Global Bifurcations of Mean Electric Field in Plasma L–H Transition Under External Bounded Noise Excitation

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
C. Nono Dueyou Buckjohn ◽  
M. Siewe Siewe ◽  
C. Tchawoua ◽  
T. C. Kofane
Keyword(s):  
Electric Field ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Power Spectra ◽  
Melnikov Method ◽  
Radial Electric Field ◽  
Model Parameters ◽  
Global Bifurcations ◽  
Bounded Noise ◽  
Noise Perturbation

In this paper, global bifurcations and chaotic dynamics under bounded noise perturbation for the nonlinear normalized radial electric field near plasma are investigated using the Melnikov method. From this analysis, we get criteria that could be useful for designing the model parameters so that the appearance of chaos could be induced (when heating particles) or run out for quiescent H-mode appearance. For this purpose, we use a test of chaos to verify our prediction. We find that, chaos could be enhanced by noise amplitude growing. The results of numerical simulations also reveal that noise intensity modifies the attractor size through power spectra, correlation function, and Poincaré map. The criterion from the Melnikov method which is used to analytically predict the existence of chaotic behavior of the normalized radial electric field in plasma could be a valid tool for predicting harmful parameters values involved in experiment on Tokamak L–H transition.

scholarly journals Generation of the sheared radial electric field by a magnetic island structure

2004 ◽  
Vol 3 (1) ◽  
pp. 19-26 ◽  
Author(s):  
Ana Mancic ◽  
Aleksandra Maluckov ◽  
Yokoyama Masayoshi ◽  
Okamoto Masao
Keyword(s):  
Electric Field ◽  
Local Treatment ◽  
Radial Electric Field ◽  
Model Parameters ◽  
Field Region ◽  
Field Profile ◽  
Magnetic Island ◽  
Island Structure ◽  
Non Local ◽  
Electric Field Profile

The effect of the presence of a magnetic island structure on the am bipolar radial electric field is studied in the context of the belt island model. It is shown that the sheared radial electric field region exists on the island position. Depending on the model parameters, the single (ion root) or multiple (one ion and two electron roots) solutions for the radial electric field are obtained at different radial positions. The radially non-local treatment is developed proposing the steady-state plasma conditions. The numerical calculations show that the diffusion of the radial electric field is significant only near the island boundaries. As a result the discontinuities in the am bipolar electric field profile are smoothed.

CONSTANT-PHASE-INDUCED CONTROL OF CHAOTIC CHARGED PARTICLES IN THE FIELD OF A WAVE PACKET

2013 ◽  
Vol 23 (06) ◽  
pp. 1330022
Author(s):  
RICARDO CHACÓN
Keyword(s):  
Charged Particle ◽  
Wave Packet ◽  
Finite Number ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Charged Particles ◽  
Melnikov Method ◽  
Wide Range ◽  
Range Of Values

It is shown that the dissipative chaotic dynamics of a charged particle in the field of a wave packet with an arbitrary but finite number of harmonics can be reliably suppressed by judiciously varying the constant phase of the main harmonic, ϕ0, while keeping null the corresponding constant phases of the remaining harmonics. The dependence of the chaotic threshold on the wave packet parameters is predicted theoretically (Melnikov method) and confirmed numerically (Lyapunov exponents). In particular, it is shown that ϕ0 is effective at suppressing the chaotic behavior existing when ϕ0 = 0 over a wide range of values of the wave packet width, while the remaining parameters are kept constant.

THEORIES OF MULTI-PULSE GLOBAL BIFURCATIONS FOR HIGH-DIMENSIONAL SYSTEMS AND APPLICATION TO CANTILEVER BEAM

2008 ◽  
Vol 22 (24) ◽  
pp. 4089-4141 ◽  
Author(s):  
W. ZHANG ◽  
M. H. YAO
Keyword(s):  
Nonlinear Systems ◽  
Cantilever Beam ◽  
Chaotic Dynamics ◽  
Adjoint Operator ◽  
Operator Method ◽  
Melnikov Method ◽  
Heteroclinic Orbits ◽  
High Dimensional ◽  
Phase Method ◽  
Global Bifurcations

The aim of this survey paper is to illustrate the perspectives on the theories of the single- and multi-pulse global bifurcations and chaotic dynamics of high-dimensional nonlinear systems and applications to several engineering problems in the past two decades. Two main methods for studying the Shilnikov type multi-pulse homoclinic and heteroclinic orbits in high-dimensional nonlinear systems, which are the energy-phase method and generalized Melnikov method, are briefly demonstrated in the theoretical frame. In addition, the theory of normal form and an improved adjoint operator method for high-dimensional nonlinear systems is also applied to describe a reducing procedure to high-dimensional nonlinear systems. The aforementioned methods are utilized to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end. How to employ these methods to analyze the Shilnikov type multi-pulse homoclinic and heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example.

