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Homoclinic and heteroclinic orbits in a modified Lorenz system

Zhong Li, Guanrong Chen, Wolfgang A. Halang
2004 Information Sciences  
This paper presents a mathematically rigorous proof for the existence of chaos in a modified Lorenz system using the theory of Shil'nikov bifurcations of homoclinic and heteroclinic orbits.  ...  Together with its dynamical behaviors, which have been extensively studied, the chaotic dynamics of the modified Lorenz system are now much better understood, providing a rigorous theoretic foundation  ...  Following the work of [1] , this paper applies the Shil'nikov theorem to a better understanding of the chaotic behavior in the modified Lorenz system through finding homoclinic and heteroclinic orbits  ... 
doi:10.1016/j.ins.2003.06.005 fatcat:fdvp5l5tcbcyjpcwfyex6s5ucq

Homoclinic puzzles and chaos in a nonlinear laser model [article]

K. Pusuluri, H.G.E. Meijer, A.L. Shilnikov
2020 arXiv   pre-print
We present a case study elaborating on the multiplicity and self-similarity of homoclinic and heteroclinic bifurcation structures in the 2D and 3D parameter spaces of a nonlinear laser model with a Lorenz-like  ...  In a symbiotic approach combining the traditional parameter continuation methods using MatCont and a newly developed technique called the Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics  ...  We thank NVIDIA Corporation for supporting us with the Tesla K40 GPUs used in this study.  ... 
arXiv:2006.13812v1 fatcat:67cz26tvabaqrjqugu664ras3i

Dynamical systems modeling of low-frequency variability in low-order atmospheric models

Renato Vitolo, H. Broer
2008 Discrete and continuous dynamical systems. Series B  
In this paper possible relations with dynamical systems theory are given, in particular through bifurcation theory.  ...  It is discussed how the latter connection may be consolidated in higher dimensional and in PDE models.  ...  The authors are indebted to Antonio Speranza for valuable suggestions and scientific discussions.  ... 
doi:10.3934/dcdsb.2008.10.401 fatcat:ihrwjjnn7ffy3azsadujbvoqo4

Leonid Shilnikov and mathematical theory of dynamical chaos [article]

Alexey Kazakov, Sergey Gonchenko, Dmitry Turaev, Andrey L Shilnikov
2021 arXiv   pre-print
This Focus Issue Global Bifurcations, Chaos, and Hyperchaos Theory and Applications is dedicated to the 85th anniversary of the great mathematician, one of the founding fathers of dynamical chaos theory  ...  ., S.G and D.T. acknowledge financial support from the Mathematics Science and Education Center "Mathematics of Future Technologies," project no. 075-02-2021-1394.  ...  ACKNOWLEDGMENTS We would like to thank all contributors to this Focus Issue for their hard work and passion. Leonid Pavlovich Shilnikov would have enjoyed your papers.  ... 
arXiv:2112.02423v2 fatcat:xaqh6lrabzcbdatuuywkmi2qpy

Spiralling dynamics near heteroclinic networks

Alexandre A.P. Rodrigues, Isabel S. Labouriau
2014 Physica D : Non-linear phenomena  
The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols  ...  trajectories connecting them transversely and a non-trivial hyperbolic, invariant and transitive set.  ...  In Kokubu and Roussarie [33] , the classic Lorenz model is considered as a particular case of a model whose flow contains a heteroclinic cycle.  ... 
doi:10.1016/j.physd.2013.10.012 fatcat:djidyyj6efgonefaflg565s2fa

Heteroclinic bifurcations in a simple model of double-diffusive convection

E. Knobloch, M. R. E. Proctor, N. O. Weiss
1992 Journal of Fluid Mechanics  
The complex dynamics is associated with a heteroclinic orbit in phase space linking a pair of saddle-foci with eigenvalues satisfying Shil'nikov's criterion.  ...  The same bifurcation structure occurs in a truncated fifthorder model and numerical experiments confirm that similar behaviour extends to a significant region of parameter space.  ...  Heteroclinicity and chaos In this section we shall investigate the Lorenz system (12) in the parameter range where 0 < a < 4 and p > 0.  ... 
doi:10.1017/s0022112092004403 fatcat:fmyk7yhxtnhctjctdxb3lxpcee

Page 5057 of Mathematical Reviews Vol. , Issue 97H [page]

1997 Mathematical Reviews  
It is also shown that exactly two limit cycles can be bi- furcated from a non-hyperbolic homoclinic loop or a heteroclinic cycle by perturbing the system within bounded quadratic systems.”  ...  in- terpreted as homoclinic orbits in the flow to the vector field.  ... 

