General Lax-representation of a new higher-dimensional system: The current-fed membrane

2014 ◽  
Vol 59 ◽  
pp. 89-98 ◽  
Author(s):  
Victor K. Kuetche
Keyword(s):  
Dimensional System ◽  
Lax Representation ◽  
Higher Dimensional

scholarly journals Inverse-square law between time and amplitude for crossing tipping thresholds

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180504 ◽  
Author(s):  
Paul Ritchie ◽  
Özkan Karabacak ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Dimensional System ◽  
Stochastic Forcing ◽  
Inverse Square Law ◽  
System A ◽  
Asymptotic Expressions ◽  
Higher Dimensional ◽  
Random Disturbances

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

scholarly journals Revealing topology with transformation optics

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Lizhen Lu ◽  
Kun Ding ◽  
Emanuele Galiffi ◽  
Xikui Ma ◽  
Tianyu Dong ◽  
...  
Keyword(s):  
Physical Space ◽  
Real Space ◽  
Virtual Space ◽  
Transformation Optics ◽  
Coordinate Space ◽  
Dimensional System ◽  
Edge States ◽  
And Topology ◽  
Higher Dimensional ◽  
Insight Into

AbstractSymmetry deepens our insight into a physical system and its interplay with topology enables the discovery of topological phases. Symmetry analysis is conventionally performed either in the physical space of interest, or in the corresponding reciprocal space. Here we borrow the concept of virtual space from transformation optics to demonstrate how a certain class of symmetries can be visualised in a transformed, spectrally related coordinate space, illuminating the underlying topological transitions. By projecting a plasmonic system in a higher-dimensional virtual space onto a lower-dimensional system in real space, we show how transformation optics allows us to construct a topologically non-trivial system by inspecting its modes in the virtual space. Interestingly, we find that the topological invariant can be controlled via the singularities in the conformal mapping, enabling the intuitive engineering of edge states. The confluence of transformation optics and topology here can be generalized to other wave realms beyond photonics.

Non-Standard Reduction of Noisy Structural Systems: Shallow Arches

2000 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Exit Time ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Shallow Arches ◽  
Higher Dimensional ◽  
Mean Exit Time ◽  
Stationary Probability Density Function ◽  
Low Dimensional

Abstract The dynamics of a shallow arch subjected to small random external and parametric excitation is invegistated in this work. We develop rigorous methods to replace, in some limiting regime, the original higher dimensional system of equations by a simpler, constructive and rational approximation – a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results such as mean exit time, stationary probability density function.

Targeting Applying ε-Bounded Orbit Correction Perturbations

1998 ◽  
Vol 08 (07) ◽  
pp. 1575-1584 ◽  
Author(s):  
M. S. Baptista
Keyword(s):  
Dimensional System ◽  
Fast Computation ◽  
Orbit Correction ◽  
Higher Dimensional ◽  
Starting Point

A low-sized chaotic trajectory is used to compute epsilon-bounded orbit correction perturbations in order to rapidly target a trajectory from the vicinity of a starting point to a target. An algorithm that allows fast computation of a set of perturbations to be applied is presented, and its performance is tested in a higher-dimensional system, the kicked double rotor.

Distinguishing quantum operations: LOCC versus separable operators

2016 ◽  
Vol 14 (06) ◽  
pp. 1640028 ◽  
Author(s):  
Indrani Chattopadhyay ◽  
Debasis Sarkar
Keyword(s):  
Dimensional System ◽  
Quantum Operation ◽  
Initial State ◽  
Quantum Operations ◽  
Higher Dimensional

In this paper, we discuss the issue of distinguishing a pair of quantum operation in general. We use Krause theorem for representing the operations in unitary form. This supports the existence of pair of quantum operations that are not locally distinguishable, but distinguishable in asymptotic sense in some higher dimensional system. The process can even be successful without any use of the entangled initial state.

scholarly journals CARTESIAN PRODUCT PARTITIONING OF MULTI-DIMENSIONAL REACHABLE STATE SPACES

2016 ◽  
Vol 30 (3) ◽  
pp. 413-430 ◽  
Author(s):  
Tuǧrul Dayar ◽  
M. Can Orhan
Keyword(s):  
State Space ◽  
Cartesian Product ◽  
Model Systems ◽  
Dimensional System ◽  
Reachable State ◽  
State Spaces ◽  
Cartesian Products ◽  
Higher Dimensional ◽  
The Matrix ◽  
Minimum Number

