Entropy Production of the Willamowski-Rössler Oscillator

2005 ◽  
Vol 60 (8-9) ◽  
pp. 599-9 ◽  
Author(s):  
Jörg W. Stucki ◽  
Robert Urbanczik
Keyword(s):  
Fixed Point ◽  
Entropy Production ◽  
Dynamic Behaviour ◽  
Basin Of Attraction ◽  
Steady States ◽  
Original Model ◽  
Core Component ◽  
Mean Values ◽  
The Mean ◽  
The Basin Of Attraction

Some properties of the Willamowski-Rössler model are studied by numerical simulations. From the original equations a minimal version of the model is derived which also exhibits the characteristic properties of the original model. This minimal model shows that it contains the Volterra-Lotka oscillator as a core component. It thus belongs to a class of generalized Volterra-Lotka systems. It has two steady states, a saddle point, responsible for chaos, and a fixed point, dictating its dynamic behaviour. The chaotic attractor is located close to the surface of the basin of attraction of the saddle node. The mean values of the variables are equal to the (unstable) steady state values during oscillations even under chaos, and the variables are always non-negative as in other generalized Volterra-Lotka systems. Surprisingly this was also the case with the original reversible Willamowski-Rössler model allowing to compare the entropy production during oscillations with the entropy production of the steady states. During oscillations the entropy production was always lower even under chaos. Since under these circumstances less energy is dissipated to produce the same output, the oscillating system is more efficient than the non-oscillatory one.

Approximations of large trunk line systems under heavy traffic

1994 ◽  
Vol 26 (04) ◽  
pp. 1063-1094
Author(s):  
Harold J. Kushner
Keyword(s):  
Fixed Point ◽  
Heavy Traffic ◽  
Service Capacity ◽  
Mean Values ◽  
Reflected Diffusion ◽  
Trunk Line ◽  
Overflow Traffic ◽  
The Mean ◽  
The Many ◽  
Fully Connected

The paper deals with large trunk line systems of the type appearing in telephone networks. There are many nodes or input sources, each pair of which is connected by a trunk line containing many individual circuits. Traffic arriving at either end of a trunk line wishes to communicate to the node at the other end. If the direct route is full, a rerouting might be attempted via an alternative route containing several trunks and connecting the same endpoints. The basic questions concern whether to reroute, and if so how to choose the alternative path. If the network is ‘large’ and fully connected, then the overflow traffic which is offered for rerouting to any trunk comes from many other trunks in the network with no one dominating. In this case one expects that some sort of averaging method can be used to approximate the rerouting requests and hence simplify the analysis. Essentially, the overflow traffic that a trunk offers the network for rerouting is in some average sense similar to the overflow traffic offered to that trunk. Indeed, a formalization of this idea involves the widely used (but generally heuristic) ‘fixed point' approximation method. One sets up the fixed point equations for appropriate rerouting strategies and then solves them to obtain an approximation to the system loss. In this paper we work in the heavy traffic regime, where the external offered traffic to any trunk is close to the service capacity of that trunk. It is shown that, as the number of links and circuits within each link go to infinity and for a variety of rerouting strategies, the system can be represented by an averaged limit. This limit is a reflected diffusion of the McKean–Vlasov (propagation of chaos) type, where the driving terms depend on the mean values of the solution of the equation. The averages occur due to the symmetry of the network and the averaging effects of the many interactions. This provides a partial justification for the fixed point method. The concrete dynamical systems flavor of the approach and the representations of the limit processes provide a useful way of visualizing the system and promise to be useful for the development of numerical methods and further analysis.

Biaccessibility in quadratic Julia sets

2000 ◽  
Vol 20 (6) ◽  
pp. 1859-1883 ◽  
Author(s):  
SAEED ZAKERI
Keyword(s):  
Fixed Point ◽  
Measure Zero ◽  
Julia Set ◽  
Basin Of Attraction ◽  
Countable Union ◽  
Common Theme ◽  
Chebyshev Map ◽  
Straight Line ◽  
The Common ◽  
The Basin Of Attraction

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$.In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve.In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.

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.

scholarly journals Exit time as a measure of ecological resilience

Science ◽  
2021 ◽  
Vol 372 (6547) ◽  
pp. eaay4895
Author(s):  
Babak M. S. Arani ◽  
Stephen R. Carpenter ◽  
Leo Lahti ◽  
Egbert H. van Nes ◽  
Marten Scheffer
Keyword(s):  
Time Series ◽  
Exit Time ◽  
Basin Of Attraction ◽  
Ecological Resilience ◽  
Average Life ◽  
Average Life Expectancy ◽  
Mean Exit Time ◽  
The Mean ◽  
Deterministic Components ◽  
The Basin Of Attraction

Ecological resilience is the magnitude of the largest perturbation from which a system can still recover to its original state. However, a transition into another state may often be invoked by a series of minor synergistic perturbations rather than a single big one. We show how resilience can be estimated in terms of average life expectancy, accounting for this natural regime of variability. We use time series to fit a model that captures the stochastic as well as the deterministic components. The model is then used to estimate the mean exit time from the basin of attraction. This approach offers a fresh angle to anticipating the chance of a critical transition at a time when high-resolution time series are becoming increasingly available.

