scholarly journals Structural Stability of Perturbed mKdV Solitons

1991 ◽  
Vol 44 (5) ◽  
pp. 495 ◽  
Author(s):  
J Roessler
Keyword(s):  
Structural Changes ◽  
Lattice Models ◽  
Melnikov Method ◽  
Homoclinic Orbits ◽  
Travelling Wave ◽  
Mkdv Equation ◽  
Reduced Form ◽  
Stable And Unstable Manifolds ◽  
Standard Application ◽  
Numerical Generation

Periodic perturbations are applied to the homoc\inic orbits corresponding to solitons of the modified Korteweg-de Vries (mKdV) equation, which is significant in plasma physics and lattice models. It is observed that for certain distinct frequencies the homoclinic orbits do not split into stable and unstable manifolds, which means absence of horseshoes and chaos. The analysis is performed on a travelling wave reduced form of the mKdV equation both by standard application of the Melnikov method as well as numerical generation of POincare maps. In particular, the geometry of the homoclinic orbits and their structural changes under perturbations is investigated.

Subharmonic bifurcation and chaos of a carbon nanotube supported by a Winkler and Pasternak foundation

2019 ◽  
Vol 33 (19) ◽  
pp. 1950207
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen ◽  
Ziman Zhao
Keyword(s):  
Carbon Nanotube ◽  
Melnikov Method ◽  
Homoclinic Orbits ◽  
Dynamic Responses ◽  
Pasternak Foundation ◽  
Stable And Unstable Manifolds ◽  
Subharmonic Bifurcation ◽  
Global Dynamic ◽  
Critical Curves ◽  
Subharmonic Bifurcations

In this paper, by employing both analytical and numerical methods, global dynamic responses including subharmonic bifurcations and chaos are investigated for a carbon nanotube supported by a Winkler and Pasternak foundation. The criteria of chaos arising from transverse intersections for stable and unstable manifolds of homoclinic orbits are proposed with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted in the parameter plane. The parameter conditions for subharmonic bifurcations are also obtained by the subharmonic Melnikov method. It is proved rigorously that the route to chaos for this model is infinite subharmonic bifurcations. The stability of subharmonic bifurcations is also studied by the characteristic multipliers. Numerical simulations are given to confirm the analytical results.

scholarly journals A Melnikov Method for Homoclinic Orbits with Many Pulses

1998 ◽  
Vol 143 (2) ◽  
pp. 105-193 ◽  
Author(s):  
Roberto Camassa ◽  
Gregor Kovačič ◽  
Siu-Kei Tin
Keyword(s):  
Melnikov Method ◽  
Homoclinic Orbits

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350074 ◽  
Author(s):  
N. HAN ◽  
Q. J. CAO ◽  
M. WIERCIGROCH
Keyword(s):  
Nonlinear System ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Viscous Damping ◽  
Approximate System ◽  
Smoothness Parameter

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

scholarly journals Explaining the environmental Kuznets curve: structural change and international agreements in reducing sulphur emissions

1997 ◽  
Vol 2 (4) ◽  
pp. 485-503 ◽  
Author(s):  
SANDER M. DE BRUYN
Keyword(s):  
Environmental Policy ◽  
Structural Changes ◽  
Decomposition Analysis ◽  
International Agreements ◽  
Reduced Form ◽  
Kuznets Curve ◽  
Explanatory Variables ◽  
Developed Economies ◽  
Form Model ◽  
Reduced Form Model

Environmental Kuznets curves have been estimated using a simple reduced-form model that gives no information on the mechanisms underlying the estimated inverted U-shaped relationship between some pollutants and income. Various intuitive appealing explanations for the observed patterns have been offered, such as structural changes and environmental policy, but these have rarely been empirically investigated. Expanding the reduced-form model with explanatory variables may introduce serious multicollinearity problems, a reason why decomposition analysis is a preferable alternative for investigating the origins of change in emissions. Applying decomposition analysis fails to find evidence for structural changes as an important determinant of the impressive reductions in SO2 emissions of developed economies during the 1980s. Environmental policy, fostered by international agreements, gives a better explanation why pollution curbs downward at high income levels. Besides the level of income, the present state of the environment seems an important, but often neglected, variable that explains the ambition of environmental policy.

Application of the Random Melnikov Method for Single-Degree-of-Freedom Vessel Rolls Motion

2010 ◽  
Author(s):  
Kaiye Hu ◽  
Yong Ding ◽  
Hongwei Wang ◽  
Jide Li
Keyword(s):  
Global Stability ◽  
Global Bifurcation ◽  
Melnikov Method ◽  
Threshold Value ◽  
Homoclinic Orbits ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Bounded Noise ◽  
Single Degree Of Freedom ◽  
Single Degree

Basing on the nonlinear dynamics theory, the global stability of ship in stochastic beam sea is researched by the global bifurcation method. In this paper, bounded noise is first briefly introduced. Bounded noise is a harmonic function with constant random frequency and phase. It has finite power and its spectral shape can be made to fit a target spectrum, such as Pierson-Moskowitz spectrum, by adjusting its parameters. This paper considered the stochastic excitation term as bounded noise and the influence of nonlinear damping and nonlinear righting moment, setup the random single degree of freedom nonlinear rolling equation. Then the random Melnikov process for the nonlinear system with homoclinic orbits under both dissipative and bounded noise perturbations is derived. The random Melnikov mean-square criterion is used to analysis the global stability of this system. The research indicates that the bounded noise can approximately simulate the wave excitation and if the noise exceeds the threshold value, the ship will undergo stochastic chaotic motion. That will lead ships to instability and even to capsizing.

Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650030 ◽  
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Homoclinic Solution ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Stable And Unstable Manifolds ◽  
Straight Line ◽  
Unstable Manifolds ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

Experimental Method of Determination of Structural Correlations in Surface Layers of Oxide Glass

1990 ◽  
Vol 216 ◽  
Author(s):  
Zenon BochyŃski
Keyword(s):  
Structural Changes ◽  
Lattice Models ◽  
Molecular Size ◽  
Oxide Glass ◽  
X Ray Diffraction ◽  
Surface Layers ◽  
X Ray ◽  
The Real ◽  
Inorganic Glasses

ABSTRACTA new method of X-ray diffraction analysis of structural inhomogeneities in the quartz/Si02/n based inorganic glasses is presented. The method enables the determination of structural changes occuring in the real nodal lattice in the regions of 10…20 Å or more as well as substructural changes in the regions 5…15 Å comparable to the molecular size of SiO2…SiO4. In consequence these changes can be correlated with approximate nodal lattice models of different degree of ordering. The applied method provided the possibility of constructing structural models of nodal lattices describing the surface and inner layers of the real glasses, changes in the local inhomogeneities as well as boundaries in water-gel associates.

Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator

2003 ◽  
Vol 9 (3-4) ◽  
pp. 281-315 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Finite Number ◽  
Control Method ◽  
Mathematical Problem ◽  
Melnikov Method ◽  
Excitation Amplitude ◽  
Homoclinic Bifurcation ◽  
Theoretical Treatment ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Basin Erosion

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikov's theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.

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