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2 edition of computation of paths of homoclinic orbits. found in the catalog.

computation of paths of homoclinic orbits.

Vincent.* Canale

# computation of paths of homoclinic orbits.

Published by University of Toronto, Dept. of Computer Science in Toronto .
Written in English

Edition Notes

Thesis (M.Sc.)--University of Toronto, 1988.

The Physical Object
Pagination76 leaves
Number of Pages76
ID Numbers
Open LibraryOL21258963M

VOL NUMBER 8 PH YSICAL REVIEW LETTERS 20 FEBRUARY Chaos from Orbit-Flip Homoclinic Orbits Generated in a Practical Circuit Hisa-Aki Tanaka Department of Electronics and Communication Engineering, Waseda University, Shinj uku ku, -Tokyo , Japan (Received 15 March ) A new class of chaotic systems is generated in a practical, nonlinear, mutually coupled phase . MATH MATHEMATICAL COMPUTATION AND SCIENTIFIC VISUALIZATION (3) LEC. 3. Pr. MATH A programming language, or departmental approval. An introduction to the computational modeling process, numerical programming tools for large-scale scientific computation, parallel and cluster computing, and to scientific visualization techniques. Abstract: In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum. We present a numerical method to locate periodic orbits near homoclinic orbits. Using a method of [X.-B. Lin, Proc. Roy. Soc. Edinburgh, A (), pp. ] and solutions of the adjoint variational equation, we get a bifurcation function for periodic orbits, whose periods are asymptotic to infinity on approaching a homoclinic orbit.

In this paper, we investigate the existence and multiplicity of homoclinic solutions for a class of nonlinear difference systems involving classical (ϕ 1, ϕ 2) $(\\phi_{1},\\phi _{2})$ -Laplacian and a parameter: { Δ (ρ 1 (n − 1) ϕ 1 (Δ u 1 (n − 1))) − ρ 3 (n) ϕ 3 (u 1 (n)) + λ ∇ u 1 F (n, u 1 (n), u 2 (n)) = f 1 (n), Δ (ρ 2 (n − 1) ϕ 2 (Δ Cited by: 1.

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### computation of paths of homoclinic orbits. by Vincent.* Canale Download PDF EPUB FB2

Discrete computation of paths of homoclinic orbits. book system. Homoclinic orbits and homoclinic points are defined in the same way for iterated functions, as the intersection of the stable set and unstable set of some fixed point or periodic point of the system.

We also have the notion of homoclinic orbit when considering discrete dynamical systems. In such a case, if: → is a diffeomorphism of a manifold, we say that is a. DECEMBER 1, Abstract This paper is a study of one of the most beautiful phenomena in dynamical systems: homoclinic orbits. We will ﬁrst deﬁne what a homoclinic orbit is, then we will study some of the properties of homoclinic points, using methods from symbolic dynamics.

In particular, we will prove that, under certain assumptions,File Size: KB. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation.

Homoclinic orbits and heteroclinic connections are important in several contexts, in particular for a proof of the existence of chaos and for the description of bifurcations of chaotic attractors. In the previous sections, we apply the method to find the exact heteroclinic orbits for some well-known systems.

In this section, the method is applied to find homoclinic orbits in a complicated system. Consider the following system () u ˙ (t) = 2 v-2 u 3 + 2 uv 2 + 5 u u 2 v Cited by: In dynamic systems, some nonlinearities generate special connection problems of non-Z 2 symmetric homoclinic and heteroclinic orbits.

Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2 symmetric homoclinic and heteroclinic Author: Jingjing Feng, Qichang Zhang, Wei Wang, Shuying Hao.

Accurate Computation and Continuation of Homoclinic and Heteroclinic Orbits for Singular Perturbation Problems M. Friedman and A. Monteiro The University of Alabama in Huntsville Huntsville, Alabama Prepared for George C.

Marshall Space Flight Center under Contract NAS N/.qA National Aeronautics and Space Administration Office of. Recently, we witness that there is a clear attention in connecting orbits (homoclinic or hetroclinic orbits) in dynamical systems [1].

The notion of a homoclinic point was first introduced by. In other words, what may be affected by these problems is the calculation of very long paths to the target interval for parameter values which are very close to homoclinic bifurcation points.

