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Ale Jan Homburg - One of the best experts on this subject based on the ideXlab platform.
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Construction of codimension one Homoclinic cycles
Dynamical Systems, 2013Co-Authors: Ale Jan Homburg, Maria Kellner, Jürgen KnoblochAbstract:We give an explicit construction of families of Dm-equivariant polynomial vector fields in possessing a codimension one Homoclinic cycle. The Homoclinic cycle consists of m Homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these Homoclinic cycles, in particular for m equal to a multiple of 4, where the bifurcations form an open problem.
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Lorenz attractors in unfoldings of Homoclinic-flip bifurcations
Dynamical Systems, 2010Co-Authors: A. Golmakani, Ale Jan HomburgAbstract:Lorenz-like attractors are known to appear in unfoldings from certain codimension two Homoclinic bifurcations for differential equations in ℝ3 that possess a reflectional symmetry. This includes Homoclinic loops under a resonance condition and the inclination-flip Homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two Homoclinic bifurcation (for Homoclinic loops to equilibria with real different eigenvalues); the orbit-flip Homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.
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Homoclinic and heteroclinic bifurcations in vector fields
Handbook of Dynamical Systems, 2010Co-Authors: Ale Jan Homburg, Björn SandstedeAbstract:An overview of Homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, Homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round Homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of Homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin’s method, Shil’nikov variables and Homoclinic centre manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinite-dimensional systems, are reviewed briefly.
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Essentially asymptotically stable Homoclinic networks
Dynamical Systems, 2009Co-Authors: R. Driesse, Ale Jan HomburgAbstract:Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835–844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for Homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable Homoclinic network. The essentially asymptotically stable Homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the Homoclinic network.
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Homoclinic-Doubling Cascades
Archive for Rational Mechanics and Analysis, 2001Co-Authors: Ale Jan Homburg, Hiroshi Kokubu, Vincent NaudotAbstract:Cascades of period-doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos. In this paper we consider routes to onset of chaos involving Homoclinic-doubling bifurcations. We show the existence of cascades of Homoclinic-doubling bifurcations which occur persistently in two-parameter families of vector fields on ℝ3. The cascades are found in an unfolding of a codimension-three Homoclinic bifurcation which occur an orbit-flip at resonant eigenvalues. We develop a continuation theory for Homoclinic orbits in order to follow Homoclinic orbits through infinitely many Homoclinic-doubling bifurcations.
Xingbo Liu - One of the best experts on this subject based on the ideXlab platform.
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Bifurcations of Double Homoclinic Loops in Reversible Systems
International Journal of Bifurcation and Chaos, 2020Co-Authors: Yuzhen Bai, Xingbo LiuAbstract:This paper is devoted to the study of bifurcation phenomena of double Homoclinic loops in reversible systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equation, we perform a detailed study to obtain fruitful results, and demonstrate the existence of the R-symmetric large Homoclinic orbit of new type near the primary double Homoclinic loops, the existence of infinitely many R-symmetric periodic orbits accumulating onto the R-symmetric large Homoclinic orbit, and the coexistence of R-symmetric large Homoclinic orbit and the double Homoclinic loops. The Homoclinic bellow can also be found under suitable perturbation. The relevant bifurcation surfaces and the existence regions are located.
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Analysis of a Shil’nikov Type Homoclinic Bifurcation
Acta Mathematica Sinica English Series, 2018Co-Authors: Xingbo LiuAbstract:The bifurcation associated with a Homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related Homoclinic bifurcation theory. It is proved that, in a neighborhood of the Homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse Homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain Homoclinic connections to the periodic orbit is proved.
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On the stability of Homoclinic loops with higher dimension
Discrete & Continuous Dynamical Systems - B, 2012Co-Authors: Xingbo Liu, Deming ZhuAbstract:In this paper the stability of Homoclinic loops of saddle equilibrium states in high dimensional systems is analyzed. By constructing local moving frame along the unperturbed Homoclinic orbit, the refined Poincare map is well established, and simple criteria are given for the stability of the saddle Homoclinic loop. Some known results are extended.
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Homoclinic flip bifurcation with a nonhyperbolic equilibrium
Nonlinear Dynamics, 2011Co-Authors: Xingbo Liu, Lina Shi, Dongmei ZhangAbstract:In this paper, the problem of Homoclinic bifurcation accompanied by a transcritical bifurcation is investigated for high-dimensional systems. With the aid of a suitable local coordinate system, the Poincare map is constructed. Under certain nongeneric conditions (orbit flip and inclination flip Homoclinic orbits), the existence, nonexistence, coexistence and uniqueness of Homoclinic and periodic orbits are studied. Some known results are extended.
