Non-Smooth Dynamics and Non-Classical Bifurcations in Impulse-Impact Oscillators

1999 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Local Level ◽  
Basins Of Attraction ◽  
Dynamical Regime ◽  
Impact Oscillator ◽  
Stable And Unstable Manifolds ◽  
Impact Oscillators ◽  
Unstable Manifolds ◽  
Local And Global Bifurcations ◽  
Initial Description ◽  
Smooth Dynamics

Abstract Some aspects of the nonlinear dynamics of an impulse-impact oscillator are investigated. After an initial description of the prototype mechanical model used to illustrate the results, attention is paid to the classical local and global bifurcations which are at the base of the changes of dynamical regime. Some non-classical phenomena due to the particular nature of the investigated system are then considered. At a local level, it is shown that periodic solutions may appear (or disappear) through a non-classical bifurcation which involves synchronization of impulses and impacts. Similarities and differences with the classical bifurcations are discussed. At a global level, the effects of the non-continuity of the orbits in the phase space on the basins of attraction topology are investigated. It is shown how this property is at the base of a non-classical homoclinic bifurcation where the homoclinic points disappear after the first touch between the stable and unstable manifolds.

GLOBAL AND LOCAL CONTROL OF HOMOCLINIC AND HETEROCLINIC BIFURCATIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 2411-2432 ◽  
Author(s):  
HONGJUN CAO ◽  
GUANRONG CHEN
Keyword(s):  
Optimal Control ◽  
Control Method ◽  
Basins Of Attraction ◽  
Stable And Unstable Manifolds ◽  
Optimal Control Method ◽  
Global Optimal ◽  
Unstable Manifolds ◽  
Global And Local ◽  
Theoretical Analyses ◽  
Optimal Control Methods

A comprehensive resonant optimal control method is developed and discussed for suppressing homoclinic and heteroclinic bifurcations of a general one-degree-of-freedom nonlinear oscillator. Based on an adjustable phase shift, the primary resonant optimal control method is presented. By solving an optimization problem for the optimal amplitude coefficients to be used as the control parameters, the force term as the controller can be designed. Three kinds of resonant optimal control methods are compared. The control mechanism of the primary resonant optimal control method is to enlarge to the largest possible degree the control region where homoclinic and/or heteroclinic transversal intersections do not occur, and this is accomplished at lowest cost. It is shown that the primary resonant optimal control method has much better performance than the superharmonic resonant optimal control method, and it works well even when the superharmonic optimal control method fails. In particular, one new global optimal control method is presented, whose central idea is to find a frequency such that the asymmetric homoclinic bifurcations or the multiple homoclinic and heteroclinic bifurcations can attain the same critical values. On the basis of these same critical bifurcation values, chaos resulting from asymmetric homoclinic or multiple homoclinic and heteroclinic bifurcations can be effectively suppressed by the primary resonant optimal control method. This is confirmed by two illustrative examples. The theoretical analyses concerning the suppression of local and global homoclinic and heteroclinic bifurcations are in agreement with the numerical simulations, including the identification of the stable and unstable manifolds and the basins of attraction.

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

Global Analysis of the Double Impact Oscillator Dynamics

1999 ◽  
Author(s):  
František Peterka
Keyword(s):  
Global Analysis ◽  
Impact Oscillator ◽  
System Parameters ◽  
Impact Oscillators ◽  
Real Value ◽  
Simulation Results ◽  
The Impact ◽  
Impact Motion ◽  
Double Impact ◽  
Phase Motion

Abstract The double impact oscillator represents two symmetrically arranged single impact oscillators. It is the model of a forming machine, which does not spread the impact impulses into its neighbourhood. The anti-phase impact motion of this system has the identical dynamics as the single system. The in-phase motion and the influence of asymmetries of the system parameters are studied using numerical simulations. Theoretical and simulation results are verified experimentally and the real value of the restitution coefficient is determined by this method.

Splitting of Separatrices in the Rapidly Forced Duffing Oscillator

1993 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Asymptotic Approximation ◽  
Duffing Oscillator ◽  
Perturbation Parameter ◽  
Negative Stiffness ◽  
Stable And Unstable Manifolds ◽  
Analytic Approximations ◽  
Melnikov Integral ◽  
Unstable Manifolds ◽  
Splitting Of Separatrices ◽  
Splitting Distance

Abstract The splitting of the stable and unstable manifolds of the rapidly forced Duffing oscillator with negative stiffness is investigated. The method used relies on the computation of analytic approximations for the orbits on the perturbed manifolds, and the asymptotic approximation of these orbits by successive integrations by parts. It is shown, that the splitting of the manifolds becomes exponentially small as the perturbation parameter tends to zero, and that the estimate for the splitting distance given by the Melnikov Integral dominates over high order corrections.

Random invariant manifolds for ill-posed stochastic evolution equations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050013
Author(s):  
Alexandra Neamţu
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Linear Part ◽  
Time Behavior ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Stable And Unstable Manifolds ◽  
Long Time ◽  
Ill Posed ◽  
Unstable Manifolds

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.

scholarly journals Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

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