---
abstract: |
    Elementary particles possess quantized values of charge and internal angular momentum or spin.  These characteristics do not change when the particles interact with other particles or fields as long as they preserve their entities.  Quantum theory does not explain this quantization.  It is introduced into the theory a priori.  An interacting particle is an open system and thus does not obey conservation laws.  However, an open system may create dynamically stable states with unchanged dynamical variables via self-organization.  In self-organized systems stability is achieved through the interplay of nonlinearity and dissipation.  Can self-organization be responsible for particle formation?  In this paper we develop and analyze a particle model based on qualitative dynamics and the Feigenbaum universality.  This model demonstrates that elementary particles can be described as self-organized dynamical systems belonging to a wide class of systems characterized by a hierarchy of period-doubling bifurcations.  This semi-qualitative heuristic model gives possible explanations for charge and action quantization, and the origination and interrelation between the strong, weak, and electromagnetic forces, as well as ${SU(2)}$ symmetry.  It also provides a basis for particle taxonomy endorsed by the Standard Model.  The key result is the discovery that the Planck constant is intimately related to elementary charge.\
author:
- 'Vladimir A. Manasson'
bibliography:
- 'Manasson\_AreParticlesSelfOrganizedSystems\_arXiv.bib'
title: 'Are Particles Self-Organized Systems?' 
---

=1

Introduction
============

Our world appears to be quantized.  The most basic creatures of matter/fields, the elementary particles, possess fixed values of strong, weak, and electromagnetic charges and angular momenta.  These fixed dynamical variables do not change when a particle interacts with the surrounding vacuum and with other particles/fields.  In classical mechanics, two different classes of systems can exhibit stability: the conservative systems and the dissipative systems [@1; @2a]. 

In *conservative systems*, dynamical variables are conserved due to the existence of symmetries (the Noether theorem).  The finite motion of a classical conserved system can be described by a closed loop trajectory that is parameterized by the corresponding conserved variable.  Different trajectories densely fill the state space, and an infinitely small perturbation can shift the system from one trajectory to another.  The new trajectory corresponds to a different value of the conserved variable and is as stable as the previous one.  This system is not asymptotically (absolutely) stable in the sense that a small perturbation does not asymptotically fade out after the interaction.  Thus, conservation is conditional and requires the conservative system to be closed (i.e. it cannot interact with the rest of the world).  However, elementary particles interact with external fields; they are open and the stability of their dynamical variables cannot be explained by the conservation laws of classical mechanics. 

In contrast, asymptotic stability is quite common in *nonlinear dissipative* systems and manifests itself as a phenomenon of self-organization [@2a].  The theory of dynamical systems, which includes nonlinear and dissipative phenomena, has succeeded in understanding the origin of numerous patterns like vortices, domain walls, pinches, various sorts of waves, Bénard cells, linear and point defects, etc.  that occur in dynamical media such as fluids, the atmosphere, chemical reactions, gaseous and solid-state plasmas, laser cavities, electric circuits, cellular automata, etc. 

*Self-organized systems* (*SOS*s) are open.  Their stability comes from the interplay between nonlinearity and dissipation.  The stability is dynamical and is achieved in states that are far from thermodynamic equilibrium.  In many cases, SOS phase portraits represent trajectories spiraling toward closed loops called *attractors*.  Attractor closure implies stability in the corresponding dynamical variables.  Near the attractor an SOS behaves like a conservative system.  However, the SOS stability is not conditional.  Due to dissipation it is asymptoticsmall perturbations fade out with time and the system returns to its original attractor [@1; @2a]. 