Markov chains (MCs) are widely used to model systems which evolve by visiting the states in their state spaces following the available transitions. When such systems are composed of interacting subsystems, they can be mapped to a multi-dimensional MC in which each subsystem normally corresponds to a different dimension. Usually the reachable state space of the multi-dimensional MC is a proper subset of its product state space, that is, Cartesian product of its subsystem state spaces. Compact storage of the matrix underlying such a MC and efficient implementation of analysis methods using Kronecker operations require the set of reachable states to be represented as a union of Cartesian products of subsets of subsystem state spaces. The problem of partitioning the reachable state space of a three or higher dimensional system with a minimum number of partitions into Cartesian products of subsets of subsystem state spaces is shown to be NP-complete. Two algorithms, one merge based the other refinement based, that yield possibly non-optimal partitionings are presented. Results of experiments on a set of problems from the literature and those that are randomly generated indicate that, although it may be more time and memory consuming, the refinement based algorithm almost always computes partitionings with a smaller number of partitions than the merge-based algorithm. The refinement based algorithm is insensitive to the order in which the states in the reachable state space are processed, and in many cases it computes partitionings that are optimal.

scholarly journals The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xiaodong Li ◽  
Weipeng Zhang ◽  
Fengjie Geng ◽  
Jicai Huang
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Dimensional System ◽  
Bifurcation Diagrams ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Higher Dimensional

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

Higher dimensional systems of differential equations obtainable by iterative use of complex methods

2015 ◽  
Vol 38 ◽  
pp. 1560077 ◽  
Author(s):  
Asghar Qadir ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equations ◽  
Iterative Procedure ◽  
Dimensional System ◽  
Symmetry Generator ◽  
Higher Dimensional ◽  
Symmetry Properties ◽  
Systems Of Odes ◽  
Systems Of Pdes ◽  
Two Dimensional System ◽  
Complex Methods

A procedure had been developed to solve systems of two ordinary and partial differential equations (ODEs and PDEs) that could be obtained from scalar complex ODEs by splitting into their real and imaginary parts. The procedure was extended to four dimensional systems obtainable by splitting complex systems of two ODEs into their real and imaginary parts. As it stood, this procedure could be extended to any even dimension but not to odd dimensional systems. In this paper, the complex splitting is used iteratively to obtain three and four dimensional systems of ODEs and four dimensional systems of PDEs for four functions of two and four variables that correspond to a scalar base equation. We also provide characterization criteria for such systems to correspond to the base equation and a clear procedure to construct the base equation. The new systems of four ODEs are distinct from the class obtained by the single split of a two dimensional system. The previous complex methods split each infinitesimal symmetry generator into a pair of operators such that the entire set of operators do not form a Lie algebra. The iterative procedure sheds some light on the emergence of these "Lie-like" operators. In this procedure the higher dimensional system may not have any or the required symmetry for being directly solvable by symmetry and other methods although the base equation can have sufficient symmetry properties. Illustrative examples are provided.

scholarly journals Simultaneous Localization and Map Building with Modified System State

10.5772/7233 ◽  
2009 ◽  
Vol 6 (3) ◽  
pp. 27
Author(s):  
Zezhong Xu ◽  
Yanbin Zhuang
Keyword(s):  
Mobile Robot ◽  
Covariance Matrix ◽  
Dimensional System ◽  
Map Building ◽  
System State ◽  
Laser Rangefinder ◽  
System Models ◽  
Higher Dimensional ◽  
Computational Requirement ◽  
Lower Dimensional

The full covariance solution to simultaneous localization and map building based on extended Kalman filter requires update time quadratic in the number of landmarks in the map. In order to improve the computational efficiency, this paper reorganizes system state vector and system models. The state of mobile robot is redefined and represented indirectly. The higher dimensional system models and covariance matrix can be represented with two lower dimensional submatrices. An optimization solution is proposed based on this property. The computational requirement and memory requirement are decreased by half. The covariance matrix is fully updated without any approximation during estimation. The optimization solution is consistent and convergent theoretically and realistically. The experiment also compares the performance of optimization solution with the full covariance solution. All these techniques have been implemented on mobile robot ATRVII equipped with 2D laser rangefinder SICK.

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