Synchronicity From Synchronized Chaos

The synchronization of loosely coupled chaotic oscillators, a phenomenon investigated intensively for the last two decades, may realize the philosophical notion of synchronicity. Effectively unpredictable chaotic systems, coupled through only a few variables, commonly exhibit a predictable relationship that can be highly intermittent. We argue that the phenomenon closely resembles the notion of meaningful synchronicity put forward by Jung and Pauli if one identifies "meaningfulness" with internal synchronization, since the latter seems necessary for synchronizability with an external system. Jungian synchronization of mind and matter is realized if mind is analogized to a computer model, synchronizing with a sporadically observed system as in meteorological data assimilation. Internal synchronization provides a recipe for combining different models of the same objective process, a configuration that may also describe the functioning of conscious brains. In contrast to Pauli's view, recent developments suggest a materialist picture of semi-autonomous mind, existing alongside the observed world, with both exhibiting a synchronistic order. Basic physical synchronicity is manifest in the non-local quantum connections implied by Bell's theorem. The quantum world resides on a generalized synchronization "manifold", a view that provides a bridge between nonlocal realist interpretations and local realist interpretations that constrain observer choice .Comment: 1) clarification regarding the connection with philosophical synchronicity in Section 2 and in the concluding section 2) reference to Maldacena-Susskind "ER=EPR" relation in discussion of role of wormholes in entanglement and nonlocality 3) length reduction and stylistic changes throughou

Synchronization of extended systems from internal coherence

A condition for the synchronizability of a pair of PDE systems, coupled through a finite set of variables, is commonly the existence of internal synchronization or internal coherence in each system separately. The condition was previously illustrated in a forced-dissipative system, and is here extended to Hamiltonian systems, using an example from particle physics. Full synchronization is precluded by Liouville's theorem. A form of synchronization weaker than "measure synchronization" is manifest as the positional coincidence of coherent oscillations ("breathers" or "oscillons") in a pair of coupled scalar field models in an expanding universe with a nonlinear potential, and does not occur with a variant of the model that does not exhibit oscillons.Comment: version accepted for publication in PRE (paragraph beginning at the bottom of pg. 5 has been rewritten to suggest unifying principle for synchronizability, applying to both forced-dissipative and Hamiltonian systems; other minor changes

Synchronicity from synchronized chaos

The synchronization of loosely-coupled chaotic oscillators, a phenomenon investigated intensively for the last two decades, may realize the philosophical concept of “synchronicity”—the commonplace notion that related eventsmysteriously occur at the same time. When extended to continuous media and/or large discrete arrays, and when general (non-identical) correspondences are considered between states, intermittent synchronous relationships indeed become ubiquitous. Meaningful synchronicity follows naturally if meaningful events are identified with coherent structures, defined by internal synchronization between remote degrees of freedom; a condition that has been posited as necessary for synchronizability with an external system. The important case of synchronization between mind and matter is realized if mind is analogized to a computer model, synchronizing with a sporadically observed system, as in meteorological data assimilation. Evidence for the ubiquity of synchronization is reviewed along with recent proposals that: (1) synchronization of different models of the same objective process may be an expeditious route to improved computational modeling and may also describe the functioning of conscious brains; and (2) the nonlocality in quantum phenomena implied by Bell’s theorem may be explained in a variety of deterministic (hidden variable) interpretations if the quantum world resides on a generalized synchronization “manifold”.publishedVersio

The 3-dimensional oscillon equation

On a bounded three-dimensional smooth domain, we consider the generalized oscillon equation with Dirichlet boundary conditions, with time-dependent damping and time-dependent squared speed of propagation. Under structural assumptions on the damping and the speed of propagation, which include the relevant physical case of reheating phase of inflation, we establish the existence of a pullback global attractor of optimal regularity, and finite-dimensionality of the kernel sections

Time-Dependent Attractor for the Oscillon Equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Klein-Gordon equation in an expanding background, in one space dimension with periodic boundary conditions, with a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular, time-dependent global attractor. The sections of the attractor at given times have finite fractal dimension.Comment: to appear in Discrete and Continuous Dynamical System

A lattice path integral for supersymmetric quantum mechanics

We report on a study of the supersymmetric anharmonic oscillator computed using a euclidean lattice path integral. Our numerical work utilizes a Fourier accelerated hybrid Monte Carlo scheme to sample the path integral. Using this we are able to measure massgaps and check Ward identities to a precision of better than one percent. We work with a non-standard lattice action which we show has an {\it exact} supersymmetry for arbitrary lattice spacing in the limit of zero interaction coupling. For the interacting model we show that supersymmetry is restored in the continuum limit without fine tuning. This is contrasted with the situation in which a `standard' lattice action is employed. In this case supersymmetry is not restored even in the limit of zero lattice spacing. Finally, we show how a minor modification of our action leads to an {\it exact}, local lattice supersymmetry even in the presence of interaction.Comment: 18 pages, 7 figures, 1 reference added, 1 correcte

A "Cellular Neuronal" Approach to Optimization Problems

The Hopfield-Tank (1985) recurrent neural network architecture for the Traveling Salesman Problem is generalized to a fully interconnected "cellular" neural network of regular oscillators. Tours are defined by synchronization patterns, allowing the simultaneous representation of all cyclic permutations of a given tour. The network converges to local optima some of which correspond to shortest-distance tours, as can be shown analytically in a stationary phase approximation. Simulated annealing is required for global optimization, but the stochastic element might be replaced by chaotic intermittency in a further generalization of the architecture to a network of chaotic oscillators.Comment: -2nd revised version submitted to Chaos (original version submitted 6/07