driven by both a low-frequency and a high-frequency periodic force in both underdamped and overdamped limits. In both cases, when the amplitude of the high-frequency force is varied, the response amplitude at the low-frequency exhibits a series of resonance peaks and approaches a limiting value. Using a theoretical approach, we analyse the mechanism of multiresonance in terms of the resonant frequency and the stability of the equilibrium points of the equation of motion of the slow variable. In the overdamped system, the response amplitude is always higher than in the absence of the high-frequency force. However, in the underdamped system, this happens only if the low-frequency is less than 1. In the underdamped system, the response amplitude is maximum when the equilibrium point around which slow oscillations take place is maximally stable and minimum at the transcritical bifurcation. And in the overdamped system, it is maximum at the transcritical bifurcation and minimum when the associated equilibrium point is maximally stable. When the periodicity of the potential is truncated, the system displays only a few resonance peaks. 相似文献