Self-organization is a ubiquitous phenomenon and has been observed at different scales of matter.  The observable part of the Universe looks like a hierarchy of self-organized structures, starting from galaxy super-clusters to stars and planets to atmospheres and ecosystems to living organisms and their organs to cells and microorganisms, and all the way down to molecules and atoms.  All of these systems are far from thermodynamic equilibriumthey are dynamical systems.  Hubbles law demonstrates the dynamical state of the visible universe at the largest scales.  Stars are born and die.  Novas explode.  Atmospheres seethe.  Plants grow.  Animals breathe.  Cells self-reproduce.  Even vacuum exists in a state of thermodynamic non-equilibriumit is filled with CMB radiation, neutrinos, and other excited fields.  We have all reason to assume that this vibrant multi-level dynamical pyramid interlaced with self-organization can be extended to the subatomic level.  But, can the phenomenon of self-organization explain the stability of elementary particles and the quantum nature of their internal angular momenta, charges, and masses? 

Traditionally, particle physics belongs to the framework of *quantum theory* (*QT*).  QT acknowledges particle *openness* and the fact that particles interact with the surrounding medium.  QT utilizes Lagrange-Hamiltonian mechanics of conservative systems but in a different way than its classical counterpart.  The major difference here is the introduction of special constraints, the quantization rules.  Quantization provides absolute stability to the particles, a stability that is absent in classical mechanics of conservative systems.  In QT, dynamical characteristics such as charge and action are postulated to have only discrete or fixed values, and these constraints dictate which dynamical trajectories are permitted and which are forbidden.  The permitted trajectories constitute a discrete set in the corresponding state space.  Now a small perturbation cannot shift a particle from its permitted trajectory to a nearby trajectory for arbitrarily long periods of time because the latter trajectory is forbidden.  Thus, the perturbation is expected to die within the time interval limited by the Heisenberg uncertainty principle.  Such processes, known as *virtual processes*, strongly resemble *dissipation*, and the most probable QT trajectories, the eigenstates, strongly resemble SOS attractors.  The possible importance of dissipation in the foundations of QT has been discussed by t Hooft [@15]. 

Despite being based on the superposition principle, QT’s framework includes elements of *nonlinearity*.  The commonly used perturbation theory is concerned with nonlinear corrections.  The renormalization technique assumes that charges (coupling constants) and masses depend on the perturbation level.  Non-Abelian Yang-Mills theories of weak and strong interactions use charged bosons that interact with one another and make the nonlinearity even more profound.  Abrupt transitions from one state to another as well the entire measurement process (the so-called collapse of the wave function) are super-nonlinearthey are discontinuous.  The latter also breaks time symmetry, and is thus a dissipative process. 

We can see that QT objects possess some important attributes of self-organized systems: *nonlinearity* and *dissipation*.  The QED triumph (one example being the striking accuracy in calculating the electron magnetic moment) would be impossible if an electron were treated strictly as a closed linear conservative system.  Despite the similarity between QT objects and SOSs, absolute stability emerges in different ways in the two frameworks.  In SOSs it is an outcome of the theory.  In QT it is introduced *a priori* via quantization rules and fundamental quantum constants like $\hbar$, $e$, $\alpha$, $\sin\left(\theta_{W}\right)$, $\sin\left(\theta_{C}\right)$, etc. 

In this paper we build a model describing elementary particles as self-organized systems.  Our approach is heuristic and phenomenological.  We are not concerned with a specific form of differential equations.  Dissipative systems are immune to small perturbations and their global behavior is not very sensitive to the details of the governing equations.  To understand the overall SOS behavior, it is often sufficient to use qualitative analysis and study a prototype-system that belongs to the proper dynamical class. 

In the Standard Model (SM) we find a number of different doublets such as particleanti-particle, spin-upspin-down, protonneutron, leptonquark, u-quarkd-quark, electronneutrino, etc.  We can even organize all the fundamental fermions as a doublet pyramid resembling a phylogenetic tree, where each fork represents a new doublet (Fig.\[fig:Fig1\]).  In many cases these doublets are seen as two states of a single particle.  The transformation from one state to the other can be viewed as a rotation in some complex internal space.  A remarkable property of these rotations is their $SU(2)$ symmetry, which requires rotation through $4\pi$ rather than the usual $2\pi$ to return to the original state.  This period doubling and the bifurcating structure of the phylogenetic tree astonishingly resemble the period-doubling bifurcation diagrams that are often found in systems with non-linear dynamics, and are typical for a large class of dissipative systems.  The similarity is clearly evident from the comparison of Fig.\[fig:Fig1\] with Fig.\[fig:Fig2\], which shows the period-3 stability window from the logistic map bifurcation diagram.  The three branches in the bifurcation diagram resemble three families in the particle diagram. 

