Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of ann-cycle (n > 1) approach the discontinuities of thenth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.     

A two-dimensional Fokker-Planck equation degenerating on a straight line

By an example of a two-dimensional hydrodynamic system, second-order Langevin equations with two correlated noise sources are investigated. It is shown that the asymptotic expression (t) for the stationary distribution functionP depends on the order in which the limiting transitions;t andN ₂₂0 (N ₂₂ is the power of one of the noises) are made. Using the method of local expansions in trigonometric form, approximate expressions are written for the distribution functionP at small but finiteN ₂₂ tending atN ₂₂0 to the known exact solution.     

The influence of noise on the logistic model

The behavior of the logistic system which is generated by the functionf(x =ax (1–x) changes in an interesting way if it is perturbed by external noise. It turns out that the chaotic behavior which was predicted by Li and Yorke for orbits of period 3, becomes visible and that a sequence of mergence transitions occurs at the critical parameter. The change of the invariant probability density and the Lyapunov exponents are examined numerically. The power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete-time analog of a Chapman-Kolmogorov equation.     

Generation of a magnetic field by convective flows in a rotating horizontal layer

The generation of a magnetic field by convective flows of a conducting fluid in a rotating plane layer is investigated numerically. The problem is considered in the complete three-dimensional nonlinear formulation. The sequence of temporal regimes that ensue as the Taylor number Ta increases from 0 (no rotation) to 2000 (the fluid motion is suppressed by rapid rotation) when the other parameters are fixed is studied. The Ta intervals on which bifurcations occur are found, and the breakdown and onset of symmetries in the attractors that arise is investigated.     

Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds

In this paper we establish an SIR model with a standard incidence rate and a nonlinear recovery rate, formulated to consider the impact of available resource of the public health system especially the number of hospital beds. For the three dimensional model with total population regulated by both demographics and diseases incidence, we prove that the model can undergo backward bifurcation, saddle-node bifurcation, Hopf bifurcation and cusp type of Bogdanov–Takens bifurcation of codimension 3. We present the bifurcation diagram near the cusp type of Bogdanov–Takens bifurcation point of codimension 3 and give epidemiological interpretation of the complex dynamical behaviors of endemic due to the variation of the number of hospital beds. This study suggests that maintaining enough number of hospital beds is crucial for the control of the infectious diseases.     

Bifurcation analysis on a survival red blood cells model

In this paper, we consider a model described the survival of red blood cells in animal. Its dynamics are studied in terms of local and global Hopf bifurcations. We show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay crosses some critical values. Using the reduced system on the center manifold, we also obtain that the periodic orbits bifurcating from the positive equilibrium are stable in the center manifold, and all Hopf bifurcations are supercritical. Further, particular attention is focused on the continuation of local Hopf bifurcation. We show that global Hopf bifurcations exist after the second critical value of time delay.     

Dynamical analysis,feedback control and synchronization of Liu dynamical system

Dynamical behaviors of Liu system is studied using Routh–Hurwitz criteria, Center manifold theorem and Hopf bifurcation theorem. Periodic solutions and their stabilities about the equilibrium points are studied by using Hsü & Kazarinoff theorem. Linear feedback control techniques are used to stabilize and synchronize the chaotic Liu system.     

Melnikov函数符号的应用

用Melnikov函数的符号判断未摄动系统是Hamilton系统的二维系统x′=f(x)+εg(x,a),0<ε<<1,a∈R的周期解的存在性和稳定性.其结果可应用于具有双重零特征值时流的余维二分支的分支集的相图构造.     

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given. Project supported by the National Natural Science Foundation of China (No: 10171044), the Natural Science Foundation of Jiangsu Province (No: BK2001024), the Foundation for University Key Teachers of the Ministry of Education of China     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.