scholarly journals A scenario of the homotopy type changing of the invariant saddle manifold closure

2020 ◽  
Vol 22 (3) ◽  
pp. 306-319
Author(s):  
Elena V. Nozdrinova
Keyword(s):  
Invariant Manifold ◽  
Homotopy Type ◽  
Invariant Manifolds ◽  
Saddle Points ◽  
Closed Curve ◽  
Surface Gradient ◽  
Nodal Points

The paper deals with surface gradient-like diffeomorphisms. The closures of the invariant manifolds of saddle points of such systems contain nodal points in their closure. In the case when there is only one such point, the closure of the invariant manifold is a closed curve that is homeomorphic to the circle. In a general case the conjugating homeomorphism changes the homotopy type of the closed curve, while the diffeomorphisms themselves may remain in the same isotopic class. This means that in the space of diffeomorphisms two such systems are connected by an arc, but every such arc necessarily undergoes bifurcations. In this paper, we describe a scenario for changing the homotopy type of the closure of the invariant saddle manifold. Moreover, the constructed arc is stable in the space of diffeomorphisms.

Invariant Manifold Detection From Phase Space Trajectories

2008 ◽  
Author(s):  
Joseph Kuehl ◽  
David Chelidze
Keyword(s):  
Phase Space ◽  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Basins Of Attraction ◽  
Detection Methods ◽  
Trajectory Data ◽  
Space Trajectories ◽  
Data Set ◽  
Phase Space Warping ◽  
Phase Space Trajectories

Invariant manifolds provide important information about the structure of flows. When basins of attraction are present, the stable invariant manifold serves as the boundary between these basins. Thus, in experimental applications such as vibrations problems, knowledge of these manifolds is essential to understanding the evolution of phase space trajectories. Most existing methods for identifying invariant manifolds of a flow rely on knowledge of the flow field. However, in experimental applications only knowledge of phase space trajectories is available. We provide modifications to several existing invariant manifold detection methods which enables them to deal with trajectory only data, as well as introduce a new method based on the concept of phase space warping. The method of Stochastic Interrogation applied to the damped, driven Duffing equation is used to generate our data set. The result is a set of trajectory data which randomly populates a phase space. Manifolds are detected from this data set using several different methods. First is a variation on manifold “growing,” and is based on distance of closest approach to a hyperbolic trajectory with “saddle like behavior.” Second, three stretching based schemes are considered. One considers the divergence of trajectory pairs, another quantifies the deformation of a nearest neighbor cloud, and the last uses flow fields calculated from the trajectory data. Finally, the new phase space warping method is introduced. This method takes advantage of the shifting (warping) experienced by a phase space as the parameters of the system are slightly varied. This results in a shift of the invariant manifolds. The region spanned by this shift, provides a means to identify the invariant manifolds. Results show that this method gives superior detection and is robust with respect to the amount of data.

scholarly journals GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

2007 ◽  
Vol 17 (09) ◽  
pp. 3295-3302 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Main Tool ◽  
Singular Points ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Heteroclinic Loop

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

Amplitude Nonlinear Normal Modes of Piecewise Linear Systems

2001 ◽  
Author(s):  
Dongying Jiang ◽  
Vincent Soumier ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Flip Bifurcation ◽  
Strong Nonlinearity ◽  
Nonlinear Modes ◽  
Nonlinear Normal

Abstract A numerical method for constructing nonlinear normal modes for piecewise linear autonomous systems is presented. Based on the concept of invariant manifolds, a Galerkin based approach is applied here to obtain nonlinear normal modes numerically. The accuracy of the constructed nonlinear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that the Galerkin based construction approach can represent the invariant manifold accurately over strong nonlinearity regions. Several interesting dynamic characteristics of the nonlinear modal motion are found and compared to those of linear modes. The stability of the nonlinear normal modes of a two-degree of freedom system is investigated using characteristic multipliers and Poincaré maps, and a flip bifurcation is found for both nonlinear modes.

Non-Linear Dynamics of a Rotor-Bearing System Based on the Invariant Manifold Approach

2003 ◽  
Author(s):  
Cristiano Viana Serra Villa ◽  
Jean-Jacques Sinou ◽  
Fabrice Thouverez
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Initial Condition ◽  
Order Model ◽  
Linear Dynamics ◽  
Reduced Order ◽  
Spin Speed ◽  
Non Linear ◽  
Bearing System

The invariant manifold approach is used to explore the dynamics of a non-linear rotor, by determining the non-linear normal modes, constructing a reduced order model and evaluating its performance in the case of response to an initial condition. The procedure to determine the approximation of the invariant manifolds is discussed and a strategy to retain the speed dependent effects on the manifolds without solving the eigenvalue problem for each spin speed is presented. The performance of the reduced system is analysed in function of the spin speed.

scholarly journals The Effect of Slow Invariant Manifold and Slow Flow Dynamics on the Energy Transfer and Dissipation of a Singular Damped System with an Essential Nonlinear Attachment

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jamal-Odysseas Maaita ◽  
Efthymia Meletlidou
Keyword(s):  
Energy Transfer ◽  
Total Energy ◽  
Invariant Manifold ◽  
Nonlinear Oscillator ◽  
Invariant Manifolds ◽  
Dissipative System ◽  
Flow Dynamics ◽  
Slow Flow ◽  
Damped System ◽  
Slow Invariant Manifold

We study the effect of slow flow dynamics and slow invariant manifolds on the energy transfer and dissipation of a dissipative system of two linear oscillators coupled with an essential nonlinear oscillator with a mass much smaller than the masses of the linear oscillators. We calculate the slow flow of the system, the slow invariant manifold, the total energy of the system, and the energy that is stored in the nonlinear oscillator for different sets of the parameters and show that the bifurcations of the SIM and the dynamics of the slow flow play an important role in the energy transfer from the linear to the nonlinear oscillator and the rate of dissipation of the total energy of the initial system.

