scholarly journals COMPUTING TWO-DIMENSIONAL GLOBAL INVARIANT MANIFOLDS IN SLOW–FAST SYSTEMS

2007 ◽  
Vol 17 (03) ◽  
pp. 805-822 ◽  
Author(s):  
J. P. ENGLAND ◽  
B. KRAUSKOPF ◽  
H. M. OSINGA
Keyword(s):  
Invariant Manifolds ◽  
Multiple Time Scales ◽  
Test Case ◽  
Multiple Time ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Boundary Problems ◽  
Case Examples ◽  
Unstable Manifolds ◽  
Global Invariant

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.

scholarly journals Structural Disorder and Collective Behavior of Two-Dimensional Magnetic Nanostructures

Nanomaterials ◽  
2021 ◽  
Vol 11 (6) ◽  
pp. 1392
Author(s):  
David Gallina ◽  
G. M. Pastor
Keyword(s):  
Interaction Energy ◽  
Time Evolution ◽  
Magnetic Nanostructures ◽  
Thermal Quenching ◽  
Structural Disorder ◽  
Square Lattice ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Two Dimensional ◽  
Energy Landscapes

Structural disorder has been shown to be responsible for profound changes of the interaction-energy landscapes and collective dynamics of two-dimensional (2D) magnetic nanostructures. Weakly-disordered 2D ensembles have a few particularly stable magnetic configurations with large basins of attraction from which the higher-energy metastable configurations are separated by only small downward barriers. In contrast, strongly-disordered ensembles have rough energy landscapes with a large number of low-energy local minima separated by relatively large energy barriers. Consequently, the former show good-structure-seeker behavior with an unhindered relaxation dynamics that is funnelled towards the global minimum, whereas the latter show a time evolution involving multiple time scales and trapping which is reminiscent of glasses. Although these general trends have been clearly established, a detailed assessment of the extent of these effects in specific nanostructure realizations remains elusive. The present study quantifies the disorder-induced changes in the interaction-energy landscape of two-dimensional dipole-coupled magnetic nanoparticles as a function of the magnetic configuration of the ensembles. Representative examples of weakly-disordered square-lattice arrangements, showing good structure-seeker behavior, and of strongly-disordered arrangements, showing spin-glass-like behavior, are considered. The topology of the kinetic networks of metastable magnetic configurations is analyzed. The consequences of disorder on the morphology of the interaction-energy landscapes are revealed by contrasting the corresponding disconnectivity graphs. The correlations between the characteristics of the energy landscapes and the Markovian dynamics of the various magnetic nanostructures are quantified by calculating the field-free relaxation time evolution after either magnetic saturation or thermal quenching and by comparing them with the corresponding averages over a large number of structural arrangements. Common trends and system-specific features are identified and discussed.

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 A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 763-791 ◽  
Author(s):  
B. KRAUSKOPF ◽  
H. M. OSINGA ◽  
E. J. DOEDEL ◽  
M. E. HENDERSON ◽  
J. GUCKENHEIMER ◽  
...  
Keyword(s):  
Vector Field ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Lorenz System ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
The Lorenz System ◽  
Global Invariant

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

2019 ◽  
Vol 29 (08) ◽  
pp. 1950105 ◽  
Author(s):  
Xiao-Le Yue ◽  
Yong Xu ◽  
Wei Xu ◽  
Jian-Qiao Sun
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Invariant Manifolds ◽  
High Efficiency ◽  
Search Algorithm ◽  
Mapping Method ◽  
Cell Mapping ◽  
Stable And Unstable Manifolds ◽  
Generalized Cell Mapping ◽  
Global Invariant

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

Chaotic Mixing of Highly Viscous Liquids With Rectangular or Elliptical Rotors

2005 ◽  
Author(s):  
M. Erol Ulucakli
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Fixed Points ◽  
Stokes Flow ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Rectangular Cell ◽  
Viscous Liquids ◽  
Unstable Manifolds ◽  
Time Periodic

The objective of this research is to experimentally investigate various mixing regions in a two-dimensional Stokes flow driven by a rectangular or elliptical rotor. Flow occurs in a rectangular cell filled with a very viscous fluid. The Reynolds number based on rotor size is in the order of 0.5. The flow is time-periodic and can be analyzed, both theoretically and experimentally, by considering the Poincare map that maps the position of a fluid particle to its position one period later. The mixing regions of the flow are determined, theoretically, by the fixed points of this map, either hyperbolic or degenerate, and their stable and unstable manifolds. Experimentally, the mixing regions are visualized by releasing a blob of a passive dye at one of these fixed points: as the flow evolves, the blob stretches to form a streak line that lies on the unstable manifold of the fixed point.

A NEW ALGORITHM FOR COMPUTING ONE-DIMENSIONAL STABLE AND UNSTABLE MANIFOLDS OF MAPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250018 ◽  
Author(s):  
HUIMIN LI ◽  
YANGYU FAN ◽  
JING ZHANG
Keyword(s):  
Bisection Method ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Numerical Examples ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Correction Scheme ◽  
Accuracy Criterion ◽  
Tangent Component ◽  
Unstable Manifolds

A new algorithm is presented to compute one-dimensional stable and unstable manifolds of fixed points for both two-dimensional and higher dimensional diffeomorphism maps. When computing the stable manifold, the algorithm does not require the explicit expression of the inverse map. The global manifold is grown from a local manifold and one point is added at each step. The new point is located with a "prediction and correction" scheme, which avoids searching the computed part of the manifold with a bisection method and accelerates the searching process. By using the fact that the Jacobian transports derivatives along the orbit of the manifold, the tangent component of the manifold is determined and a new accuracy criterion is proposed to check whether the new point that defines the manifold is acceptable. The performance of the algorithm is demonstrated with several numerical examples.

Heteroclinic Bifurcations and Invariant Manifolds in Rocking Block Dynamics

1991 ◽  
Vol 46 (6) ◽  
pp. 481-490 ◽  
Author(s):  
B. Bruhn ◽  
B. P. Koch
Keyword(s):  
Invariant Manifolds ◽  
Melnikov Method ◽  
Heteroclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Smale Horseshoe ◽  
Driven Systems ◽  
Rocking Block ◽  
Block Motion ◽  
Analytical Formulas ◽  
Unstable Manifolds

Abstract A simple model of rigid block motion under the influence of external perturbations is discussed. For periodic forcings we prove the existence of Smale horseshoe chaos in the dynamics. For slender blocks a heteroclinic bifurcation condition is calculated exactly, i.e. without using perturbation methods. That means that our results are valid for arbitrary excitation amplitudes. Furthermore, analytical formulas for the first pieces of the stable and unstable manifolds are derived not only for periodically but also for transiently driven systems. In the case of small excitation and damping the Melnikov method is used to treat the full nonlinear problem

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