scholarly journals Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

2020 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhouchao Wei ◽  
Fanrui Wang ◽  
Huijuan Li ◽  
Wei Zhang
Keyword(s):  
Stability Analysis ◽  
Periodic Orbit ◽  
Impulsive Control ◽  
Hidden Attractor ◽  
Geometric Methods ◽  
Onset Of Chaos ◽  
Specific Parameter ◽  
The Matrix ◽  
Parameter Values ◽  
Deviation Vector

<p style='text-indent:20px;'>In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.</p>

Jacobi analysis of a segmented disc dynamo system

2020 ◽  
Vol 17 (14) ◽  
pp. 2050205
Author(s):  
Aimin Liu ◽  
Biyu Chen ◽  
Yuming Wei
Keyword(s):  
Angular Velocity ◽  
Periodic Orbit ◽  
Finsler Geometry ◽  
Small Perturbations ◽  
Stable Periodic Orbit ◽  
Onset Of Chaos ◽  
Internal Parameters ◽  
Geometric Objects ◽  
Stable Periodic ◽  
Deviation Vector

In this paper, Jacobi stability of a segmented disc dynamo system is geometrically investigated from viewpoint of Kosambi–Cartan–Chern (KCC) theory in Finsler geometry. First, the geometric objects associated to the reformulated system are explicitly obtained. Second, the Jacobi stability of equilibria and a periodic orbit are discussed in the light of deviation curvature tensor. It is shown that all the equilibria are always Jacobi unstable for any parameters, a Lyapunov stable periodic orbit falls into both Jacobi stable regions and Jacobi unstable regions. The considered system is not robust to small perturbations of the equilibria, and some fragments of the periodic orbit are included in fragile region, indicating that the system is extremely sensitive to internal parameters and environment. Finally, the dynamics of the deviation vector and its curvature near all the equilibria are presented to interpret the onset of chaos in the dynamo system. In a physical sense, magnetic fluxes and angular velocity can show irregular oscillations under some certain cases, these oscillations may reveal the irregularity of magnetic field’s evolution and reversals.

scholarly journals Hexagonal-Shaped Spin Crossover Nanoparticles Studied by Ising-Like Model Solved by Local Mean Field Approximation

2021 ◽  
Vol 7 (5) ◽  
pp. 69
Author(s):  
Catherine Cazelles ◽  
Jorge Linares ◽  
Mamadou Ndiaye ◽  
Pierre-Richard Dahoo ◽  
Kamel Boukheddaden
Keyword(s):  
Spin Crossover ◽  
Hexagonal Lattice ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Ligand Field ◽  
Order And Disorder ◽  
The Matrix ◽  
Local Mean ◽  
Parameter Values

The properties of spin crossover (SCO) nanoparticles were studied for five 2D hexagonal lattice structures of increasing sizes embedded in a matrix, thus affecting the thermal properties of the SCO region. These effects were modeled using the Ising-like model in the framework of local mean field approximation (LMFA). The systematic combined effect of the different types of couplings, consisting of (i) bulk short- and long-range interactions and (ii) edge and corner interactions at the surface mediated by the matrix environment, were investigated by using parameter values typical of SCO complexes. Gradual two and three hysteretic transition curves from the LS to HS states were obtained. The results were interpreted in terms of the competition between the structure-dependent order and disorder temperatures (TO.D.) of internal coupling origin and the ligand field-dependent equilibrium temperatures (Teq) of external origin.

scholarly journals State estimation for bilinear impulsive control systems under uncertainties

2018 ◽  
pp. 35-41 ◽  
Author(s):  
Oxana G. Matviychuk
Keyword(s):  
Differential Equations ◽  
State Estimation ◽  
Control Systems ◽  
Uncertain Systems ◽  
Impulsive Control ◽  
Compact Set ◽  
The Matrix ◽  
Initial States ◽  
Impulsive Control Systems ◽  
The Given

The state estimation problem for uncertain impulsive control systems with a special structure is considered. The initial states are taken to be unknown but bounded with given bounds. We assume here that the coefficients of the matrix included in the differential equations are not exactly known, but belong to the given compact set in the corresponding space. We present here algorithms that allow to find the external ellipsoidal estimates of reachable sets for such bilinear impulsive uncertain systems.

