Nonlinear Dynamics of a Rigid Block on a Rigid Base

1996 ◽  
Vol 63 (1) ◽  
pp. 55-61 ◽  
Author(s):  
R. N. Iyengar ◽  
D. Roy
Keyword(s):  
Stability Boundary ◽  
Three Dimensional ◽  
Melnikov Function ◽  
Rigid Base ◽  
Rigid Block ◽  
Stable And Unstable Manifolds ◽  
Melnikov Functions ◽  
Unstable Manifolds ◽  
Simulation Results ◽  
The Stability

The planar rocking of a prismatic rectangular rigid block about either of its corners is considered. The problem of homoclinic intersections of the stable and unstable manifolds of the perturbed separatrix is addressed to and the corresponding Melnikov functions are derived. Inclusion of the vertical forcing in the Hamiltonian permits the construction of a three-dimensional separatrix. The corresponding modified Melnikov function of Wiggins for homoclinic intersections is derived. Further, the 1-period symmetric orbits are predicted analytically using the method of averaging and compared with the simulation results. The stability boundary for such orbits is also established.

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.

Strange Attractors in a Three-Dimensional Autonomous Polynomial Equation

2014 ◽  
Vol 24 (09) ◽  
pp. 1450111 ◽  
Author(s):  
Fengjuan Chen ◽  
Jibin Li
Keyword(s):  
Three Dimensional ◽  
Small Neighborhood ◽  
Polynomial Equation ◽  
Strange Attractors ◽  
Stable And Unstable Manifolds ◽  
Small Perturbations ◽  
Unstable Manifolds ◽  
Comprehensive Study

We studied a three-dimensional autonomous polynomial equation. This equation is with a parameter, which we denote as μ. At μ = 0, the system has two saddles, the stable and unstable manifolds of which coincide. We present a comprehensive study on the dynamics of the system for small μ ≠ 0 in a small neighborhood of the unperturbed stable and unstable manifolds, where one of the heteroclinic connections of the two saddles are broken by small perturbations and strange attractors are created.

scholarly journals Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps

2017 ◽  
Vol 27 (02) ◽  
pp. 1730010 ◽  
Author(s):  
David J. W. Simpson ◽  
Christopher P. Tuffley
Keyword(s):  
Periodic Solutions ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Stable And Unstable Manifolds ◽  
Homoclinic Connections ◽  
Linear Maps ◽  
Continuous Maps ◽  
Piecewise Linear Maps ◽  
Stable Periodic ◽  
Unstable Manifolds

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for [Formula: see text]-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.

Multiple Attractor Bifurcation in Three-Dimensional Piecewise Linear Maps

2018 ◽  
Vol 28 (10) ◽  
pp. 1830032 ◽  
Author(s):  
Mahashweta Patra
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Chaotic Orbit ◽  
Stable And Unstable Manifolds ◽  
Period 2 ◽  
Coexistence Of Attractors ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Unstable Manifolds ◽  
Fundamental Uncertainty

Multiple attractor bifurcations lead to simultaneous creation of multiple stable orbits. This may be damaging for practical systems as there is a fundamental uncertainty regarding which orbit the system will follow after a bifurcation. Such bifurcations are known to occur in piecewise smooth maps, which model many practical and engineering systems. So far the occurrence of such bifurcations have been investigated in the context of 2D piecewise linear maps. In this paper, we investigate multiple attractor bifurcations in a three-dimensional piecewise linear normal form map. We show the occurrence of different types of multiple attractor bifurcations in the system, like the simultaneous creation of a period-2 orbit, a period-3 orbit and an unstable chaotic orbit; a mode-locked torus, an ergodic torus and periodic orbits; a one-loop torus and a two-loop torus; a one-loop mode-locked torus and a two-loop mode-locked torus; a one-piece chaotic orbit and a 3-piece chaotic orbit, etc. As orbits lie on unstable manifolds of fixed points, the structure of unstable manifold plays an important role in understanding the coexistence of attractors. In this work, we show that interplay between 1D and 2D stable and unstable manifolds plays an important role in global bifurcations that can give rise to multiple coexisting attractors.

Three-dimensional instability of the flow over a forward-facing step

2012 ◽  
Vol 695 ◽  
pp. 390-404 ◽  
Author(s):  
Daniel Lanzerstorfer ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Stability Boundary ◽  
Temporal Stability ◽  
Three Dimensional ◽  
Plane Channel ◽  
Separated Flow ◽  
Critical Mode ◽  
Flow Deceleration ◽  
Forward Facing Step ◽  
Adjoint Analysis ◽  
The Stability

AbstractThe global, temporal stability of the two-dimensional, incompressible flow over a forward-facing step in a plane channel is investigated numerically. The geometry is varied systematically covering constriction ratios (step-to-inlet height) from 0.23 to 0.965. A three-dimensional linear stability analysis shows that the stability boundary is a smooth continuous function of the constriction ratio. If the critical Reynolds and wavenumbers are scaled appropriately, they approach a linear asymptotic behaviour for large step heights. The critical mode is found to be stationary and confined to the region of separated flow downstream of the step for all constriction ratios. An energy-transfer analysis reveals that the basic flow becomes unstable due to a combined effect involving lift-up and flow deceleration, leading to a critical mode exhibiting steady streaks. Moreover, the receptivity of the flow to initial as well as to structural perturbations is studied by means of an adjoint analysis.

