Sensitivity and Property P in Non-Autonomous Systems
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Sensitivity and Property P in Non-Autonomous Systems Mohammad Salman and Ruchi Das Abstract. We prove equivalence of different stronger forms of sensitivity under certain conditions for non-autonomous systems and provide examples wherever equivalence is not true in general. We also prove that if a nonminimal, topologically transitive non-autonomous system having the set of almost periodic points dense converges uniformly, then it is thickly syndetically sensitive. Moreover, we introduce the notion of property P for non-autonomous systems, study it in general and on certain induced systems. Mathematics Subject Classification. 37B20, 37B55, 54B20, 54H20. Keywords. Non-autonomous systems, sensitivity, multi-sensitivity, property P .
1. Introduction Dynamical systems theory is an effective mathematical mechanism which describes time-fluctuating phenomena. Its broad application area differs from simple pendulum motion to complicated climate prediction models in physics and complex signal processes in biological cells. One of the most remarkable components of dynamical systems theory is sensitive dependence on initial conditions, also a key ingredient in chaos theory, which is closely linked to different variants of mixing. Most of the parameters in real-world problems such as weather predictions, brain patterns and population dynamics, whose behavior is influenced by external factors, are time dependent. As a result, many of the methods, concepts and results of autonomous dynamical systems are not applicable. Therefore, we require more general systems known as non-autonomous discrete dynamical systems which are widely useful in physics, mathematical biology, engineering, cf., [10,25]. Such systems comprise the crucial mechanism for non-stationary output signals. In addition, for suitable time-dependence of the external factor, it might be possible that Mohammad Salman is supported by Ministry of Minority Affairs, Government of India, Maulana Azad National Fellowship (F.No. 61-1/2019 (SA-III)). 0123456789().: V,-vol
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M. Salman, and R. Das
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a non-autonomous dynamics is unchanging. Hence, non-autonomous systems act as functional generators for stationary and non-stationary dynamical systems. We consider the following non-autonomous discrete dynamical system or difference equation in which the right-hand side is dependent specifically on the current time, xn+1 = fn (xn ), n ≥ 1,
(1.1)
where X is a metric space and each fn : X → X is a continuous map, for n ≥ 1. In such systems the trajectory of a point x ∈ X is the composition of different continuous maps fi ’s applied at x. Special cases of non-autonomous systems include uniformly convergent, finitely generated, periodic, etc. For a continuous map f , taking fn = f , for all n ≥ 1, the above system becomes an autonomous dynamical system (X, f ). In 1996, non-autonomous discrete dynamical systems were introduced by Kolyada and Snoha [7]. A dynamical system has two natural induced systems; one induced on the hyperspaces and the second one on the probability
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