scholarly journals Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families Information

2021 ◽  
Vol 20 ◽  
pp. 186-195
Author(s):  
Orge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Bladimir Blanco Montes ◽  
Primitivo B. Acosta Humánez
Keyword(s):  
Critical Points ◽  
Stable Manifold ◽  
Phase Portraits ◽  
Infinite Plane ◽  
Finite Plane ◽  
Quadratic Systems ◽  
Liénard Equations ◽  
The Stability ◽  
Transcritical Bifurcations

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis, moreover Algebraic aspects are also included such that hamiltonian cases and Galois differential groupes. It should be noted that these families have associated oscillating type problems given their similarity to the Liénard equations.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Phase space correspondence between Jordan and Einstein frames

2021 ◽  
pp. 2150087
Author(s):  
Amin Salehi
Keyword(s):  
Phase Space ◽  
Critical Point ◽  
Critical Points ◽  
Dynamical Behavior ◽  
Cosmological Parameters ◽  
Jordan Frame ◽  
Field Equations ◽  
Theories Of Gravity ◽  
The Stability ◽  
Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

scholarly journals Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

2020 ◽  
Vol 55 (4) ◽  
Author(s):  
Jorge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Juliana Vargas Sánchez
Keyword(s):  
Linear System ◽  
Critical Point ◽  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Critical Points ◽  
Global Dynamics ◽  
Parametric Surfaces ◽  
Critical Points At Infinity ◽  
Global Phase Portrait ◽  
The Stability

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

scholarly journals Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

2021 ◽  
Vol 29 (3) ◽  
Author(s):  
Murtala Bello Aliyu ◽  
Mohd Hafiz Mohd ◽  
Mohd Salmi Md. Noorani
Keyword(s):  
Bifurcation Analysis ◽  
Species Coexistence ◽  
Real Life ◽  
Period Doubling ◽  
Type Model ◽  
Coexistence Mechanisms ◽  
Multiple Species ◽  
The Stability ◽  
Transcritical Bifurcations ◽  
Multiple Interactions

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

scholarly journals Theory and applications of new fractional-order chaotic system under Caputo operator

2021 ◽  
Vol 12 (1) ◽  
pp. 20-38
Author(s):  
Ndolane Sene
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Caputo Derivative ◽  
Phase Portraits ◽  
Chaotic Circuit ◽  
Chaotic Model ◽  
The Stability ◽  
The Impact ◽  
Schematic Circuit ◽  
Good Agreement

This paper introduces the properties of a fractional-order chaotic system described by the Caputo derivative. The impact of the fractional-order derivative has been focused on. The phase portraits in different orders are obtained with the aids of the proposed numerical discretization, including the discretization of the Riemann-Liouville fractional integral. The stability analysis has been used to help us to delimit the chaotic region. In other words, the region where the order of the Caputo derivative involves and where the presented system in this paper is chaotic. The nature of the chaos has been established using the Lyapunov exponents in the fractional context. The schematic circuit of the proposed fractional-order chaotic system has been presented and simulated in via Mutltisim. The results obtained via Multisim simulation of the chaotic circuit are in good agreement with the results with Matlab simulations. That provided the fractional operators can be applied in real- worlds applications as modeling electrical circuits. The presence of coexisting attractors for particular values of the parameters of the presented fractional-order chaotic model has been studied.

Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms

2008 ◽  
Vol 51 (3) ◽  
pp. 448-459 ◽  
Author(s):  
Toru Sasahara
Keyword(s):  
Critical Points ◽  
Harmonic Map ◽  
Biharmonic Maps ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
The Stability ◽  
Biharmonic Map ◽  
Legendrian Submanifolds

AbstractBiharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

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