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Homoclinic flip bifurcations accompanied by transcritical bifurcation
Chinese Annals of Mathematics Series B, 2011Co-Authors: Xingbo LiuAbstract:The bifurcations of orbit flip Homoclinic loop with nonhyperbolic equilibria are investigated. By constructing local coordinate systems near the unperturbed Homoclinic orbit, Poincare maps for the new system are established. Then the existence of Homoclinic orbit and the periodic orbit is studied for the system accompanied with transcritical bifurcation.
Maoan Han - One of the best experts on this subject based on the ideXlab platform.
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hopf and Homoclinic bifurcations for near hamiltonian systems
Journal of Differential Equations, 2017Co-Authors: Yun Tian, Maoan HanAbstract:Abstract We study Homoclinic bifurcation of limit cycles in perturbed planar Hamiltonian systems. Suppose that a Homoclinic loop is defined by H = h s . Our main result is that a new method is established for computing the coefficients of the expansion of Melnikov functions at h = h s . Then by using those coefficients, more limit cycles would be found around Homoclinic loops. An example is also provided to illustrate our method.
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bifurcation of limit cycles from generalized Homoclinic loops in planar piecewise smooth systems
Journal of Differential Equations, 2013Co-Authors: Feng Liang, Maoan Han, Xiang ZhangAbstract:Abstract The generalized Homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades. For planar piecewise smooth differential systems, there are concrete examples showing that under suitable perturbations of a generalized Homoclinic loop one or two limit cycles can appear. But up to now there is no a general theory to study the cyclicity of a generalized Homoclinic loop, that is, the maximal number of limit cycles which are bifurcated from it. In this paper, we provide some sufficient conditions on the cyclicity of some Homoclinic loops. Especially we prove the existence of one or two limit cycles which are bifurcated from a generalized Homoclinic loop of an unperturbed piecewise smooth differential systems. Also we obtain a class of piecewise differential system in which two limit cycles can be bifurcated from a generalized Homoclinic loop with multiplicity one. The phenomena cannot happen in the smooth differential systems. Finally we provide five concrete piecewise smooth differential systems showing the applications of our theories on this phenomenon.
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limit cycle bifurcations near a double Homoclinic loop with a nilpotent saddle
International Journal of Bifurcation and Chaos, 2012Co-Authors: Maoan Han, Junmin Yang, Dongmei XiaoAbstract:Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a Homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double Homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a Homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double Homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the Homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.
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Limit cycles near generalized Homoclinic and double Homoclinic loops in piecewise smooth systems
Chaos Solitons & Fractals, 2012Co-Authors: Feng Liang, Maoan HanAbstract:Abstract In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized Homoclinic or generalized double Homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) Homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized Homoclinic loop and a central symmetric piecewise smooth system with a generalized double Homoclinic loop.
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The loop quantities and bifurcations of Homoclinic loops
Journal of Differential Equations, 2007Co-Authors: Maoan Han, Huaiping ZhuAbstract:Abstract The stability and bifurcations of a Homoclinic loop for planar vector fields are closely related to the limit cycles. For a Homoclinic loop of a given planar vector field, a sequence of quantities, the Homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the Homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double Homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double Homoclinic loop by using the method of stability-switching.
Jun Liu - One of the best experts on this subject based on the ideXlab platform.
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New Rational Homoclinic and Rogue Waves for Davey-Stewartson Equation
Abstract and Applied Analysis, 2014Co-Authors: Changfu Liu, Chuan-jian Wang, Zhengde Dai, Jun LiuAbstract:A new method, Homoclinic breather limit method (HBLM), for seeking rogue wave solution of nonlinear evolution equation is proposed. A new family of Homoclinic breather wave solution, and rational Homoclinic solution (Homoclinic rogue wave) for DSI and DSII equations are obtained using the extended Homoclinic test method and Homoclinic breather limit method (HBLM), respectively. Moreover, rogue wave solution is exhibited as period of periodic wave in Homoclinic breather wave approaches to infinite. This result shows that rogue wave can be generated by extreme behavior of Homoclinic breather wave for higher dimensional nonlinear wave fields.
Howard Weiss - One of the best experts on this subject based on the ideXlab platform.
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A Geometric Criterion for Positive Topological Entropy II: Homoclinic Tangencies
Communications in Mathematical Physics, 1999Co-Authors: Ale Jan Homburg, Howard WeissAbstract:In a series of important papers [GS1,GS2] Gavrilov and Shilnikov established a topological conjugacy between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic Homoclinic tangencies and the full shift on two symbols, thus exhibiting horseshoes near a tangential Homoclinic point. In this note, which should be viewed of as an addendum to [BW] we extend this result by showing that such a diffeomorphism with a one-sided isolated Homoclinic tangency having any order contact, possible with infinite order contact, possesses a horseshoe near the Homoclinic point.