Jacobi Stability of Simple Chaotic Systems with One Lyapunov Stable Equilibrium

2021 ◽  
Author(s):  
Changzhi Li ◽  
Biyu Chen ◽  
Aimin Liu ◽  
Huanhuan Tian
Keyword(s):  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Geometrical Structure ◽  
Chaotic Behavior ◽  
Internal Environment ◽  
Small Perturbations ◽  
Chaotic Flows ◽  
Dynamical Behaviors ◽  
Onset Of Chaos ◽  
Deviation Vector

Abstract This paper presents Jacobi stability analysis of 23 simple chaotic systems with only one Lyapunov stable equilibrium by Kosambi-Cartan-Chern (KCC) theory, and analyzes the chaotic behavior of these systems from the geometric viewpoint. Different from Lyapunov stability, the unique equilibrium for each system is always Jacobi unstable. Moreover, the dynamical behaviors of deviation vector near equilibrium are discussed to reveal the onset of chaos for these 23 systems, and show furtherly the coexistence of unique Lyapunov stable equilibrium and chaotic attractor for each system geometrically. The obtaining results show that these chaotic systems are not robust to small perturbations of the equilibrium, indicating that the systems are extremely sensitive to internal environment. This reveals that the chaotic flows generated by these systems may be related to Jacobi instability of the equilibrium. It is hoped that the study of this paper can help reveal the true geometrical structure of hidden chaotic attractors.

scholarly journals Connected Stability Analysis of Delay Systems via the Matrix-Valued Lyapunov Function

2013 ◽  
Vol 313-314 ◽  
pp. 1288-1292
Author(s):  
Jian Fei Sun ◽  
Xiao Li Wang ◽  
Chen Chen
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Comparison Theorem ◽  
Lyapunov Stability ◽  
Lyapunov Functional ◽  
Delay Systems ◽  
The Matrix

Bythe method of unioning thematrix-valued Lyapunov functional and comparison theorem,delay big system’sconnected Lyapunov stability is studied deeply.A series of new sufficientconditions are proposed.These results have not only theory meaning but alsopractical value.

Aspects of Osillations of Beams and Plates

2005 ◽  
Author(s):  
Mariya A. Zarubinska ◽  
W. T. van Horssen
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boundary Values ◽  
Internal Resonances ◽  
Specific Parameter ◽  
Plate Equations ◽  
Initial Boundary Value ◽  
Parameter Values

In this paper some initial boundary value problems for beam and plate equations will be studied. These initial boundary values problems can be regarded as simple models describing free oscillations of plates on elastic foundations or describing coupled torsional and vertical oscillations of a beam. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For the plate on an elastic foundation it turns out that complicated internal resonances can occur for specific parameter values.

Jacobi Stability Analysis of the Chen System

2019 ◽  
Vol 29 (10) ◽  
pp. 1950139 ◽  
Author(s):  
Qiujian Huang ◽  
Aimin Liu ◽  
Yongjian Liu
Keyword(s):  
Periodic Orbit ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chen System ◽  
Nonlinear Connection ◽  
Berwald Connection ◽  
The Lorenz System ◽  
Nonzero Equilibrium ◽  
Deviation Vector

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation

2002 ◽  
Author(s):  
Leslie Ng ◽  
Richard Rand
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Stability Analysis ◽  
Linear Stability Analysis ◽  
Perturbation Methods ◽  
Nonlinear Effects ◽  
Free Vibrations ◽  
The Stability ◽  
Parameter Values ◽  
Two Degree Of Freedom

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomena of coexistence. The differential equation studied governs the stability mode of vibration in an unforced conservative two degree of freedom system used to model the free vibrations of a thin elastica. Using perturbation methods, we show that at parameter values corresponding to coexistence, nonlinear terms can cause the origin to become nonlinearly unstable, even though linear stability analysis predicts the origin to be stable. We also investigate the bifurcations associated with this instability.

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