# Synchronicity from Synchronized Chaos

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. Highly Intermittent Synchronization in Loosely-Coupled Chaotic Systems

_{1}= X, Y

_{1}, Z

_{1}), respectively, synchronize rapidly: as t → ∞, Y

_{1}(t) − Y (t) → 0, Z

_{1}(t) − Z(t) → 0, as shown in Figure 2 (synchronization also occurs if the slave system is driven by the master Y variable instead of the X variable, but not if driven by the Z variable). Various schemes to use chaos synchronization for cryptography were motivated by the thought that variables analogous to X in (1) could be used as carrier signals that would be difficult to distinguish from noise [16]: the signal between the two systems defined by X is broadband and has no characteristic frequency. The Takens–Mañé Theorem [17,18] can still be used to infer information about the encoding from a segment of a signal that is sufficiently long, but as one considers higher-dimensional analogues of (1), it becomes increasingly difficult to do so.

_{1}. In this limit, the system reduces to a bidirectionally coupled version of (1), which indeed synchronizes. In the general case of the coupled System (2) with finite Γ, the subsystems exchange information more slowly: if X and X

_{1}are slowly varying, then S asymptotes to X − X

_{1}over a time scale 1/Γ. Thus, Γ is an inverse time lag in the coupling dynamics.

_{1}vs. time, for decreasing values of Γ. For large Γ, the case represented in Figure 4a, the subsystems synchronize. As Γ is decreased in Figure 4b–d, corresponding to increased time lag, increasingly frequent bursts of desynchronization are observed, until in Figure 4d (uncoupled systems), no portion of the trajectory is synchronized. The bursting behavior can be understood as an instance of on-off intermittency [19,24], the phenomenon that can occur when an invariant manifold containing an attractor loses stability, so that the attractor is no longer an attractor for the entire phase space, but is still effective in portions of the phase space. Trajectories then spend finite periods very close to the invariant manifold, interspersed with bursts away from it.

## 3. Machine/Human Perception as a Synchronization of Reality and Model

_{i}, as defined in [5]. The forcing induces a relaxation to a jet-like background flow ψ

^{∗}(Figure 5a,b) with q

^{*}≡ q(ψ

^{*}), injecting energy into the system.

^{A}/Dt = F

^{A}+D

^{A}and Dq

^{B}/Dt = F

^{B}+D

^{B}, were coupled diffusively through a modified forcing term ${F}_{\mathrm{k}}^{B}={\mu}_{\mathrm{k}}^{c}[{q}_{\mathrm{k}}^{A}-{q}_{\mathrm{k}}^{B}]+{\mu}_{\mathrm{k}}^{ext}[{q}_{\mathrm{k}}^{*}-{q}_{\mathrm{k}}^{B}]$, where the flow has been decomposed spectrally and the subscript k on each quantity indicates the wave number k spectral component. The two sets of coefficients ${\mu}_{\mathrm{k}}^{c}$ and ${\mu}_{\mathrm{k}}^{ext}$ were chosen to couple the two channels only in some medium range of wavenumbers.

## 4. Internal Sync vs. Mind-Matter Sync and the Role of Meaning

## 5. Sync as an Organizational Principle in Mind and in Computational Modeling

_{i}, y

_{i}, z

_{i}) i = 1, 2, 3 are the three models. An extra term μ is present in the models, but not in the real system. Because of the relatively small number of variables available in this toy system, all possible directional couplings among corresponding variables in the three Lorenz systems were considered, giving 18 connection coefficients ${C}_{ij}^{A}$ A = x, y, z i, j = 1, 2, 3 i ≠ j. The constants K

_{A}A = x, y, z are chosen arbitrarily so as to effect “data assimilation” from the “real” Lorenz system into the three coupled “model” systems. The configuration is schematized in Figure 9.

