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2.
We seize some new dynamics of a Lorenz-like system: $dot{x} = a(y - x)$, quad $dot{y} = dy - xz$, quad $dot{z} = - bz + fx^{2} + gxy$, such as for the Hopf bifurcation, the behavior of non-isolated equilibria, the existence of singularly degenerate heteroclinic cycles and homoclinic and heteroclinic orbits. In particular, our new discovery is that the system has also two heteroclinic orbits for $bg = 2a(f + g)$ and $a > d > 0$ other than known $bg > 2a(f + g)$ and $a > d > 0$, whose proof is completely different from known case. All the theoretical results obtained are also verified by numerical simulations. 相似文献
3.
软弹簧型Duffing方程在摄动下分支出的极限环 总被引：5，自引：0，他引：5
在这篇文章中,作者用Melnikov函数方法分析了软弹簧型Duffing方程[1]在摄动下异宿轨道破裂后稳定流形与不稳定流形的相对位置,给出了方程在不同摄动下分支出极限环的条件与极限环的位置. 相似文献
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The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What\"s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor. 相似文献
6.
The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature [Nonlinear Dyn (2010) 60(3): 443--457] with the entitle ``A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems'' and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby. 相似文献
7.
§ 1 IntroductionThe study of nonlinear dynamics is a fascinating problem which is at the very heartofthe understanding ofmany importantproblems ofthe natural sciences.Infinite dimensionaldynamical systems are very important in nonlinear dynamics.For an infinite dimensionaldynamical system,we mainly study the existence and the structure ofthe attractors.Thereare detailed discussions in [1 ] .Itis especially mentioned there thatthe attractors of gradi-ent systems are of simple structure in s… 相似文献
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Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated
existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and bifurcation surfaces of the two-fold
periodic orbit are also obtained. At last, these bifurcation results are applied to the fine heteroclinic loop for the planar
system, which leads to some new and interesting results. 相似文献
10.
Bifurcations of heteroclinic loops 总被引：14，自引：0，他引：14
By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system
is established in a neighborhood of the heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits.
Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature
concerned with the 1-hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the
1-per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1-hom and 1-per orbit and the inexistence
of the 2-hom and 2-per orbit are also obtained.
Project supported by the National Natural Science Foundation of China (Grant No. 19771037) and the National Science Foundation
of America # 9357622. This paper was completed when the first author was visiting Northwestern University. 相似文献
11.
本文用Melnikov函数方法讨论了一类扩张了的软弹簧型Duffing方程(k=1,2,3,…)在周期激励下的紊动现象.给出了出现二阶同宿切的条件.文中所采用的方法对于不能给出并宿轨道的显式的系统的研究是非常有用的. 相似文献
12.
There has been growing interest in analyzing stability in design controls of stochastic systems. This interest arises out of the need to develop robust control strategies for systems with uncertain dynamics. This paper is concerned with the examination of conditions under which the desired structure of a stochastic interval system with time dependent parameters is stabilizable. Necessary and sufficient condition under which two-level preconditioner guarantees quadratic mean exponential stability of the desired structure of uncontrolled stochastic interval system is presented. Sufficient condition for exponential stability of the equilibrium solution of uncontrolled stochastic interval system is also presented. 相似文献
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Guojun Xin Shida Liu Shikou Liu Fuming Liang 《Communications in Nonlinear Science & Numerical Simulation》1996,1(4):23-26
In this paper, KdV-Burgers equation can be regarded as the normal equation of atmospheric turbulence in the stable boundary layer. On the basis of the travelling wave analytic solution of KdV-Burgers equation, the turbulent spectrum is obtained. We observe that the behavior of the spectra is consistent with actual turbulent spectra of stable atmospheric boundary layer. 相似文献
15.
Makoto Hayashi 《Mathematica Slovaca》2009,59(4):505-515
In this paper, the non-existence of limit cycles of a Liénard system ẋ = y−F(x), ẏ = −g(x) is discussed. By using the transformation y = z+ϕ(x) to the system, the new system has two special isoclines. We call the curves Vertical isocline or Horizontal isocline, respectively. It shall be shown that the existence of these isoclines play an important role in the non-existence of limit cycles of the system. The results are applied to many examples, and the known results are improved in certain cases. The results were announced at Annual Meeting of Mathematical Society of Japan on September 19 of 2005. Also they were published at the Poster Competition of ICM(Madrid) on August 22–30 of 2006. 相似文献
16.
研究一类含有两个参数和有理奇性平面哈密顿系统的同宿与异宿轨道,该问题来源于一个关于聚合物流体剪切流动特性的研究．借助常微定性理论和不变流形分析的方法,文中给出了系统存在同宿与异宿轨道的条件,并通过数值计算检验了所得理论结果。 相似文献
17.
Tibor Krisztin 《Periodica Mathematica Hungarica》2008,56(1):83-95
In this survey paper the delay differential equation (t) = −μx(t) + g(x(t − 1)) is considered with μ ≥ 0 and a smooth real function g satisfying g(0) = 0. It is shown that the dynamics generated by this simple-looking equation can be very rich. The dynamics is completely
understood only for a small class of nonlinearities. Open problems are formulated.
Supported in part by the Hungarian NFSR, Grant No. T049516. 相似文献
18.
Gamaliel Blé Víctor Castellanos Jaume Llibre 《Mathematical Methods in the Applied Sciences》2016,39(14):3996-4006
We are interested in the coexistence of three species forming a tritrophic food chain model. Considering a linear grow for the lowest trophic species, Holling III and Holling II functional response for the predator and the top‐predator, respectively. We prove that this model has stable periodic orbits for adequate values of its parameters. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
19.
Vittorino Pata 《Central European Journal of Mathematics》2009,7(3):487-492
A classical result on the existence of global attractors for gradient systems is extended to the case of a semigroup S(t) lacking strong continuity, but satisfying the weaker property of being a closed map for every fixed t ≥ 0.
相似文献
20.
Seasonality is pervasive in nature and WNV is a complex disease which appears to be transmitted periodically. Lacking of vaccine and anti-virus treatment brings culling to be an effective strategy of controlling the spread of WNV. This paper proposes a piecewise smooth model of WNV transmission between mosquitoes and birds with periodic forcing by employing the threshold policy of culling mosquitoes and birds when the number of infected birds exceeds the critical level. We firstly investigate the dynamical behaviours of the periodic subsystems and illustrate the existence and global stability of the disease-free and positive periodic solutions. Then by assuming a periodic variation in the total number of mosquitoes and birds with and without culling strategy respectively in the ecosystem, the subsystems are reduced, respectively. Further, the sliding mode domain and sliding mode dynamics are derived by using the Filippov convex method and the existence of the sliding periodic solution is illustrated. Finally, numerical investigations are carried out to address how the periodic forcing and culling strategy affect the pattern and amplitude of the novel periodic solution. Our results show that a suitable combination of threshold value and culling rates should be properly chosen to ensure that the ideal periodic solution with a low number of infected birds/mosquitoes oscillating in a small range is globally stable and consequently is approximated, which suggests useful strategies of curbing WNV transmission by setting proper threshold level and culling rates. 相似文献