scholarly journals Energy function for Ω-stable flows without limit cycles on surfaces

2019 ◽  
Vol 21 (4) ◽  
pp. 460-468 ◽  
Author(s):  
Anna E. Kolobyanina ◽  
Vladislav E. Kruglov
Keyword(s):  
Finite Number ◽  
Limit Cycles ◽  
Energy Function ◽  
Complex Structure ◽  
Saddle Points ◽  
Hyperbolic Fixed Points ◽  
Smale Flows ◽  
Further Development ◽  
Flows On Surfaces ◽  
Arbitrary Flow

The paper is devoted to the study of the class of Ω-stable flows without limit cycles on surfaces, i.e. flows on surfaces with non-wandering set consisting of a finite number of hyperbolic fixed points. This class is a generalization of the class of gradient-like flows, differing by forbiddance of saddle points connected by separatrices. The results of the work are the proof of the existence of a Morse energy function for any flow from the considered class and the construction of such a function for an arbitrary flow of the class. Since the results are a generalization of the corresponding results of K. Meyer for Morse-Smale flows and, in particular, for gradient-like flows, the methods for constructing the energy function for the case of this article are a further development of the methods used by K. Meyer, taking in sense the specifics of Ω-stable flows having a more complex structure than gradient-like flows due to the presence of the so-called "chains" of saddle points connected by their separatrices.

scholarly journals Morse-Bott energy function for surface Ω-stable flows

2020 ◽  
Vol 22 (4) ◽  
pp. 434-441
Author(s):  
Anna E. Kolobyanina ◽  
Vladislav E. Kruglov
Keyword(s):  
Lyapunov Function ◽  
Finite Number ◽  
Fixed Points ◽  
Structural Stability ◽  
Limit Cycles ◽  
Energy Function ◽  
Hyperbolic Fixed Points ◽  
Smale Flows ◽  
Flows On Surfaces

In this paper, we consider the class of Ω-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of Ω -stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of saddle connections that do not form cycles. The authors have constructed the Morse-Bott energy function for any such flow. The results obtained are an ideological continuation of the classical works of S. Smale, who proved the existence of the Morse energy function for gradient-like flows, and K. Meyer, who established the existence of the Morse-Bott energy function for Morse-Smale flows. The specificity of Ω-stable flows takes them beyond the framework of structural stability, but the decrease along the trajectories of such flows is still tracked by the regular Lyapunov function.

Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

2021 ◽  
Vol 29 (6) ◽  
pp. 835-850
Author(s):  
Vladislav Kruglov ◽  
◽  
Olga Pochinka ◽  
◽  
Keyword(s):  
Finite Number ◽  
Limit Cycle ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Smooth Flow ◽  
Direction Of Movement ◽  
Smale Flows ◽  
Phase Surface ◽  
Flows On Surfaces

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another.

scholarly journals Memory Dynamics in Attractor Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Guoqi Li ◽  
Kiruthika Ramanathan ◽  
Ning Ning ◽  
Luping Shi ◽  
Changyun Wen
Keyword(s):  
Energy Function ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Memory Systems ◽  
Initial Stimulus ◽  
Attractor Network ◽  
Local Maxima ◽  
Attractor Networks ◽  
New Energy

As can be represented by neurons and their synaptic connections, attractor networks are widely believed to underlie biological memory systems and have been used extensively in recent years to model the storage and retrieval process of memory. In this paper, we propose a new energy function, which is nonnegative and attains zero values only at the desired memory patterns. An attractor network is designed based on the proposed energy function. It is shown that the desired memory patterns are stored as the stable equilibrium points of the attractor network. To retrieve a memory pattern, an initial stimulus input is presented to the network, and its states converge to one of stable equilibrium points. Consequently, the existence of the spurious points, that is, local maxima, saddle points, or other local minima which are undesired memory patterns, can be avoided. The simulation results show the effectiveness of the proposed method.

Integrating PDF interface into Java application

2014 ◽  
Vol 32 (3) ◽  
pp. 495-508 ◽  
Author(s):  
Quan Lu ◽  
Gao Liu ◽  
Jing Chen
Keyword(s):  
Digital Library ◽  
Complex Structure ◽  
Portable Document Format ◽  
Content Type ◽  
Service Specification ◽  
Java Application ◽  
Novel Approach ◽  
Extensible Markup ◽  
Further Development ◽  
Practical Implications

Purpose – The purpose of this paper is to propose a novel approach to integrate portable document format (PDF) interface into Java-based digital library application. It bridges the gap between conducting content operation and viewing on PDF document asynchronously. Design/methodology/approach – In this paper, the authors first review some related research and discuss PDF and its drawbacks. Next, the authors propose the design steps and implementation of three modes of displaying PDF document: PDF display, image display and extensible markup language (XML) display. A comparison of these three modes has been carried out. Findings – The authors find that the PDF display is able to completely present the original PDF document contents and thus obviously superior to the other two displays. In addition, the format specification of PDF-based e-book does not perform well; lack of standardization and complex structure is exposed to the publication. Practical implications – The proposed approach makes viewing the PDF documents more convenient and effective, and can be used to retrieve and visualize the PDF documents and to support the personalized function customization of PDF in the digital library applications. Originality/value – This paper proposes a novel approach to solve the problem between content operation and the view of PDF synchronously, providing users a new tool to retrieve and reuse the PDF documents. It contributes to improve the service specification and policy of viewing the PDF for digital library. Besides, the personalized interface and public index make further development and application more feasible.

