N-dimensional Dynamical System Research Articles

Statistical mechanics of multi-dimensional Cantor sets, Gödel theorem and quantum spacetime

Two different descriptions of an abstract n-dimensional dynamical system are discussed: a Sierpinski space setting and a statistical cellular space setting. The results suggest that in four dimensions the phase space dynamics is peano-like and resembles an Anosov diffeomorphism of a compact manifold which is dense and quasi-ergodic. The Hausdorff capacity dimension in this case is d (4) C = 3.981 ≅ 4 and we conjecture that the simplest fully developed turbulence is related to d (5) C ≅ 6.3. The corresponding Shannon information entropy of the second analysis are l (4) S = 3.68 and l (5) S = 6.12. The implications of the results for quantum spacetime are outlined and found to be consistent with Heisenberg uncertainty relationship and Bekenstein-Hawking entropy. Finally, the connection between strange nonchaotic behaviour and Gödel theorem is discussed.

Lyapunov exponents as a nonparametric diagnostic for stability analysis

SUMMARY The common observation made in the empirical nonlinear dynamics literature is the constraints imposed by the availability of a limited number of observations in the implementation of the existing algorithms of Lyapunov exponents. The algorithm discussed here can estimate all n Lyapunov exponents of an unknown n-dimensional dynamical system accurately with limited number of observations. This makes the algorithm attractive for applications to economic as well as financial time-series data. The implementation of the algorithm is carried out by multilayer feedforward networks which are capable of approximating any function and its derivatives to any degree of accuracy.

An algorithm for the n Lyapunov exponents of an n-dimensional unknown dynamical system

An algorithm for estimating Lyapunov exponents of an unknown dynamical system is designed. The algorithm estimates not only the largest but all Lyapunov exponents of the unknown system. The estimation is carried out by a multivariate feedforward network estimation technique. We focus our attention on deterministic as well as noisy system estimation. The performance of the algorithm is very satisfactory in the presence of noise as well as with limited number of observations.

C*-ALGEBRAS OF SINGULAR INTEGRAL OPERATORS AND TOEPLITZ OPERATORS ASSOCIATED WITH n-DIMENSIONAL FLOWS

For a given covariant representation π of an n-dimensional dynamical system (X, R n, α), we study the C*-algebras [Formula: see text] of abstract singular integral opertors and Toeplitz operators generated by π(C(X)) and Hilbert transforms corresponding to n linearly independent directions. Such an algebra has a chain of ideals [Formula: see text]. We compute all [Formula: see text] and show that for 0≤k≤n−1 each of these quotients is the direct sum of C*-algebras which can be thought of as [Formula: see text] for flows of lower dimensions. The ideal structure of [Formula: see text] is carefully studied. We also determine precisely when [Formula: see text] is a C*-algebra of type I.

Multi-dimensional Cantor sets in classical and quantum mechanics

We give two different descriptions of an abstract n-dimensional dynamical system. First we use a Sierpinski space setting and subsequently we use a statistical cellular space setting. The results of the analysis elucidate certian universal behaviour which was observed in a wide category of cellular automata. The results further show that in four dimensions the phase space dynamics is Peano-like and resembles an Anosov diffeomorphism of a compact manifold which is dense and quasi-ergodic. The fractal dimension in this case is d c (4) = 3.981 ≅ 4 and we conjecture that fully developed turbulence is related to d c (5) = 6.3. The corresponding Shannon information entropy of the second analysis are δ s (4) = 3.68 and δ s (5) = 6.12. In the case of eight dimensional phase space both descriptions lead to almost identical numerical results. Possible implications of these theoretical results to physical spatio-temporal chaos and the reduction of complexity are discussed. In conclusion, the relevance of Cantor-like space-time for the Copenhagen interpretation of quantum mechanics and the connection to non-standard analysis and Boscovich covariance are touched upon.

Invariants of a certain non-linear N-dimensional dynamical system

We consider a certain second order non-linear system of N dimensions represented by coupled ordinary differential equations and employ a direct and simple method to obtain its invariants. Illustrative examples are also discussed.

Dense memory with high order neural networks

General high order neural networks [LD…] (models which are multinomial as opposed to linear in gain functions) offer high storage capacity [PPH] and fast convergence [GM] at the expense combinatorial growth of connections. This paper presents a specific high order neural network design which can store using n neurons any number M,1≤M≤2 n, of any of the binomial n-strings; in a schematic representation the model requires only 5n+M(1+2n) edges. Each stored n-string represents a memory as a constant attractor trajectory of an n-dimensional differential equation dynamical system, a memory model neural network. With sufficiently high gains, the only stable attractors are the memories. Thus the memory model amounts to a solution of a version of the fundamental memory problem posed in [LMP] and elsewhere. The memory model can be used in error-correcting decoding of any binary string code and can accept noisy nonbinary signals.

Necessary and sufficient conditions for existence of periodic solutions of n-dimensional dynamical systems

On considere le systeme dynamique a 2 N dimensions: x˙=f(x,y), y˙=g(x,y) ou x∈R N et y∈R N , f,g∈C 2 (,R n ), est un domaine de R N ×R N , O(0,0)∈int. On donne une condition necessaire et suffisante pour l'existence de solutions periodiques en appliquant le theoreme de point fixe de Schauder