Remarks on cycles. The algorithm described above is designed for finding homoclinic orbits and heteroclinic connections of Cited by: 7. Finding Homoclinic and Heteroclinic Orbits Remark If d dt F(x,y,z)| F=0 =0, the surface S is called without contact for trajectories of the system or a transversally section to the ﬂow of the system.

If d dt F(x,y,z)| F=0 =0 then the trajectory is included in S or tangent to it. 2 Tracing the Unstable Manifold Wu We will apply the method for detecting homoclinic orbits only but File Size: KB.

Section 2 discusses the fast-slow decomposition of the homoclinic orbits of the FitzHugh-Nagumo equation in the region I. This decomposition has been used to prove the existence of homoclinic orbits in the system for su ciently small [2, 17, 21, 20, 25], but previous work only applies to File Size: KB.

the super-homoclinic orbits and the creation of infinite series of homoclinic loops seems to be quite general phenomenon. Note that the object studied here was originally seen [6] in a system describing a plane high-frequency electro-magnetic field in non-linear non-dissipative me­ dia, the detection of a super-homoclinic orbit was based.

Homoclinic orbits to saddle xed points of planar di eomorphisms generically imply complicated dynamics due to Smale Horseshoes. Such orbits can be computed only numerically, which is time-consuming. The aim of this project is to explore an alternative method to compute homoclinic orbits near degenerate xed points of codimension 2 with.

which is of the form () with representing the wave speed c. Any solution of () gives a travelling wave with speed cof the PDE (). In particular, homoclinic orbits of. Abstract. We have added the functionality for continuing homoclinic orbits to cl_matcont, a user-friendly matlab package for the study of dynamical systems and their bifurcations.

It is now possible to continue homoclinic-to-hyperbolic-saddle and homoclinic-to-saddle-node by: 9. () Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof. Journal of Differential EquationsWilliam D. Kalies, Shane Kepley. Abstract. We study the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems, where is unnecessarily positive definite for all, and is of at most linear growth and satisfies some twist condition between the origin and the infinity.

Introduction. Consider the following second-order non-autonomous Hamiltonian system where is a symmetric matrix-valued function Author: Qi Wang, Qingye Zhang. We consider a periodically forced dynamical system possessing a small parameter, in arbitrary dimension.

When the parameter is zero the system is autonomous with an explicitly known homoclinic orbit; we develop a criterion for this homoclinic orbit to persist for small, nonzero values of the parameter. The theory is applied to an example arising from a magnetized spherical theory Cited by: Homoclinic orbits.

Ask Question Asked 5 years, 3 months ago. Active 5 years, 3 months ago. Viewed times 1 $\begingroup$ Consider an autonomous vector field on the plane having a hyperbolic fixed point with a homoclinic orbit connecting the hyperbolic fixed point.

Can a trajectory starting at any point on the homoclinic orbit reach the. orbits is necessarily infinite. In this paper we give an accurate, robust, and systematic method for computing entire families of orbits connecting two saddle points.

In a forthcoming paper [g] we shall consider, more generally, the computation of manifolds connecting two fixed points in Iw”.File Size: KB.

Finally, there is an introduction to chaos. Beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. MSC:Keywords: Ordinary differential equations, Dynamical systems, Sturm-Liouville equations.

The existence of the homoclinic orbit then follows from a Melnikov argument combined with methods from geometric singular perturbation theory. Next these homoclinic orbits are constructed, and studied, numerically with a bifurcation by: Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP E.

Barrabes⁄, J.M. Mondelo y, M. Olle z December 1, Abstract We consider the planar Restricted Three-Body problem and the collinear equilibrium point L3, as an example of a center£saddle equilibrium point in a Hamiltonian with two degrees of freedom.

homoclinic orbits for a perturbed nonlinear Schrodinger equation (PNLS) iqt = qrr + 2 [yq - u2] 4 + ic [ 64 - where 6 is a bounded dissipative operator and c > 0.

These results were announced at the ICM in Zurich [50]. Our methods are based on infinite- dimensional versions of. The existence of homoclinic orbits plays an important role in the study of the behavior of dynamical systems. If a system has transversely intersected homoclinic orbits, then it must be chaotic.

If it has smoothly connected homoclinic orbits, then it cannot stand the perturbation, and its perturbed system probably produces chaotic by: 2.

Homoclinic and heteroclinic orbits arise in the study of bifurcation and chaos phenom-ena (see e.g. [1—7], [10], [48, 49] and [55]) as well as their applications in mechanics, biomathematics and chemistry (see e.g.

