Gluing Bifurcations in Chua Oscillator

Roy, Prodyot Kumar; Dana, Syamal K.

doi:10.1142/s021812740601694x

Cited by 11 publications

(7 citation statements)

References 12 publications

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“…In proposition 5.4 we have shown why (1.1) has a ghost unstable homoclinic loop. Interestingly, a gluing bifurcation occurs in the small neighborhood of V, see [9,19]. Gluing bifurcation is an important class of homoclinic bifurcation, see [1,2,5,19], and the references therein.…”

Section: Examples and Simulationsmentioning

confidence: 99%

Global study of Rayleigh–Duffing oscillators

Chen

Zou

2016

J. Phys. A: Math. Theor.

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In this paper we investigate the global dynamics of Rayleigh-Duffing oscillators with global parameters, including equilibria at both finity and infinity, existences and coexistence of limit cycles and homoclinic loops. In fact, this oscillator will occur Hopf bifurcations, homoclinic bifurcations and double limit cycle bifurcations. Moreover, we find that the homoclinic bifurcation of this oscillator is special which is a gluing bifurcation. The global bifurcation diagram and all phase portrait are given, and numerical simulations are shown to verify our analysis finally.

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Section: Examples and Simulationsmentioning

confidence: 99%

Global study of Rayleigh–Duffing oscillators

Chen

Zou

2016

J. Phys. A: Math. Theor.

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“…Most occurrences of gluing scenarios in families of dynamical systems were reported in theoretical and numerical studies: in the context of hydrodynamics [7,8,9], nematic liquid crystals [10,11] and optothermal devices [12]. In the experiments, separate gluing bifurcations were identified in the Taylor-Couette flow of the viscous fluid between two cylin-ders [13] and in the Chua oscillators [14]. Here, we report on our experimental investigation of the sequence of gluing bifurcations in an analog electronic circuit.…”

Section: Introductionmentioning

confidence: 85%

Sequences of gluing bifurcations in an analog electronic circuit

Akhtanov¹,

Zhanabaev²,

Zaks³

2013

Physics Letters A

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We report on the experimental investigation of gluing bifurcations in the analog electronic circuit which models a dynamical system of the third order: Lorenz equations with an additional quadratic nonlinearity. Variation of one of the resistances in the circuit changes the coefficient at this nonlinearity and enables transition from the Lorenz route to chaos to a different scenario which leads, through the sequence of homoclinic bifurcations, from periodic oscillations of the voltage to the chaotic state. A single bifurcation "glues" in the phase space two stable periodic orbits and creates a new one, with the doubled length: a bifurcation sequence results in the birth of the chaotic attractor.

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“…For the case of a uniform horizontal magnetic field along the y-axis, the equations for momentum transfer and the induced magnetic field [Eqs. [8][9] are replaced by,…”

Section: B Relevant Equations With a Horizontal Magnetic Fieldmentioning

confidence: 99%

“…The symmetry of a field under interchange of indices l and m is broken by a horizontal magnetic field. The integration of the two hydromagnetic systems [ (8)(9)(10)(11) and (10)(11)(13)(14)] are done on 64×64×64 spatial grid points using a standard fourth order Runge-Kutta (RK4) method with a time step δt ≤ 0.001.…”

Section: Direct Numerical Simulationsmentioning

confidence: 99%

“…Homoclinic bifurcations are observed in several fluid dynamical systems including liquid crystals 1,2 , Taylor-Couette flow 3 , Rayleigh-Bénard convection in low-Prandtl-number fluids with and without rotation [4][5][6] . They are also known to occur in biological systems 7 , electrical systems 8,9 and optical systems 10 . A homoclinic bifurcation may lead to the possibility of Shilnikov wiggle 11 , three-frequency quasi-periodic orbits 12,13 , spontaneous gluing of two limit cycles into a large one or spontaneous breaking of a limit cycle into two smaller ones 2,[4][5][6] or homoclinic chaos.…”

Section: Introductionmentioning

confidence: 99%

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Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid

Basak

Kumar

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Effects of a uniform magnetic field on homoclinic bifurcations in Rayleigh-Bénard convection in a fluid of Prandtl number P r = 0.01 are investigated using direct numerical simulations (DNS). A uniform magnetic field is applied either in the vertical or in the horizontal direction. For a weak vertical magnetic field, the possibilities of both forward and backward homoclinic bifurcations are observed leading to a spontaneous merging of two limit cycles into one as well as a spontaneous breaking of a limit cycle into two for lower values of the Chandrasekhar's number (Q ≤ 5). A slightly stronger magnetic field makes the convective flow time independent giving the possibility of stationary patterns at the secondary instability. For horizontal magnetic field, the x ⇌ y symmetry is destroyed and neither a homoclinic gluing nor a homoclinic breaking is observed. Two low-dimensional models are also constructed: one for a weak vertical magnetic field and another for a weak horizontal magnetic field. The models qualitatively capture the features observed in DNS and help understanding the unfolding of bifurcations close to the onset of magnetoconvection. Dissipative structures spontaneously appear in several continuum mechanical systems when they are subjected to a uniform external forcing. The dynamics of such structures depends upon the nature of the underlying bifurcations. A dynamical system having symmetrically placed saddle and unstable fixed points in its phase space may show the possibility of either homoclinic or heteroclinic bifurcations. Several fluid dynamical systems show homoclinic gluing of two limit cycles into one. This may lead to the phenomenon of pattern bursting, where a dissipative structure disappears for a finite time and then reappears again for a fixed value of the bifurcation parameter. Fluid patterns in Rayleigh-Bénard convection in low-Prandtl-number fluids with stress-free boundaries show homoclinic as well as heteroclinic dynamics. A uniform applied magnetic field produces Lorentz force which is likely to affect this behavior significantly. The role of a small magnetic field on such behavior is not known. Direct numerical simulations and low-dimensional models for magnetoconvection show that a small vertical magnetic field allows homoclinic gluing. However, even a very weak horizontal magnetic field destroys homoclinic gluing, as it breaks the rotational symmetry of the flow structure about a vertical axis. The unfolding of bifurcations near the instability onset is quite different for a vertical magnetic field from that for a horizontal magnetic field.

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Gluing Bifurcations in Chua Oscillator

Cited by 11 publications

References 12 publications

Global study of Rayleigh–Duffing oscillators

Global study of Rayleigh–Duffing oscillators

Sequences of gluing bifurcations in an analog electronic circuit

Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid

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