### Abstract

Mathematical setting for discrete dynamics is a state space, X, and a map S: X → X (the evolution operator) which defines the change of a state over one time step. Dynamics with choice, as we define it in [2], is a generalization of discrete dynamics where at every time step there is not one but several available maps that can transform the current state of the system. Many real life processes, from autocatalytic reaction systems to switched systems to cellular biochemical processes, exhibit the properties described by dynamics with choice. We are interested in the long-term behavior of such systems. In [2] we studied dynamics with choice with a finite number of available maps, S _{0}, S_{1},..., S_{n-1}. The orbit of a point x ∈ X then depends on the infinite sequence of symbols from the set J = {0,1,..., N - 1} encoding the order of maps Sj applied at each step. Denote by Σ the space of all one-sided infinite sequences of symbols from J and denote by σ the shift operator that erases the first symbol in sequences. We define the dynamics on the state space X with the choice of the maps S_{0}, S_{1},..., S_{n-1} as the discrete dynamics on the state space X = X x Σ with the evolution operator S: (x,w) → (S _{w(0)}(x),σ(w)), where w(0) is the first symbol in the sequence w. In this paper we address the case when there is possibly a continuum of available maps parameterized by points from the interval [0,1] or any metric compact J. (Think of a system of equations with parameters, where each parameter may change from step to step while staying within a prescribed interval.) We say that there is a range of choice. We give mathematical description of dynamics with a range of choice and prove general results on the existence and properties of global compact attractors in such dynamics. One of practical consequences of our results is that when the parameters of a nonlinear discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

Original language | English (US) |
---|---|

Pages (from-to) | 290-299 |

Number of pages | 10 |

Journal | Reliable Computing |

Volume | 15 |

Issue number | 4 |

State | Published - Jul 2011 |

### Fingerprint

### Keywords

- Discrete-time dynamics
- Global attractors
- Symbolic dynamics

### ASJC Scopus subject areas

- Software
- Applied Mathematics
- Computational Mathematics

### Cite this

*Reliable Computing*,

*15*(4), 290-299.

**Dynamics with a range of choice.** / Kapitanski, Lev; Živanović, Sanja.

Research output: Contribution to journal › Article

*Reliable Computing*, vol. 15, no. 4, pp. 290-299.

}

TY - JOUR

T1 - Dynamics with a range of choice

AU - Kapitanski, Lev

AU - Živanović, Sanja

PY - 2011/7

Y1 - 2011/7

N2 - Mathematical setting for discrete dynamics is a state space, X, and a map S: X → X (the evolution operator) which defines the change of a state over one time step. Dynamics with choice, as we define it in [2], is a generalization of discrete dynamics where at every time step there is not one but several available maps that can transform the current state of the system. Many real life processes, from autocatalytic reaction systems to switched systems to cellular biochemical processes, exhibit the properties described by dynamics with choice. We are interested in the long-term behavior of such systems. In [2] we studied dynamics with choice with a finite number of available maps, S 0, S1,..., Sn-1. The orbit of a point x ∈ X then depends on the infinite sequence of symbols from the set J = {0,1,..., N - 1} encoding the order of maps Sj applied at each step. Denote by Σ the space of all one-sided infinite sequences of symbols from J and denote by σ the shift operator that erases the first symbol in sequences. We define the dynamics on the state space X with the choice of the maps S0, S1,..., Sn-1 as the discrete dynamics on the state space X = X x Σ with the evolution operator S: (x,w) → (S w(0)(x),σ(w)), where w(0) is the first symbol in the sequence w. In this paper we address the case when there is possibly a continuum of available maps parameterized by points from the interval [0,1] or any metric compact J. (Think of a system of equations with parameters, where each parameter may change from step to step while staying within a prescribed interval.) We say that there is a range of choice. We give mathematical description of dynamics with a range of choice and prove general results on the existence and properties of global compact attractors in such dynamics. One of practical consequences of our results is that when the parameters of a nonlinear discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

AB - Mathematical setting for discrete dynamics is a state space, X, and a map S: X → X (the evolution operator) which defines the change of a state over one time step. Dynamics with choice, as we define it in [2], is a generalization of discrete dynamics where at every time step there is not one but several available maps that can transform the current state of the system. Many real life processes, from autocatalytic reaction systems to switched systems to cellular biochemical processes, exhibit the properties described by dynamics with choice. We are interested in the long-term behavior of such systems. In [2] we studied dynamics with choice with a finite number of available maps, S 0, S1,..., Sn-1. The orbit of a point x ∈ X then depends on the infinite sequence of symbols from the set J = {0,1,..., N - 1} encoding the order of maps Sj applied at each step. Denote by Σ the space of all one-sided infinite sequences of symbols from J and denote by σ the shift operator that erases the first symbol in sequences. We define the dynamics on the state space X with the choice of the maps S0, S1,..., Sn-1 as the discrete dynamics on the state space X = X x Σ with the evolution operator S: (x,w) → (S w(0)(x),σ(w)), where w(0) is the first symbol in the sequence w. In this paper we address the case when there is possibly a continuum of available maps parameterized by points from the interval [0,1] or any metric compact J. (Think of a system of equations with parameters, where each parameter may change from step to step while staying within a prescribed interval.) We say that there is a range of choice. We give mathematical description of dynamics with a range of choice and prove general results on the existence and properties of global compact attractors in such dynamics. One of practical consequences of our results is that when the parameters of a nonlinear discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

KW - Discrete-time dynamics

KW - Global attractors

KW - Symbolic dynamics

UR - http://www.scopus.com/inward/record.url?scp=80053895091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053895091&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:80053895091

VL - 15

SP - 290

EP - 299

JO - Reliable Computing

JF - Reliable Computing

SN - 1385-3139

IS - 4

ER -