THE BASIN OF ATTRACTION OF LOZI MAPPINGS

2009 ◽  
Vol 19 (03) ◽  
pp. 1043-1049 ◽  
Author(s):  
DIOGO BAPTISTA ◽  
RICARDO SEVERINO ◽  
SANDRA VINAGRE
Keyword(s):  
Strange Attractor ◽  
Basin Of Attraction ◽  
Parameter Values ◽  
The Basin Of Attraction

For parameter values which assure its existence, we characterize the basin of attraction for the Lozi map strange attractor.

scholarly journals Capture Zones of the Family of Functions λzmexp(z)

2003 ◽  
Vol 13 (09) ◽  
pp. 2623-2640 ◽  
Author(s):  
Núria Fagella ◽  
Antonio Garijo
Keyword(s):  
Fixed Point ◽  
Critical Point ◽  
Basin Of Attraction ◽  
Capture Zone ◽  
Connected Component ◽  
Dynamical Plane ◽  
Capture Zones ◽  
The Family ◽  
Parameter Values ◽  
The Basin Of Attraction

We consider the family of entire transcendental maps given by Fλ,m(z)=λzm exp (z) where m≥2. All functions Fλ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

Intermingled basins for the triangle map

1996 ◽  
Vol 16 (4) ◽  
pp. 651-662 ◽  
Author(s):  
James C. Alexander ◽  
Brian R. Hunt ◽  
Ittai Kan ◽  
James A. Yorke
Keyword(s):  
Lebesgue Measure ◽  
Basin Of Attraction ◽  
The Other ◽  
Positive Lebesgue Measure ◽  
Quadratic Maps ◽  
Open Set ◽  
Parameter Values ◽  
The Basin Of Attraction

AbstractA family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

scholarly journals A New Circumscribed Self-Excited Spherical Strange Attractor

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ramesh Ramamoorthy ◽  
Sajjad Shaukat Jamal ◽  
Iqtadar Hussain ◽  
Mahtab Mehrabbeik ◽  
Sajad Jafari ◽  
...  
Keyword(s):  
Strange Attractor ◽  
Unique Feature ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Spherical Coordinates ◽  
Chaotic Flows ◽  
Pass Through ◽  
Third Dimension ◽  
The Basin Of Attraction

Studying new chaotic flows with specific characteristics has been an open-ended field of exploring nonlinear dynamics. Investigation of chaotic flows is an area of research that has been taken into consideration for many years; thus, it helps in a better understanding of the chaotic systems. In this paper, an original chaotic 3D system, which has not been investigated yet, is presented in spherical coordinates. A unique feature of the proposed system is that its velocity becomes zero for a specific value of the radius variable. Hence, the system’s attractor is expected to be stuck on one side of a plane in spherical coordinates and inside or outside a sphere in the corresponding Cartesian coordinates. It means that the attractor cannot pass through the sphere or even touch it. The introduced system owns two unstable equilibria and a self-excited strange attractor. The 1D and 2D system’s bifurcation diagrams concerning the alteration of two bifurcation parameters are plotted to investigate the system’s dynamical properties. Moreover, the system’s Lyapunov exponents in the corresponding period of bifurcation parameters are calculated. Then, two 2D basins of attraction for two different third dimension values are explored. Based on the basin of attraction, it can be found that the sphere has attraction itself, partially, and some initial conditions are led to the sphere, not to the strange attractor. Ultimately, the connecting curves of the proposed system are explored to find an informative 1D set in addition to the system’s equilibria.

scholarly journals Equilibrium switching in nonlinear biological interaction networks with concurrent antagonism

2014 ◽  
Author(s):  
Jomar Rabajante ◽  
Cherryl O. Talaue
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Gene Interaction ◽  
Oscillatory Behavior ◽  
Interaction Networks ◽  
Dominant Component ◽  
Maximal Growth Rate ◽  
Decision Making Model ◽  
Parameter Values ◽  
The Basin Of Attraction

Concurrent decision-making model (CDM) of interaction networks involving more than two antagonistic components can represent various biological systems, such as gene interaction, species competition and mental perception. The model assumes sigmoid kinetics where every component stimulates itself but concurrently represses the others. Here we prove general dynamical properties of the CDM (e.g., location and stability of steady states) for any dimension of the state space even if the reciprocal antagonism between two components is asymmetric. Significant modifications in parameter values serve as biological regulators for inducing steady state switching by leading a temporal state to escape an undesired equilibrium. Increasing the maximal growth rate and decreasing the decay rate expand the basin of attraction of a steady state with the desired dominant component. Perpetually adding an external stimulus can shut down multi-stability of the system that increases the robustness of the system against stochastic noise. We further show that asymmetric interaction that forms a repressilator-type network generates oscillatory behavior.

scholarly journals Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 865 ◽  
Author(s):  
Julian Gonzalez-Ayala ◽  
Moises Santillán ◽  
Maria Santos ◽  
Antonio Calvo Hernández ◽  
José Mateos Roco
Keyword(s):  
Local Stability ◽  
Production Efficiency ◽  
Basin Of Attraction ◽  
Heat Engine ◽  
Time Constraints ◽  
Thermal Gradients ◽  
Heat Engines ◽  
Low Dissipation ◽  
The Basin Of Attraction

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

scholarly journals Time Delay and Feedback Control of an Inverted Pendulum With Stick Slip Friction

2007 ◽  
Author(s):  
Sue Ann Campbell ◽  
Stephanie Crawford ◽  
Kirsten Morris
Keyword(s):  
Time Delay ◽  
Basin Of Attraction ◽  
Critical Time ◽  
Viscous Friction ◽  
Experimental System ◽  
Friction Model ◽  
Stable Equilibrium Point ◽  
Feedback Controllers ◽  
Stick Slip ◽  
The Basin Of Attraction

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart which can move in one dimension. There is stick slip friction between the cart and the track on which it moves. Using two different models for this friction we design feedback controllers to stabilize the pendulum in the upright position. We show that controllers based on either friction model give better performance than one based on a simple viscous friction model. We then study the effect of time delay in this controller, by calculating the critical time delay where the system loses stability and comparing the calculated value with experimental data. Both models lead to controllers with similar robustness with respect to delay. Using numerical simulations, we show that the effective critical time delay of the experiment is much less than the calculated theoretical value because the basin of attraction of the stable equilibrium point is very small.

Stable running of a planar underactuated biped robot

Robotica ◽  
2010 ◽  
Vol 29 (5) ◽  
pp. 657-665 ◽  
Author(s):  
Yong Hu ◽  
Gangfeng Yan ◽  
Zhiyun Lin
Keyword(s):  
Limit Cycle ◽  
Stance Phase ◽  
Control Strategies ◽  
Basin Of Attraction ◽  
Biped Robot ◽  
State Spaces ◽  
Continuous State ◽  
Event Based ◽  
Telescopic Legs ◽  
The Basin Of Attraction

SUMMARYThis paper investigates the stable-running problem of a planar underactuated biped robot, which has two springy telescopic legs and one actuated joint in the hip. After modeling the robot as a hybrid system with multiple continuous state spaces, a natural passive limit cycle, which preserves the system energy at touchdown, is found using the method of Poincaré shooting. It is then checked that the passive limit cycle is not stable. To stabilize the passive limit cycle, an event-based feedback control law is proposed, and also to enlarge the basin of attraction, an additive passivity-based control term is introduced only in the stance phase. The validity of our control strategies is illustrated by a series of numerical simulations.

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