Using Energy Shaping and Regulation for Limit Cycle Stabilization, Generation, and Transition in Simple Locomotive Systems

2021 ◽  
Author(s):  
Mark Yeatman ◽  
Robert D. Gregg
Keyword(s):  
Limit Cycles ◽  
Energy Function ◽  
Inverted Pendulum ◽  
Control Parameters ◽  
Energy Shaping ◽  
Constant Energy ◽  
Gait Characteristics ◽  
Hopping Robot ◽  
Control Framework ◽  
Globally Stable

Abstract This paper explores new ways to use energy shaping and regulation methods in walking systems to generate new passive-like gaits and dynamically transition between them. We recapitulate a control framework for Lagrangian hybrid systems, and show that regulating a state varying energy function is equivalent to applying energy shaping and regulating the system to a constant energy value. We then consider a simple 1-dimensional hopping robot and show how energy shaping and regulation control can be used to generate and transition between nearly globally stable hopping limit cycles. The principles from this example are then applied on two canonical walking models, the spring loaded inverted pendulum (SLIP) and compass gait biped, to generate and transition between locomotive gaits. These examples show that piecewise jumps in control parameters can be used to achieve stable changes in desired gait characteristics dynamically/online.

On stochastic behavior of perturbed Hamiltonian systems

2000 ◽  
Vol 20 (1) ◽  
pp. 55-76 ◽  
Author(s):  
MICHAEL BRIN ◽  
MARK FREIDLIN
Keyword(s):  
Stochastic Process ◽  
Finite Number ◽  
Hamiltonian Systems ◽  
Level Sets ◽  
Local Minimum ◽  
Slow Component ◽  
Saddle Points ◽  
Connected Components ◽  
Stochastic Behavior ◽  
Sensitive Behavior

We consider deterministic perturbations $\ddot q^\varepsilon(t)+F'(q^\varepsilon(t))=\varepsilon b(\dot q^\varepsilon(t),q^\varepsilon(t))$ of an oscillator $\ddot q+F'(q)=0$, $q\in{\mathbb R}^1$. Assume that $\lim_{|q|\to\infty}F(q)=\infty$ and that $F'(q)$ has a finite number of nondegenerate zeros. For a generic $F$, if $\partial b/\partial\dot q<0$ (as in the case of friction), then typical orbits are attracted to points where $F$ has a local minimum. For $0<\varepsilon\ll1$, the equilibrium to which the trajectory is attracted is ‘random’. To study this randomness, which is caused by the sensitive behavior of trajectories near the saddle points, we consider the graph $\Gamma$ homeomorphic to the space of connected components of the level sets of the Hamiltonian $H(p,q)=p^2/2+F(q)$. We show that, as $\varepsilon\to0$, the slow component of $(p^\epsilon(t/\epsilon),q^\epsilon(t/\epsilon))$ tends to a certain stochastic process on $\Gamma$ which is deterministic inside the edges and branches at the interior vertices into adjacent edges with probabilities which can be calculated through the Hamiltonian $H$ and the perturbation $b$.

scholarly journals Modeling of stimuli-responsive nanoreactors: rational rate control towards the design of colloidal enzymes

2020 ◽  
Vol 5 (3) ◽  
pp. 602-619
Author(s):  
Matej Kanduč ◽  
Won Kyu Kim ◽  
Rafael Roa ◽  
Joachim Dzubiella
Keyword(s):  
Optimal Design ◽  
Rate Control ◽  
Complex Structure ◽  
Stimuli Responsive ◽  
Structure Property ◽  
Multi Scale ◽  
Scale Modeling ◽  
Responsive Polymer ◽  
Multi Scale Modeling ◽  
Further Development

Responsive polymer-based nanoreactors exhibit complex structure-property-function relationships which require multi-scale modeling and simulation approaches for optimal design and a further development towards 'colloidal enzymes'.

Realization of Arbitrary Configuration of Limit Cycles of Piecewise Linear Systems

2020 ◽  
Vol 30 (09) ◽  
pp. 2050133
Author(s):  
Shaoqing Wang ◽  
Jiazhong Yang
Keyword(s):  
Linear System ◽  
Finite Number ◽  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Continuous Curve ◽  
Jordan Curves ◽  
Piecewise Linear Systems ◽  
Piecewise Linear System ◽  
Arbitrary Configuration

In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of finite number of Jordan curves can be realized by a discontinuous piecewise linear system with two zones separated by a continuous curve.

scholarly journals An effective method of counting the number of limit cycles

1979 ◽  
Vol 76 ◽  
pp. 35-114 ◽  
Author(s):  
Kazuo Yamato
Keyword(s):  
Finite Number ◽  
Limit Cycles

We are interested in determining, after a finite number of procedures, the number and the approximate positions of limit cycles for a given system.

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