[29], [—]). Many works related to these topics have been done in recent years. In this paper, we intend to survey some of theFile Size: KB.

Under consideration for publication in Nonlinearity Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function Thomas J. Bridges1, Fr ed eric Chardard2 1 Department of Mathematics, University of Surrey, Guildford GU2 7XH England 2 D epartement de Math ematiques, Institut Camille Jordan, Universit e Jean Monnet, 23, rue du docteur Paul Michelon, Saint.

Lee, Seunghee and Park, Junmi Expansive homoclinic classes of generic C 1-vector Mathematica Sinica, English Series, Vol. 32, Issue. 12, p. Cited by: 2. The homoclinic orbits found by Buffoni () are of type n(1,1) and hence they have Maslov index 2n, and they were shown to exist for all P∈(−2,0].

One can deduce from this that homoclinic orbits of the ODE () exist with Maslov index of every natural number greater than or equal to by: 9.

$\begingroup$ is it possible a limit cycle to contain homoclinic orbits finishing assimptotically in it. $\endgroup$ – Herr Schrödinger Feb 3 '18 at $\begingroup$ Not quite sure what you mean--something like having the unit circle as an attracting periodic orbit while having a homoclinic point at the origin.

$\endgroup$ – erfink Feb. We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JH z (t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in by: You can write a book review and share your experiences.

Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.

Research Article Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems XiaoyanLin, 1 Qi-MingZhang, 2 3 Department of Mathematics, Huaihua College, Huaihua, Hunan, China CollegeofScience,HunanUniversityofTechnology,Zhuzhou,Hunan,C hina.

periodic orbits that have earlier bifurcated from the homoclinic loops of the saddle at r hom ˇ As rincreases to r AH ˇ the saddle periodic orbits shrink the attraction basins and collapse onto the stable foci O 1;2 through a subcritical Andronov-Hopf bifurcation.

For r AH File Size: 2MB. It is known that the existence of homoclinic orbits is a signature of global changes in the dynamics. In two-dimensional systems studied by Andronov et al. [1], the onset of a homoclinic orbit causes the sudden appearance of periodic orbits.

In the Lorenz system, homoclinic orbits can be associated with the bifurcations of a periodic set. Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part II.

Symbolic Dynamics Y. Li 1 and S. Wiggins2 1 Department of Mathematics, University of California at Los Angeles, Los Angeles, CAUSA Present Address: Department of Mathematics,Massachusetts Institute of Technology, Cambridge, MAUSA. homoclinic orbits emanating from 0.

In other words, we consider the existence of solutions of (HS) such that z(t) + 0 as ItI + () We remark that 0 is an equilibrium point of (HS). The existence of homoclinic orbits is studied by Coti-Zelati, Ekeland, and S&C: [2] and Hofer and Wysocki [6]. computation of the one-dimensional manifolds are implemented in dstool [8,20] and dynamics [34,35], while those for the continuation of homoclinic orbits and their tangencies using the projection asymptotic boundary condi-tions [2] are implemented in an auto-driver [33].

In this paper, we present new or improved methods to continue heteroclinic. Bursting phenomena are found in a wide variety of fast–slow systems. In this article, we consider the Hindmarsh–Rose neuron model, where, as it is known in the literature, there are homoclinic bifu.

"Homoclinic orbits of the FitzHugh-Nagumo equation: bifurcations in the full system'' J. Guckenheimer and C. Kuehn: SIAM Journal on Applied Dynamical Systems, Vol. 9, No. 1, pp.J4 "From first Lyapunov coefficients to maximal canards" C.

Kuehn. Homoclinic orbits play an important role in the study of qualitative behavior of dynamical systems. Such kinds of orbits have been studied since the time of Poincar e.

In this paper, we discuss how to use varia-tional methods to study the existence of homoclinic orbits of Hamiltonian systems. Introduction.TJB [] Book review of Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures by W.

David McComb (OUP ). In Contemporary Physics 56 F. Chardard & TJB [] Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function, Nonlinearity 28 Dimensional considerations show that orbits homoclinic to the periodic motions of the center manifold are more likely to exist.

The existence of such homoclinics has been studied in [4], [14] (see also [11], [9], [10], [7], [12]) by perturbation methods, and in [2] by global methods. In .