Heteroclinic bifurcations in damped rigid block motion

1992 ◽  
Vol 439 (1905) ◽  
pp. 155-162 ◽  
Keyword(s):  
Linear System ◽  
Perturbation Methods ◽  
Piecewise Linear ◽  
Exact Expression ◽  
Rigid Block ◽  
Stable And Unstable Manifolds ◽  
Piecewise Linear System ◽  
Block Motion ◽  
Rocking Motion ◽  
Unstable Manifolds

It is shown that heteroclinic bifurcations are present in a piecewise-linear system of ordinary differential equations that describe the rocking motion of a slender rigid block with damping. An exact expression is given for the bifurcation amplitude. Stable and unstable manifolds are analytically extended to explicitly reveal the intersections. As the damping increases, these bifurcations occur only at increasingly large forcing amplitudes, as manifolds move further apart. No perturbation methods are used in this analysis.

HETEROCLINIC BIFURCATIONS AND OPTIMAL CONTROL IN THE NONLINEAR ROCKING DYNAMICS OF GENERIC AND SLENDER RIGID BLOCKS

2005 ◽  
Vol 15 (06) ◽  
pp. 1901-1918 ◽  
Author(s):  
STEFANO LENCI ◽  
GIUSEPPE REGA
Keyword(s):  
Nonlinear Dynamics ◽  
Optimal Control ◽  
System Response ◽  
Rigid Block ◽  
Stable And Unstable Manifolds ◽  
Small Perturbations ◽  
Numerical Investigations ◽  
Nonlinear Dynamics And Chaos ◽  
Heteroclinic Connection ◽  
Unstable Manifolds

A method for controlling nonlinear dynamics and chaos, previously developed by the authors, is applied to the rigid block on a moving foundation. The method consists in modifying the shape of the excitation in order to eliminate, in an optimal way, the heteroclinic intersections embedded in the system dynamics. Two different cases are examined: (i) generic block under small perturbations and (ii) slender block under generic perturbations, and they are investigated analytically either by a perturbation analysis (former case) or exactly (latter case). Two different strategies are proposed: (i) one-side control, which consists in eliminating the intersections of a single heteroclinic connection, and (ii) global control, which consists in simultaneously eliminating the intersections of both heteroclinic connections. The best excitations permitting the maximum distance between stable and unstable manifolds are determined in both cases. Finally, some numerical investigations aimed at highlighting meaningful aspects of system response under controlled (optimal) and noncontrolled (harmonic) excitations are performed.

scholarly journals BARTLETT CORRECTION IN THE STABLE AR(1) MODEL WITH INTERCEPT AND TREND

2009 ◽  
Vol 25 (3) ◽  
pp. 857-872 ◽  
Author(s):  
Noud P.A. van Giersbergen
Keyword(s):  
Likelihood Ratio ◽  
Linear Trend ◽  
Likelihood Ratio Statistic ◽  
Stability Boundary ◽  
Small Samples ◽  
Bartlett Correction ◽  
Testing Hypotheses ◽  
Autoregressive Parameter ◽  
Simulation Results ◽  
The Stability

Bartlett corrections are derived for testing hypotheses about the autoregressive parameter ρ in the stable (a) AR(1) model, (b) AR(1) model with intercept, (c) AR(1) model with intercept and linear trend. The correction is found explicitly as a function of ρ. In the models with deterministic terms, the correction factor is asymmetric in ρ. Furthermore, the Bartlett correction is monotonically increasing in ρ and tends to infinity when ρ approaches the stability boundary of + 1. Simulation results indicate that the Bartlett corrections are useful in controlling the size of the likelihood ratio statistic in small samples, although these corrections are not the ultimate panacea.

Chaos and Three-Dimensional Horseshoes in Slowly Varying Oscillators

1988 ◽  
Vol 55 (4) ◽  
pp. 959-968 ◽  
Author(s):  
Stephen Wiggins ◽  
Steven W. Shaw
Keyword(s):  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Equation Of Motion ◽  
Necessary Condition ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Unstable Manifolds ◽  
Additional Equation

We present general results pertaining to chaotic motions in a class of systems termed slowly varying oscillators which consist of weakly perturbed single-degree-of-freedom systems in which parameters vary slowly in time according to an additional equation of motion. Our results include an analytical method for detecting transversal intersections of stable and unstable manifolds (typically a necessary condition for chaotic motions to exist) and a detailed description of the chaotic dynamics that occur when this situation exists.

A Liquid Metal Flow Between Two Coaxial Cylinders System

2019 ◽  
Vol 11 (1) ◽  
pp. 79-86
Author(s):  
Abdelkrim Merah ◽  
Ridha Kelaiaia ◽  
Faiza Mokhtari
Keyword(s):  
Couette Flow ◽  
Metal Flow ◽  
Three Dimensional ◽  
Liquid Sodium ◽  
Taylor Vortex ◽  
Turbulent Regime ◽  
Coaxial Cylinders ◽  
Concentric Cylinders ◽  
Ideal Tool ◽  
The Stability

Abstract The Taylor-Couette flow between two rotating coaxial cylinders remains an ideal tool for understanding the mechanism of the transition from laminar to turbulent regime in rotating flow for the scientific community. We present for different Taylor numbers a set of three-dimensional numerical investigations of the stability and transition from Couette flow to Taylor vortex regime of a viscous incompressible fluid (liquid sodium) between two concentric cylinders with the inner one rotating and the outer one at rest. We seek the onset of the first instability and we compare the obtained results for different velocity rates. We calculate the corresponding Taylor number in order to show its effect on flow patterns and pressure field.

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