THE SLOW INVARIANT MANIFOLD OF A CONSERVATIVE PENDULUM-OSCILLATOR SYSTEM

1996 ◽  
Vol 06 (04) ◽  
pp. 673-692 ◽  
Author(s):  
IOANNIS T. GEORGIOU ◽  
IRA B. SCHWARTZ
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Slow Manifold ◽  
Equilibrium States ◽  
Periodic Motions ◽  
Nonlinear Normal Mode ◽  
Oscillator System ◽  
Slow Invariant Manifold ◽  
Nonlinear Normal ◽  
Global Invariant

We analyze the motions of a conservative pendulum-oscillator system in the context of invariant manifolds of motion. Using the singular perturbation methodology, we show that whenever the natural frequency of the oscillator is sufficiently larger than that of the pendulum, there exists a global invariant manifold passing through all static equilibrium states and tangent to the linear eigenspaces at these equilibrium states. The invariant manifold, called slow, carries a continuum of slow periodic motions, both oscillatory and rotational. Computations to various orders of approximation to the slow invariant manifold allow analysis of motions on the slow manifold, which are verified with numerical experiments. Motion on the slow invariant manifold is identified with a slow nonlinear normal mode.

scholarly journals Numerical investigation of lagrangian coherent structures in steady rotation vortex shedding control

2021 ◽  
Author(s):  
Muhammad Hashir ◽  
◽  
Tauseef -ur-Rehman ◽  
Aamir Sohail ◽  
Muhammad Yasar Javaid ◽  
...  
Keyword(s):  
Dynamical System ◽  
Vortex Shedding ◽  
Coherent Structures ◽  
Invariant Manifolds ◽  
Saddle Points ◽  
Lagrangian Coherent Structures ◽  
Speed Ratio ◽  
Cylinder Wall ◽  
Steady Rotation ◽  
Autonomous Dynamical System

In this paper, vortex shedding and suppression are numerically investigated as autonomous and non-autonomous dynamical systems respectively. Lagrangian coherent structures (LCSs) are used as a numerical tool to analyze these systems. These structures are ridges of Finite time Lyapunov exponent (FTLE) which act as material surfaces that are transport barriers within the flow. Initially, the utility of LCSs is explored for revealing the coherent structures of these systems. Finally, an active flow control method, steady rotation is applied to the non-autonomous dynamical system with different speed ratios to mitigate vortex shedding magnitude. This will eventually turn the system into an autonomous system. Fixed saddle points, separation profiles essentially as unstable time variant manifolds attached to cylinder wall and evolution of other unstable manifolds with variant speed ratios are analyzed with reference to LCSs. It is revealed that speed ratio of 2.1 fully suppresses the von Karman vortex street at Reynolds number of 100 and system turns into an autonomous dynamical system with fixed saddle points and time-invariant manifolds.

scholarly journals On embedding of invariant manifolds of the simplest Morse-Smale flows with heteroclinical intersections

2018 ◽  
Vol 20 (4) ◽  
pp. 378-383
Author(s):  
E. Ya. Gurevich ◽  
D. A. Pavlova
Keyword(s):  
Phase Space ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
Classification Problem ◽  
Closed Curve ◽  
Topological Classification ◽  
Dimensional Phase Space ◽  
Smale Flows ◽  
Source Sink ◽  
The Basin Of Attraction

We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. More precisely, we consider a class G(S4) of Morse-Smale flows on the sphere S4 such that for any flow f∈G(S4) its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows contains finite number of heteroclinical curves that belong to intersection of invariant manifolds of saddle equilibria. We describe a topology of embedding of saddle equilibria’s invariant manifolds; that is the first step in the solution of topological classification problem. In particular, we prove that the closures of invariant manifolds of saddle equlibria that do not contain heteroclinical curves are locally flat 2-sphere and closed curve. These manifolds are attractor and repeller of the flow. In set of orbits that belong to the basin of attraction or repulsion we construct a section that is homeomoprhic to the direct product S2×S1. We study a topology of intersection of saddle equlibria’s invariant manifolds with this section.

Invariant manifolds in singular perturbation problems for ordinary differential equations

1990 ◽  
Vol 116 (1-2) ◽  
pp. 45-78 ◽  
Author(s):  
Kunimochi Sakamoto
Keyword(s):  
Differential Equations ◽  
Singular Perturbation ◽  
Ordinary Differential Equations ◽  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Singular Perturbation Problems ◽  
Manifold Theory ◽  
Perturbation Problems ◽  
Geometric Idea ◽  
Normally Hyperbolic Invariant Manifolds

SynopsisBased on Fenichel's geometric idea, invariant manifold theory is applied to singular perturbation problems. This approach clarifies the nature of outer and inner solutions. A specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds. A method to approximate the connections is also presented.

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