scholarly journals Bifurcation and global dynamical behavior of the f(T) theory

2014 ◽  
Vol 29 (07) ◽  
pp. 1450033 ◽  
Author(s):  
Chao-Jun Feng ◽  
Xin-Zhou Li ◽  
Li-Yan Liu
Keyword(s):  
Dynamical System ◽  
Critical Points ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Nonlinear Effects ◽  
Local Analysis ◽  
Local Method ◽  
Initial Values ◽  
Bifurcation Phenomenon

Usually, in order to investigate the evolution of a theory, one may find the critical points of the system and then perform perturbations around these critical points to see whether they are stable or not. This local method is very useful when the initial values of the dynamical variables are not far away from the critical points. Essentially, the nonlinear effects are totally neglected in such kind of approach. Therefore, one cannot tell whether the dynamical system will evolute to the stable critical points or not when the initial values of the variables do not close enough to these critical points. Furthermore, when there are two or more stable critical points in the system, local analysis cannot provide the information on which the system will finally evolute to. In this paper, we have further developed the nullcline method to study the bifurcation phenomenon and global dynamical behavior of the f(T) theory. We overcome the shortcoming of local analysis. And, it is very clear to see the evolution of the system under any initial conditions.

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

2018 ◽  
Vol 32 (05) ◽  
pp. 1850043 ◽  
Author(s):  
Feibiao Zhan ◽  
Shenquan Liu ◽  
Jing Wang ◽  
Bo Lu
Keyword(s):  
Purkinje Cell ◽  
Mixed Mode ◽  
Cell Model ◽  
Dynamical Behavior ◽  
Characteristic Index ◽  
Mixed Mode Oscillations ◽  
Codimension One ◽  
Bifurcation Phenomenon ◽  
Lyapunov Coefficient ◽  
Potential Link

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

Dynamical Constraints on Using Precise Spike Timing to Compute in Recurrent Cortical Networks

2008 ◽  
Vol 20 (4) ◽  
pp. 974-993 ◽  
Author(s):  
Arunava Banerjee ◽  
Peggy Seriès ◽  
Alexandre Pouget
Keyword(s):  
Dynamical System ◽  
Initial Conditions ◽  
Internal State ◽  
Spike Timing ◽  
Spiking Neurons ◽  
Spatiotemporal Patterns ◽  
Cortical Networks ◽  
Temporal Precision ◽  
Cortical Areas ◽  
Dynamical Constraints

Several recent models have proposed the use of precise timing of spikes for cortical computation. Such models rely on growing experimental evidence that neurons in the thalamus as well as many primary sensory cortical areas respond to stimuli with remarkable temporal precision. Models of computation based on spike timing, where the output of the network is a function not only of the input but also of an independently initializable internal state of the network, must, however, satisfy a critical constraint: the dynamics of the network should not be sensitive to initial conditions. We have previously developed an abstract dynamical system for networks of spiking neurons that has allowed us to identify the criterion for the stationary dynamics of a network to be sensitive to initial conditions. Guided by this criterion, we analyzed the dynamics of several recurrent cortical architectures, including one from the orientation selectivity literature. Based on the results, we conclude that under conditions of sustained, Poisson-like, weakly correlated, low to moderate levels of internal activity as found in the cortex, it is unlikely that recurrent cortical networks can robustly generate precise spike trajectories, that is, spatiotemporal patterns of spikes precise to the millisecond timescale.

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

The Nonstationary Effects in Asymmetric, Angle-Ply Laminated Composite Plates

1993 ◽  
Author(s):  
Ross M. Evan-Iwanowski ◽  
J. F. Nayfeh ◽  
C. H. Lu
Keyword(s):  
Initial Conditions ◽  
Laminated Composite ◽  
Composite Plates ◽  
Bench Mark ◽  
Laminated Composite Plates ◽  
Composite Columns ◽  
Decisive Effect ◽  
Initial Values ◽  
Excitation Frequencies ◽  
Nonstationary Effects

Abstract The system in the title has been subjected to a parametric, nonstationary (NS) linear v(t) = v0 + βt, and cyclic v(t) = v0 ± γ sin βt excitations. The NS linear responses settle on the initial, constant values for extended values of the excitation frequencies, thus they stabilize the stationary (ST) response. This is true for the initial conditions taken on the stationary curve, and for different ply-angles. For the initial conditions (I.C.) beyond the ST plots, the NS responses stay also near the initial conditions, but they have the wavy forms, which increase slightly for the lower values of the forcing frequencies. For the cyclic parametric excitations, the NS responses are cyclic contained within the ranges of the excitation amplitudes ±β and (finite) response amplitudes above the ST initial values, raising up and down within these limits. It appears that they cover the whole area within the above described constraints. The decisive effect of the cyclic NS inputs, i.e., almost instantaneous cyclic responses replacing the ST responses regardless of the I.C. and ST responses, is a bench mark of the cyclic NS. This behavior is distinctly different from the NS cyclic responses of the composite columns.

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 On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

scholarly journals Soil nutrient cycles as a nonlinear dynamical system

2004 ◽  
Vol 11 (5/6) ◽  
pp. 589-598 ◽  
Author(s):  
S. Manzoni ◽  
A. Porporato ◽  
P. D'Odorico ◽  
F. Laio ◽  
I. Rodriguez-Iturbe
Keyword(s):  
Dynamical System ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Soil Nutrient ◽  
Steady State Solution ◽  
Basins Of Attraction ◽  
Climatic Conditions ◽  
Stable Node ◽  
Nutrient Cycles ◽  
Stable Focus

Abstract. An analytical model for the soil carbon and nitrogen cycles is studied from the dynamical system point of view. Its main nonlinearities and feedbacks are analyzed by considering the steady state solution under deterministic hydro-climatic conditions. It is shown that, changing hydro-climatic conditions, the system undergoes dynamical bifurcations, shifting from a stable focus to a stable node and back to a stable focus when going from dry, to well-watered, and then to saturated conditions, respectively. An alternative degenerate solution is also found in cases when the system can not sustain decomposition under steady external conditions. Different basins of attraction for "normal" and "degenerate" solutions are investigated as a function of the system initial conditions. Although preliminary and limited to the specific form of the model, the present analysis points out the importance of nonlinear dynamics in the soil nutrient cycles and their possible complex response to hydro-climatic forcing.

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