see for example J. Guckenheimer and P. Holmes, Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1986).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....akin to bounding a real number by a sequence of rational approximants; we converge toward the strange set under investigation by a sequence of self similar Cantor sets. A self similar Cantor set (in the sense in which we use the word here) is a Cantor set equipped with a subshift of finite type[1, 9], i.e. the corresponding grammar can be stated as a finite number of pruning rules, each forbidding a finite symbol subsequence. The structure of the cycle expansions is more transparent if the pruning rules are implemented by redefining the alphabet; we shall give an example in sect. 4. The main ....

see for example J. Guckenheimer and P. Holmes, Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1986).

.... with that sequence as its itinerary This set of sequences is called the symbolic dynamics of the map F , and can be a very useful way to classify the system; often the partition can be chosen so that the map between points and sequences is one to one, allowing us to enumerate its periodic points [20] and calculate quantities like entropies, escape rates and Liapunov exponents [1] As another example, consider a cellular automaton (CA) in one dimension. This is a dynamical system on sequences where each site is updated according to some local rule, as a function of its state and those of its ....

....Church Turing thesis, is computationally universal. In fact, examples all up and down this hierarchy can be found in dynamical 2 systems theory. The languages generated by many simple hyperbolic systems are regular; this corresponds to the existence of a nite Markov partition for their dynamics [20]. At phase transitions such as the period doubling xed point, however, they can have a complicated scale invariant structure, and belong to an intermediate class called indexed context free [9] and the iteration of smooth maps in the plane can correspond to universal Turing machines [43] ....

see for instance Ch. 5 of J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. SpringerVerlag, 1983.

....akin to bounding a real number by a sequence of rational approximants; we converge toward the strange set under investigation by a sequence of self similar Cantor sets. A self similar Cantor set (in the sense in which we use the word here) is a Cantor set equipped with a subshift of finite type [3, 25] symbol dynamics, i.e. the corresponding grammar [26, 27] can be stated as a finite number of pruning rules, each forbidding a finite subsequence ffl 1 ffl 2 : ffl n . Here the notation ffl 1 ffl 2 : ffl n stands for n consecutive symbols ffl 1 , ffl 2 , ffl n , preceeded and ....

See for example J. Guckenheimer and P. Holmes, Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, 1986).

.... with that sequence as its itinerary This set of sequences is called the symbolic dynamics of the map F , and can be a very useful way to classify the system; often the partition can be chosen so that the map between points and sequences is one to one, allowing us to enumerate its periodic points [17] and calculate quantities like entropies, escape rates and Liapunov exponents [1] As another example, consider a cellular automaton (CA) in one dimension. This is a dynamical system on sequences where each site is updated according to some local rule, as a function of its state and those of its ....

....Church Turing thesis, is computationally universal. In fact, examples all up and down this hierarchy can be found in dynamical systems theory. The languages generated by many simple hyperbolic systems are regular; this corresponds to the existence of a finite Markov partition for their dynamics [17]. At phase transitions such as the period doubling fixed point, however, they can have a complicated scale invariant structure, and belong to an intermediate class called indexed context free [9] and the iteration of smooth maps in the plane can correspond to universal Turing machines [38] ....

see for instance Ch. 5 of J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. SpringerVerlag, 1983.

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