Attractors associated to a family of hyperbolic -adic plane automorphisms

2020 ◽  
pp. 1-16
Author(s):  
CLAYTON PETSCHE
Keyword(s):  
Hausdorff Dimension ◽  
Affine Plane ◽  
Borel Measure ◽  
Basin Of Attraction ◽  
Locally Compact ◽  
Forward Orbit ◽  
Shift Map ◽  
Almost All ◽  
The Basin Of Attraction ◽  
Two Parameter

We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.

Analysis of Coexistence and Extinction in a Two-Species Competition Model

2020 ◽  
Vol 30 (16) ◽  
pp. 2050248
Author(s):  
Sohrab Karimi ◽  
F. H. Ghane
Keyword(s):  
Population Biology ◽  
Basin Of Attraction ◽  
Stability Index ◽  
Competition Model ◽  
Growth Functions ◽  
Stable Attractor ◽  
The Stability ◽  
Almost All ◽  
The Basin Of Attraction ◽  
Normal Parameter

We study a competition model of two competing species in population biology having exponential and rational growth functions described by Alexander et al. [1992]. They observed that, for some choice of parameters, the competition model has a chaotic attractor [Formula: see text] for which the basin of attraction is riddled. Here, we give a detailed analysis to illustrate what happens when the normal parameter in this model changes. In fact, by varying the normal parameter, we discuss how the geometry of the basin of attraction of [Formula: see text], the region of coexistence or extinction, changes and illustrate the transitions between the set [Formula: see text] being an asymptotically stable attractor (extinction of rational species), a locally riddled basin attractor and a normally repelling chaotic saddle (extinction of exponential species). Additionally, we show that the riddling and the blowout bifurcation occur. Numerical simulations are presented graphically to confirm the validity of our results. In particular, we verify the occurrence of synchronization for some values of parameters. Finally, we apply the uncertainty exponent and the stability index to quantify the degree of riddling basin. Our observation indicates that the stability index is positive for Lebesgue for almost all points whenever the riddling occurs.

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.

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

Exponential Estimates for the Conjugate Function on Locally Compact Abelian Groups

1990 ◽  
Vol 33 (1) ◽  
pp. 34-44
Author(s):  
Nakhlé Habib Asmar
Keyword(s):  
Compact Support ◽  
Weak Type ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Exponential Estimates ◽  
Compact Abelian Groups ◽  
The Hilbert Transform ◽  
Almost All

AbstractLet G be a locally compact Abelian group, with character group X. Suppose that X contains a measurable order P. For the conjugate function of f is the function whose Fourier transform satisfies the identity for almost all χ in X where sgnp(χ) = - 1 , 0, 1, according as We prove that, when f is bounded with compact support, the conjugate function satisfies some weak type inequalities similar to those of the Hilbert transform of a bounded function with compact support in ℝ. As a consequence of these inequalities, we prove that possesses strong integrability properties, whenever f is bounded and G is compact. In particular, we show that, when G is compact and f is continuous on G, the function is integrable for all p > 0.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

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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