Shilnikov saddle-node bifurcation

Leonid Shilnikov, Andrey Shilnikov
2008 Scholarpedia  
gives rise to complex dynamics in a system after a merger of two saddles connected globally by ( ) heteroclinic orbits.  ...  [1977] On the appearance and structure of Lorenz attractor, DAN SSSR, 234, [336] [337] [338] [339] Figure 6 : Saddle-saddle with a pair of the homoclinic orbits in the modified Morioka-Simizu model  ... 
doi:10.4249/scholarpedia.4789 fatcat:6ilzvspdlfddndjjr7e6kjfche

Homoclinic orbits in a piecewise system and their relation with invariant sets

Rene O. Medrano-T., Murilo S. Baptista, Iberê L. Caldas
2003 Physica D : Non-linear phenomena  
Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits.  ...  In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems.  ...  Acknowledgements This work was supported by FAPESP and CNPq.  ... 
doi:10.1016/j.physd.2003.08.002 fatcat:3m46brcgcrfwfp57esecbclqn4

Analysis of a New Three-Dimensional Quadratic Chaotic System

Z. Elhadj
2008 Radioengineering  
This analysis shows that the system has complex dynamics with some interesting characteristics in which there are several periodic regions, but each of them has quite different periodic orbits.  ...  This paper has reported the finding of a new simple three dimensional quadratic chaotic system with three nonlinearities obtained by adding a cross-product nonlinear term to the first equation of the Lu  ...  Hence, for the points (P i ) 1 ≤ i ≤ 4 , there is a possibility to get a homoclinic or heteroclinic orbit for the system (1).  ... 
doaj:85c85799b8a341f0b91a4702d72a1477 fatcat:hhrabotibjg3jn43dd2o3tdjfy

Noisy homoclinic pulse dynamics

T. S. Eaves, Neil J. Balmforth
2016 Chaos  
The effect of stochastic perturbations on nearly homoclinic pulse trains are considered for three model systems: a Duffing oscillator, the Lorenz-like Shimizu-Morioka model, and a co-dimension-three normal  ...  The dynamics of these stochastic maps is then explored to examine how noise influences the sequence of bifurcations that take place adjacent to homoclinic connections in Lorenz-like and Shilnikov-type  ...  Stone & Holmes 3,4 thereby argued that stochastic perturbations of homoclinic (and heteroclinic) cycles take place near the fixed points and constructed a theory for how noise modified pulse timing (see  ... 
doi:10.1063/1.4945794 pmid:27131483 fatcat:ymu62an5wzfejhn63gtpcghqky

Homoclinic Dynamics: A Scenario for Atmospheric Ultralow-Frequency Variability

Daan T. Crommelin
2002 Journal of the Atmospheric Sciences  
In this subsystem, strong evidence for the existence of a homoclinic orbit is found.  ...  In this paper, a link will be established between atmospheric ultralow-frequency variability (ULFV) and the occurrence of homoclinic dynamics in models of large-scale atmospheric flow.  ...  I also wish to thank Theo Opsteegh and Ferdinand Verhulst for many valuable suggestions and comments during all stages of the project.  ... 
doi:10.1175/1520-0469(2002)059<1533:hdasfa>2.0.co;2 fatcat:oyv5vjrovbg3bkrs3asai73jta

Computation and Continuation of Homoclinic and Heteroclinic Orbits with Arclength Parameterization

Lixin Liu, Gerald Moore, Robert D. Russell
1997 SIAM Journal on Scientific Computing  
In this paper, we study a numerical method for the computation and continuation of homoclinic and heteroclinic orbits based upon the arclength parameterization of the orbits.  ...  Unlike most other methods, this method utilizes the geometric structure of the homoclinic and heteroclinic orbits and does not require solving a boundary value problem on an in nite interval.  ...  Numerical computation of homoclinic and heteroclinic orbits is of interest in a variety of contexts.  ... 
doi:10.1137/s1064827595288218 fatcat:jumxl6duyvejnndglcvjgd2vhq

Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

Sergey Gonchenko, Dmitry Turaev, Leonid Shilnikov
2007 Nonlinearity  
We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving  ...  diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.  ...  Figure 3 . 3 The global map T 1 in the case of homoclinic (a) and heteroclinic (b) tangency. The points M + ∈ W s loc and M − ∈ W u loc belong to the same homoclinic or heteroclinic orbit.  ... 
doi:10.1088/0951-7715/20/2/002 fatcat:2s5tapwonbbtbip3ijwwvf455i

Chaotification in the stretch-twist-fold (STF) flow

BaoZeng Yue, Muhammad Aqeel
2013 Chinese Science Bulletin  
It proposes a mechanism, by which a celestial bodies, such as earth and sun, can maintain and amplify the magnetic field continuously.  ...  As a consequence, the Sil'nikov criterion guarantees that STF flow has Smale horseshoes chaos. stretch-twist-fold flow, parameter analysis, undetermined coefficient method, heteroclinic orbit, Smale horseshoes  ...  It is an analytic technique, which is used for the existence of heteroclinic or homoclinic orbits of the Sil'nikov type in the dynamical systems.  ... 
doi:10.1007/s11434-013-5754-x fatcat:2xotmmjxzzhedb7zhq4c6goaha
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