_{ij}linking the three model systems can be chosen using yet a further extension of the synchronization paradigm: if two systems synchronize when their parameters match, then under some weak assumptions, as was proven in [62], it is possible to prescribe a dynamical evolution law for general parameters in one of the systems, so that the parameters of the two systems, as well as the states will converge. In the present case, the tunable parameters are taken to be the connection coefficients (not the parameters of the separate Lorenz systems), and they are tuned under the peculiar assumption that reality itself is a similar suite of connected Lorenz systems. The general result [62] gives the following adaptation rule for the couplings:

_{max}) for some small number δ. Without recourse to the formal result on parameter adaptation, the rule (6) has a simple interpretation: time integrals of the first terms on the right-hand side of each equation give correlations between truth-model synchronization error, $x-\frac{1}{3}{\displaystyle {\sum}_{k}xk}$, and inter-model “nudging”, x

_{j}−x

_{i}. We indeed want to increase or decrease the inter-model nudging, for a given pair of corresponding variables, depending on the sign and magnitude of this correlation (The learning algorithm we have described resembles a supervised version of Hebbian learning. In that scheme, “cells that fire together wire together.” Here, corresponding model components “wire together” in a preferred direction, until they “fire” in concert with reality.). The procedure will produce a set of values for the connection coefficients that is at least locally optimal in the multidimensional space of connection values.

_{1}in Model 1) differed even more from “reality” than the average output of the three models (Figure 10c). Therefore, it is unlikely that any ex post facto weighting scheme applied to the three outputs would give results equaling those of the synchronized suite. Internal synchronization within the multi-model “mind” is essential. Third, the choice of semi-autonomous models to be combined is not essential, as long as the “gene pool” of models is diverse. In a case where no model had the “correct” equation for any variable, results deteriorated only slightly (Figure 10d).

## 6. Sync in Quantum Theory

_{max}→ 0 as N → ∞ (the system’s “metric entropy”, the sum of the positive Lyapunov exponents, $\sum}_{{h}_{i}>0}{h}_{i$, is constant as N → ∞). In other words, the higher the dimension, the less chaotic the system. Such behavior is suspect in a system intended to represent unpredictable quantum fluctuations. Taking the GRS behavior as N → ∞ to be generic, one must reconcile its increasingly mild character with the requirement that the nonlocal “signal” be perfectly masked through chaos. It was noted [80] that the issue is resolved if the GRS is viewed as a spatially asymptotic description of an intrinsically faster dynamics in a highly curved space-time. For reference, recall that an object falling into a black hole is perceived by an observer at a distance from the hole as approaching the horizon with decreasing velocity, but never reaching it. If the physical system that the GRS describes lives in the vicinity of a micro-black hole or wormhole, the variables in the asymptotic description will be slowed, but the actual physical processes will be realistically violent and can couple to each other through “signals” that are perfectly masked. More generally, the synchronizing subsystems can be expected to behave more wildly than the usual physical systems defined by PDEs on a continuum. Some form of granularity in state-space and/or physical space-time is indicated, in agreement with the models of ’t Hooft [78] and of Palmer [72,81].

#### 6.1. Physical vs. Virtual Non-Locality

^{N}and y ∊ R

^{N}. The dynamics are modified, so as to couple the systems:

^{N}→ R

^{N}, such that ||Φ(x) − y|| → 0 as t → ∞. Then, the coupled dynamics are also defined by the two autonomous systems:

## 7. Summary and Concluding Remarks

## Appendix

## A. On the Possibility of Micro-Wormholes

#### A.1. Implications of the Weak-Energy Condition in Ordinary and Higher-Derivative Gravity

^{a}= dx

^{a}/dζ, an averaged energy along the geodesic must be positive:

_{αβ}is the stress-energy tensor. Traversable wormholes can exist only if (11) is violated for some null geodesics passing through the wormhole, implying the existence of “exotic matter” with negative energy density in the “rest frame” of a light beam described by the null geodesic. The negative energy density is required, in one sense, to hold the wormhole open.

_{n}are dimensionless constants and we have included a cosmological constant Λ for full generality. If L = L

_{P}, the Planck length, then the new terms in the extended Theory (12) are negligible on macroscopic scales. They only need be considered if curvature is significant at the Planck length scale. Any metric that solves the ordinary Einstein equations after the substitution ${T}_{\mu \nu}\to {T}_{\mu \nu}-(1/8\pi ){\displaystyle {\sum}_{n>2}{c}_{n}{L}^{n-2}{R}_{\mu \nu}^{(n)}}$ solves (12) for given T

_{μν}. It is plausible that the modified stress-energy tensor ${T}_{\mu \nu}-(1/8\pi ){\displaystyle {\sum}_{n>2}{c}_{n}{L}^{n-2}{R}_{\mu \nu}^{(n)}}$ can be made to violate the weak energy condition if the signs of the constants c

_{n}are chosen appropriately and, thus, that a traversable micro-wormhole solution is possible.

