Narrow Results

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ². As a sample of our results, we prove that N is biholomorphic to ℂ² if H²(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form where λ₁/λ₂ ∈ ℚ₊. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

In this paper, we systematically investigate the periodic solutions of the Rössler equations up to certain topological length. To overcome the difficulties for a return map that is multivalued and non-invertible in the nonlinear system, we propose a new approach that establishes one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is numerically stable for cycle searching, and two-orbit fragments can be used as basic building blocks to initialize the system. The topological classification based on the whole orbit structure seems more effective than partitioning the Poincaré surface of section. The current research supplies an interesting framework for a systematic classification of periodic orbits in a chaotic flow.

The nonlinear dynamical behaviors of single-walled carbon nanotubes (SWCNTs) with the initial axial stress are investigated based on Donnell's cylindrical shell model. By using the bifurcation theory of dynamical systems on a vibration equation of SWCNTs, the existence of centers, saddles, families of periodic orbits, homoclinic orbits and/or heteroclinic orbits is shown. Under different initial conditions, various sufficient conditions to guarantee the existence of the above solutions are given and the dynamical behaviors of SWCNTs are well-known.

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

In this paper, a novel method for locating and stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems is proposed. The main idea of the method is to formulate the UPO locating problem as an optimization issue by using some inherent properties of UPOs of chaotic systems. The global optimal solution of this problem yields the desired UPO. To avoid a local optimal solution, the state of the controlled chaotic system is absorbed into the initial condition of the optimization problem. The ergodicity of chaotic dynamics guarantees that the optimization process does not stay forever at any local optimal solution. When the chaotic orbit approaches the global optimal solution, which is the desired UPO, the controller will stabilize it at the UPO, and the optimization process will cease simultaneously. The method has been developed for both discrete-time and continuous-time systems, and validated for some typical chaotic systems such as the Hénon map and the Duffing oscillator, among others.

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion is based on the collision between the periodic and the hyperbolic fixed point involved in the bifurcation. The second uses the infinity condition of the period of the periodic orbit. For illustration a specific system is investigated.

We consider the class of two-dimensional maps of the form F(x,y) = (g(y),f(x)), (x,y) ∈ [0,1] × [0,1] = I², where f and g are continuous interval maps. The paper deals with the structure of minimal sets for this class of maps. We give a complete description of finite minimal sets and prove some partial results concerning the infinite case.

We show that a discrete dynamical system with delayed arguments that describe the computational performance of a neural network with a loop structure can exhibit the coexistence of huge number of periodic solutions, and we describe the domains of attraction for the stable periodic orbits. This demonstrates a great potential of discrete time delayed neural nets with a loop structure for the purpose of storage and retrieval of periodic patterns.

Bifurcations of homoclinic orbit with orbit-flip and resonant eigenvalues corresponding to the tangent directions are investigated in a four-dimensional system. The existence, number, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are given, and the bifurcation surfaces and the existence regions are also located.

Some techniques based on delay coordinates embedding for detecting unstable periodic orbits from chaotic time series work successfully in several applications but frequently yield false periodic orbits. We present a simple method for excluding false ones from candidates of periodic orbits obtained from chaotic time series. We demonstrate our method in numerical simulations for the Ikeda map and experiments for periodically forced, single and coupled pendula.

Consider a three-dimensional system having an invariant surface and a weak focus of order k on the invariant surface. By using bifurcation techniques and analyzing the solutions of bifurcation equations, we study the spatial bifurcation phenomena near the weak focus and obtain the conditions for the existence of periodic orbits near the weak focus. An application of our results to a SIRS epidemiological model is given.

In this work, the evolution of a family of open-to-closed curves of saddle-node bifurcations of periodic orbits is numerically studied in Chua's equation. This family is organized by open and closed curves of Shil'nikov homoclinic connections in the presence of a nontransversal T-point. We remark the important role of cusp catastrophes of periodic orbits in the complete mechanism of creation/destruction of such closed curves of saddle-node bifurcations.

In this paper we describe a rigorous method to find for a continuous system all low-period cycles enclosed in a certain region of the state space. As an example of the application of this technique we present the complete analysis of the Rössler attractor in terms of short periodic orbits. We construct the trapping region for the associated Poincaré map. The trapping region encloses the numerically observed attractor. We find all periodic orbits of the Poincaré map up to period 20. We prove that these periodic orbits correspond to all cycles of the flow with period shorter than 120. Using the number of short periodic orbits we compute estimates for topological entropy of the Poincaré map and the flow.

Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2ⁿ-homoclinic orbit, 1-periodic orbit and 2ⁿ-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

We study the bifurcation of hyperbolic periodic orbits from a four-dimensional nonlinear center in a class of differential systems. The tool for proving these results is the averaging theory.

In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behavior. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p₁ and one hyperbolic saddle p₂ are investigated, where p₁ is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p₁ (resp. p₂) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p₁ splits into two hyperbolic saddles and , a heteroclinic loop connecting and p₂, homoclinic loop with (resp. p₂) and heteroclinic orbit joining and (resp. and p₂; p₂ and ) are found. The results achieved here can be extended to higher dimensional systems.

The local moving frame approach is employed to study the bifurcation of a degenerate heterodimensional cycle with orbit-flip in its nontransversal orbit. Under some generic hypotheses, we provide the conditions for the existence, uniqueness and noncoexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. And we also present the coexistence conditions for the homoclinic orbit and the periodic orbit. But it is impossible for the coexistence of the periodic orbit and the persistent heterodimensional cycle or the coexistence of the homoclinic loop and the persistent heterodimensional cycle. Moreover, the double and triple periodic orbit bifurcation surfaces are established as well. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. An example of application is also given to demonstrate our main results.

We provide an algorithm for studying invariant tori fulfilled by periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in p : q resonance. We illustrate the algorithm with an application.