Multipulse Chaotic Dynamics of Six-Dimensional Nonautonomous Nonlinear System for a Honeycomb Sandwich Plate

2014 ◽  
Vol 24 (11) ◽  
pp. 1450138 ◽  
Author(s):  
W. L. Hao ◽  
W. Zhang ◽  
M. H. Yao
Keyword(s):  
Nonlinear System ◽  
Rectangular Plate ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
Global Bifurcations ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Extended Melnikov Method ◽  
Honeycomb Sandwich

This paper studies the global bifurcations and multipulse chaotic dynamics of a four-edge simply supported honeycomb sandwich rectangular plate under combined in-plane and transverse excitations. Based on the von Karman type equation for the geometric nonlinearity and Reddy's third-order shear deformation theory, the governing equations of motion are derived for the four-edge simply supported honeycomb sandwich rectangular plate. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional nonautonomous nonlinear system is simplified to a three-order standard form by using the normal form method. The extended Melnikov method is improved to investigate the six-dimensional nonautonomous nonlinear dynamical system in a mixed coordinate. The global bifurcations and multipulse chaotic dynamics of the four-edge simply supported honeycomb sandwich rectangular plate are studied by using the improved extended Melnikov method. The multipulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.

Global Bifurcations and Multipulse-Type Chaotic Dynamics in the Interactions of Two Flexural Modes of a Cantilever Beam

2009 ◽  
Author(s):  
Shuangbao Li ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Cantilever Beam ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Melnikov Method ◽  
Global Bifurcations ◽  
Flexural Mode ◽  
Extended Melnikov Method ◽  
Suitable Form ◽  
Parametric And External Excitations ◽  
Two Degree Of Freedom

Global bifurcations and multipulse-type chaotic dynamics in the interactions of two flexural modes of a cantilever beam are studied using the extended Melnikov method. The cantilever beam studied is subjected to a harmonic axial excitation and transverse excitations at the free end. After the governing nonlinear equations of nonplanar motion with parametric and external excitations are given, the Galerkin’s procedure based on the first flexural mode in each direction is applied to the partial differential governing equation to obtain a two-degree-of-freedom non-autonomous nonlinear system. The resonant case considered here is one-to-one internal resonance, principal parametric resonance-1/2 subharmonic resonance. The method of multiple scales is used to derive four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of two interacting modes. After transforming the modulations equations into a suitable form, the extended Melnikov method is employed to show the existence of chaotic dynamics by identifying Silnikov-type multipulse jumping orbits in the perturbed phase space. We are able to obtain the explicit restrictions on the damping, forcing, and the detuning parameters, under which multipulse-type chaotic dynamics is to be expected. Physically, such jumping means sudden, large-amplitude departures of the beam from its planar oscillations. Numerical simulations indicate that there chaotic responses and jumping phenomenon in the nonlinear nonplanar oscillations of the cantilever beam.

scholarly journals Investigation of turbulence rotation and radial electric field in the island divertor and plasma edge at W7-X

2019 ◽  
Vol 61 (5) ◽  
pp. 054003 ◽  
Author(s):  
A Krämer-Flecken ◽  
X Han ◽  
T Windisch ◽  
J Cosfeld ◽  
P Drews ◽  
...  
Keyword(s):  
Electric Field ◽  
Radial Electric Field ◽  
Plasma Edge

Multi-Pulse Chaotic Dynamics of Four-Dimensional Non-Autonomous Nonlinear System for a Truss Core Sandwich Plate

2014 ◽  
Author(s):  
Wei Zhang ◽  
Qi-liang Wu
Keyword(s):  
Nonlinear System ◽  
Chaotic Dynamics ◽  
Sandwich Plate ◽  
Melnikov Method ◽  
Order Mode ◽  
Simply Supported ◽  
Truss Core ◽  
Extended Melnikov Method ◽  
Kagome Truss ◽  
Truss Core Sandwich Plate

In this paper, an extended high-dimensional Melnikov method is used to investigate global and chaotic dynamics of a simply supported 3D-kagome truss core sandwich plate subjected to the transverse and the in-plane excitations. Based on the motion equation derived by Zhang and the method of multiple scales, the averaged equation is obtained for the case of principal parametric resonance and 1:2 sub-harmonic resonance for the first-order mode and primary resonance for the second-order mode. From the averaged equation obtained, the system is simplified to a three order standard form with a double zero and a pair of pure imaginary eigenvalues by using the theory of normal form. Then, the extended Melnikov method is utilized to investigate the Shilnikov-type multi-pulse heteroclinic bifurcations and existence of chaos. The analysis of the extended Melnikov method demonstrates that there exist the Shilnikov-type multi-pulse heteroclinic bifurcations and chaos in the four-dimensional non-autonomous nonlinear system. Finally, the results of numerical simulations also show that for the nonlinear system of simply supported 3D-kagome truss core sandwich plate with the transverse and the in-plane excitations, the Shilnikov-type multi-pulse motion of chaos can happen and further verify the result of theoretical analysis.

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