#### A.2. Vacuum Recirculation Effects for Narrow Wormholes

_{μν}, as the loops of a spiral world line pile up to form a closed timelike curve. The derivation of this controversial result is as follows: for each passage of a virtual particle through the wormhole, the contribution to the two-point function $<\mathrm{\Psi}|\widehat{\varphi}(x)\widehat{\varphi}({x}^{\prime})+\widehat{\varphi}({x}^{\prime})\widehat{\varphi}(x)|\mathrm{\Psi}>$ from a trajectory that contains that passage is attenuated by a factor b/D, where b is the wormhole width and D is the spatial length of a geodesic through the wormhole, as measured in the frame of an “observer” traveling along the geodesic from the vicinity of x and x′ through the wormhole once and back to the same vicinity. Here, x and x′ are nearby points in space-time, |Ψ > is the quantum state and $\widehat{\varphi}$ is the field operator associated with the field ϕ. The contribution to the two-point function is found to behave as (b/D)

^{k}× 1/σ, where σ is 1/2 the square of the proper distance between x and x′ along the geodesic connecting them through the wormhole, and the power k depends on the number of times that the trajectory traverses the wormhole (contributions from trajectories with k = 0 or 1, that traverse the wormhole only once, dominate). One finds σ ~ DΔt, where Δt is the proper time between x and the nearest null geodesic that passes through the wormhole. As x′ → x, the contribution diverges if x can be joined to itself by a null geodesic that passes through the wormhole. The stress-energy tensor can be expressed in terms of the two-point function [99] and also diverges as σ → 0 or Δt → 0. Specifically, one finds T

_{μν}~ (b/D)

^{k}× 1/D(Δt)

^{3}in natural units, or in dimensional units,

_{P}, the Planck length, so we only need consider the magnitude of T

_{μν}for Δt ≥ L

_{P}. At these scales, referring to (13), T

_{μν}≤ L

_{P}/D in natural units of ${m}_{P}/{L}_{P}^{3}$, giving energy densities that are far too weak to destroy the wormhole, or have other noticeable effects, for macroscopic D.

_{μν}on scales larger than Hawking’s reduced length scale would still cause collapse of the wormhole, the instant that recirculation becomes possible.

_{i}n

_{i}δ(ω − ω

_{i}), where ω

_{i}is a discrete set of frequencies and {n

_{i}} is a set of positive integers. There is a problem from the weak energy condition if any ω

_{i}> ω

_{cutoff}(with n

_{i}≥ 1), for ω

_{cutoff}sufficiently large as to cancel the negative-energy contributions to T

^{00}. In a path integral, taken both over particle trajectories and over geometries, one need only consider histories in which more energetic particles either collapse the wormhole or are reflected and do not traverse it. In contrast, for wormholes of macroscopic width, histories must be included in the path integral for which the energies of recirculating virtual quanta outside the wormhole are anomalously large (treating the geometry itself classically). The cutoff in the former case implies that the term 1/σ in the two-point function is replaced by a term like ${\int}_{{\omega}_{k}<{\omega}_{cutoff}}{d}^{4}k}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[ik\cdot (x-{x}^{\prime})]/{k}^{2$, which does not diverge.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagram constructed by Carl Jung, later modified by Wolfgang Pauli, to suggest relationships based on synchronicity as an “acausal connecting principle”, existing alongside causal relationships [8].

**Figure 2.**The trajectories of the synchronously-coupled Lorenz systems in the Pecora–Carroll complete replacement scheme (1) rapidly converge

**(a)**. Differences between corresponding variables approach zero

**(b)**.

**Figure 3.**Transition from identical to generalized synchronization, illustrated by the relationship between a pair of corresponding variables x and x′: projections of the synchronization manifold onto the (x, x′) plane are shown for

**(a)**identical synchronization,

**(b)**generalized synchronization with near-identical correspondence and

**(c)**generalized synchronization with a correspondence function that is nowhere smooth.

**Figure 4.**The difference between the simultaneous states of two Lorenz systems with time-lagged coupling (2), with σ = 10., ρ = 28., and β = 8/3, represented by Z(t) − Z

_{1}(t) vs. t for various values of the inverse time lag Γ illustrating complete synchronization

**(a)**, intermittent or “on-off” synchronization

**(b)**, partial synchronization

**(c)**and de-coupled systems

**(d)**. The average Euclidean distance ⟨D⟩ between the states of the two systems in X, Y, Z-space is also shown. A histogram of the lengths of periods of “synchronicity”, such as the one indicated by the arrow in (c), is shown in

**(e)**for the time-delayed coupling case (solid line) and a case of two unrelated Lorenz trajectories (dashed line), where synchronicity intervals are periods during which |Z(t) − Z

_{1}(t)| < 5.

