scholarly journals Global Bifurcation Map of the Homogeneous States in the Gray–Scott Model

2017 ◽  
Vol 27 (07) ◽  
pp. 1730024 ◽  
Author(s):  
Joaquín Delgado ◽  
Lucía Ivonne Hernández-Martínez ◽  
Javier Pérez-López
Keyword(s):  
Global Bifurcation ◽  
Time Dependent ◽  
Numerical Continuation ◽  
Local Bifurcation ◽  
Bautin Bifurcation ◽  
Spatially Homogeneous ◽  
Two Parameters

We study the spatially homogeneous time-dependent solutions and their bifurcations in the Gray–Scott model. We find the global map of bifurcations by a combination of rigorous verification of the existence of Takens–Bogdanov and a Bautin bifurcation, in the space of two parameters [Formula: see text]–[Formula: see text]. With the aid of numerical continuation of local bifurcation curves we give a global description of all the possible bifurcations.

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives

2020 ◽  
Vol 30 (05) ◽  
pp. 2050072 ◽  
Author(s):  
Yingjuan Yang ◽  
Guoyuan Qi ◽  
Jianbing Hu ◽  
Philippe Faradja
Keyword(s):  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
The Self ◽  
Numerical Continuation ◽  
Chaotic Attractors ◽  
Phase Portrait Analysis ◽  
The Stability ◽  
Two Parameters ◽  
The Basin Of Attraction

A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

scholarly journals A meeting point of entropy and bifurcations in cross-diffusion herding

2016 ◽  
Vol 28 (2) ◽  
pp. 317-356 ◽  
Author(s):  
ANSGAR JÜNGEL ◽  
CHRISTIAN KUEHN ◽  
LARA TRUSSARDI
Keyword(s):  
Gradient Flow ◽  
Stationary Solutions ◽  
Steady States ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
System Modelling ◽  
Local Bifurcation ◽  
Diffusion Term ◽  
Cross Diffusion ◽  
The Cross

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

scholarly journals Comparison of Stochastic Forecasting Models

2021 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Brownian Motion ◽  
Stock Prices ◽  
Absolute Error ◽  
Geometric Brownian Motion ◽  
Time Dependent ◽  
Confidence Bounds ◽  
History Of ◽  
Brownian Motion Model ◽  
Stochastic Forecasting ◽  
Two Parameters

In this work, we compare several stochastic forecasting techniques like Stochastic Differential Equations (SDE), ARIMA, the Bayesian filter, Geometric Brownian motion (GBM), and the Kalman filter. We use historical daily stock prices of Microsoft (MSFT), Target (TGT) and Tesla (TSLA) and apply all algorithms to try to predict 54 days ahead. We find that there are instances in which all algorithms do well, or do poorly. We find that all three stocks have a strong auto-correlation and a high Hurst factor which shows that it is possible to predict future prices based on a short history of past prices. In our geometric Brownian motion model, we have two parameters for drift and diffusion which are not time dependent. In our more general SDE model (TDNGBM), we have time-dependent drift and time-dependent diffusion terms which makes it more effective than GBM. We measure all algorithms on the correlation between the predicted and actual values, the mean absolute error (MAE) and also the confidence bounds generated by the methods. Confidence intervals are more important than point forecasts, and we see that TDNGBM and ARIMA produce good bounds.

scholarly journals Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Changtong Li ◽  
Hao Sun ◽  
Yuzhen Wang
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Bifurcation Theory ◽  
A Priori Estimates ◽  
Global Bifurcation ◽  
Local Bifurcation ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
Steady State Solutions

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

Sign in / Sign up

Export Citation Format

Share Document