scholarly journals Superattracting fixed points of quasiregular mappings

2014 ◽  
Vol 36 (3) ◽  
pp. 781-793 ◽  
Author(s):  
ALASTAIR FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Fixed Point ◽  
Spherical Shell ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Basin Of Attraction ◽  
Quasiregular Mapping ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Polynomial Type ◽  
The Basin Of Attraction

We investigate the rate of convergence of the iterates of an $n$-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

Approximations of large trunk line systems under heavy traffic

1994 ◽  
Vol 26 (4) ◽  
pp. 1063-1094 ◽  
Author(s):  
Harold J. Kushner
Keyword(s):  
Fixed Point ◽  
Heavy Traffic ◽  
Service Capacity ◽  
Mean Values ◽  
Reflected Diffusion ◽  
Trunk Line ◽  
Overflow Traffic ◽  
The Mean ◽  
The Many ◽  
Fully Connected

The paper deals with large trunk line systems of the type appearing in telephone networks. There are many nodes or input sources, each pair of which is connected by a trunk line containing many individual circuits. Traffic arriving at either end of a trunk line wishes to communicate to the node at the other end. If the direct route is full, a rerouting might be attempted via an alternative route containing several trunks and connecting the same endpoints. The basic questions concern whether to reroute, and if so how to choose the alternative path. If the network is ‘large’ and fully connected, then the overflow traffic which is offered for rerouting to any trunk comes from many other trunks in the network with no one dominating. In this case one expects that some sort of averaging method can be used to approximate the rerouting requests and hence simplify the analysis. Essentially, the overflow traffic that a trunk offers the network for rerouting is in some average sense similar to the overflow traffic offered to that trunk. Indeed, a formalization of this idea involves the widely used (but generally heuristic) ‘fixed point' approximation method. One sets up the fixed point equations for appropriate rerouting strategies and then solves them to obtain an approximation to the system loss. In this paper we work in the heavy traffic regime, where the external offered traffic to any trunk is close to the service capacity of that trunk. It is shown that, as the number of links and circuits within each link go to infinity and for a variety of rerouting strategies, the system can be represented by an averaged limit. This limit is a reflected diffusion of the McKean–Vlasov (propagation of chaos) type, where the driving terms depend on the mean values of the solution of the equation. The averages occur due to the symmetry of the network and the averaging effects of the many interactions. This provides a partial justification for the fixed point method. The concrete dynamical systems flavor of the approach and the representations of the limit processes provide a useful way of visualizing the system and promise to be useful for the development of numerical methods and further analysis.

scholarly journals Strategies for an Efficient Official Publicity Campaign

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Juan Neirotti
Keyword(s):  
Fixed Point ◽  
Public Opinion ◽  
Phase Space ◽  
Fixed Points ◽  
Basin Of Attraction ◽  
Opinion Formation ◽  
Publicity Campaign ◽  
The Basin Of Attraction

AbstractWe consider the process of opinion formation, in a society where there is a set of rules B that indicates whether a social instance is acceptable. Public opinion is formed by the integration of the voters’ attitudes which can be either conservative (mostly in agreement with B) or liberal (mostly in disagreement with B and in agreement with peer voters). These attitudes are represented by stable fixed points in the phase space of the system. In this article we study the properties of a perturbative term, mimicking the effects of a publicity campaign, that pushes the system from the basin of attraction of the liberal fixed point into the basin of the conservative point, when both fixed points are equally likely.

Accessible saddles on fractal basin boundaries

1992 ◽  
Vol 12 (3) ◽  
pp. 377-400 ◽  
Author(s):  
Kathleen T. Alligood ◽  
James A. Yorke
Keyword(s):  
Fixed Point ◽  
Basin Of Attraction ◽  
Open Disk ◽  
Fractal Basin Boundaries ◽  
Basin Boundary ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Simply Connected ◽  
Basin Boundaries ◽  
The Basin Of Attraction

AbstractFor a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods.

Predator–prey model with diffusion and indirect prey-taxis

2016 ◽  
Vol 26 (11) ◽  
pp. 2129-2162 ◽  
Author(s):  
J. Ignacio Tello ◽  
Dariusz Wrzosek
Keyword(s):  
Space Dimension ◽  
Basin Of Attraction ◽  
Steady States ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Random Dispersal ◽  
Dimension Formula ◽  
Predator Prey Models ◽  
Chemotaxis System ◽  
The Basin Of Attraction

We analyze predator–prey models in which the movement of predator searching for prey is the superposition of random dispersal and taxis directed toward the gradient of concentration of some chemical released by prey (e.g. pheromone), Model II, or released from damaged or injured prey due to predation (e.g. blood), Model I. The logistic O.D.E. describing the dynamics of prey population is coupled to a fully parabolic chemotaxis system describing the dispersion of chemoattractant and predator’s behavior. Global-in-time solutions are proved in any space dimension and stability of homogeneous steady states is shown by linearization for a range of parameters. For space dimension [Formula: see text] the basin of attraction of such a steady state is characterized by means of nonlinear analysis under some structural assumptions. In contrast to Model II, Model I possesses spatially inhomogeneous steady states at least in the case [Formula: see text].

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