**Figure 5.**Stream function (in units of 1.48 × 10

^{9}m

^{2}s

^{−1}) describing the forcing ψ

^{*}

**(a,b)**, and the evolving flow ψ

**(c–f)**, in a parallel channel model with coupling of medium-scale modes for which |k

_{x}| > k

_{x}

_{0}= 3 or |k

_{y}| > k

_{y}

_{0}= 2, and |k| ≤ 15, for the indicated numbers n of time steps in a numerical integration (generalizing to bidirectional coupling, for convenience). Parameters are as described previously [5]. An average stream function for the two vertical layers i = 1, 2 is shown. Synchronization occurs by the last time shown (e,f), despite differing initial conditions (c,d).

**Figure 6.**Stream function (in units of 1.48 × 10

^{9}m

^{2}s

^{−1}) describing a typical blocked flow state

**(a)**and a typical zonal flow state

**(b)**in the two-layer quasi-geostrophic channel model. An average stream function for the two vertical layers i = 1, 2 is shown.

**Figure 7.**Energy density ρ = (1/2)e

^{−}

^{Ht}(ϕ

_{x})

^{2}+ (1/2)e

^{Ht}(ϕ

_{t})

^{2}+ e

^{Ht}V (ϕ) vs. position x for a numerical simulation of the Scalar Field Equation (4) with the potential V(ϕ) = (1/2)ϕ

^{2}− (1/4)ϕ

^{4}+ (1/6)ϕ

^{6}, exhibiting localized oscillons

**(a)**, and a simulation of the same equation, but with a different potential V(ϕ) = (1/2)ϕ

^{2}+ (1/4)ϕ

^{4}+ (1/6)ϕ

^{6}, for which oscillons do not occur

**(b)**.

**Figure 8.**The local energy density ρ vs. x for two simulations (solid and dashed lines) of the Scalar Field Equation (4), coupled to one another only through modes of wavenumber k ≤ 64, where modes up to k

_{max}= 2

^{14}are realized numerically (ρ for the second system (dashed line) is also shown reflected across the x-axis for the ease of comparison). The coincidence of oscillon positions is apparent.

**Figure 9.**“Model” Lorenz systems are linked to each other, generally in both directions and to “reality” in one direction. Separate links between models, with distinct values of the connection coefficients ${C}_{l}^{ij}$, are introduced for different variables and for each direction of possible influence.

**Figure 10.**Difference z

_{m}−z between “model” and “real” z vs. time for a Lorenz system with ρ = 28, β = 8/3, σ = 10.0 and an interconnected suite of models with ρ

_{1,2,3}= ρ, β

_{1}= β, σ

_{1}= 15.0, μ

_{1}= 30.0, β

_{2}= 1.0, σ

_{2}= σ, μ

_{2}= −30.0, β

_{3}= 4.0,σ

_{3}= 5.0, μ

_{3}= 0. The synchronization error is shown for

**(a)**the average of the coupled suite z

_{m}= (z

_{1}+z

_{2}+z

_{3})/3 with couplings ${C}_{ij}^{A}$ adapted according to (6) for 0 < t < 500 and held constant for 500 < t < 1,000;

**(b)**the same average z

_{m}, but with all ${C}_{ij}^{A}=0$;

**(c)**z

_{m}= z

_{1}, the output of the model with the best z equation, with ${C}_{ij}^{A}=0$;

**(d)**as in (a), but with β

_{1}= 7/3, σ

_{2}= 13.0, and μ

_{3}= 8.0, so that no equation in any model is “correct” (analogous comparisons for x and y give similar conclusions).

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Duane, G.S.
Synchronicity from Synchronized Chaos. *Entropy* **2015**, *17*, 1701-1733.
https://doi.org/10.3390/e17041701

**AMA Style**

Duane GS.
Synchronicity from Synchronized Chaos. *Entropy*. 2015; 17(4):1701-1733.
https://doi.org/10.3390/e17041701

**Chicago/Turabian Style**

Duane, Gregory S.
2015. "Synchronicity from Synchronized Chaos" *Entropy* 17, no. 4: 1701-1733.
https://doi.org/10.3390/e17041701