![ \[fig:Fig1\] Three fermion families can be organized as bifurcation diagrams.](Fig1.pdf){width="3.25in"}

![ \[fig:Fig2\] Period-3 window from a bifurcation diagram of the logistic map.](Fig2.pdf){width="3.25in"}

Our observation suggests that particles may belong to a class of systems in which the dynamics are characterized by period-doubling bifurcation diagrams.  The class is large, and includes a number of dynamical systems that not only possess similar bifurcation diagrams, but even share the same scaling properties.  This universality, originally discovered by Feigenbaum [@3a; @3b; @3c], allows us to add to our qualitative analysis some quantitative results that can be obtained when we represent an elementary particle with the universal Feigenbaum function or with another function from the above-mentioned class. 

Building a Model
================

Let us take the electron as an example.  To visualize the complexity of the internal electron dynamics, let us imagine a hypothetical situation that can arise as a result of a negatively charged fluctuation in vacuum.  The fluctuation polarizes the surrounding vacuum and creates a positively charged halo around itself.  The halo lowers the local electric potential, and the original fluctuation becomes denser and more confined.  This in turn affects the halo.  The positive feedback described above competes with the negative feedback caused by charge diffusion and self-repellence.  Thus, the symmetric halo becomes unstable and breaks up into separate positively charged fragments, which then create secondary negatively charged halos around themselves.  This process repeats itself ad infinitum at smaller and smaller scales.  Moving fragments create currents and magnetic fields, adding more complexity to this turbulent system.  Moving positive and negative fragments experience attractive and repellant forces from their neighbors.  Both are proportional to the fragment charges and currents.  Due to the finite distance between fragments and the limit of the speed of light, feedback is delayed.  Depending on the strength of the perturbation, the original fluctuation either relaxes to a state of thermodynamic equilibrium or the system bifurcates to another state where the delayed interplay between attraction and repulsion gives birth to a *dynamically* stable spatiotemporal pattern.  We suggest that an electron is one such self-organized system. 

Due to the fractal structure and openness of this system, there can be infinitely many interactions among the particle parts.  To describe these dynamics we need an infinite-dimensional state space.  The complexity of this situation resembles a gas ensemble, or rather, a turbulent fluid.  A practical approach to this type of complex system is to find a few collective variables such as temperature, concentration, pressure, or convection velocity, that effectively describe the dynamical state of the system in a low-dimensional state space [@2a].  In our case, it is most natural to assign this role to charge.  Thus, we consider a low-dimensional state space as a nonlinear vector field $\left\{\vec{\psi}\right\}$ where $q$ is the effective collective dynamical variable, and attempt to understand why $q$ is quantized.  In QT, electrical charge is the coupling constant between the particle and the external electromagnetic field.  Since our model is primarily concerned with internal dynamics, we also assume that charge is the coupling constantalbeit a running coupling constantin the interaction between the particle and its *internal field*, the field that is created by the particle itself.  In other words, *charge defines the feedback* that is responsible for the particle’s self-organization. 

We assume that as an SOS, an electron possesses asymptotic stability.  It exercises *finite* motion in the corresponding state space, which can be represented by dynamical trajectories in that state space.  Dissipation ensures that such trajectories spiral toward their limit cycles, the attractors.  We assume that the attractors describing dynamical equilibrium can be parameterized by charge $q$. To make this parameterization sensible, we also assume that the system possesses some inertia, which means that external perturbations, though capable of changing the $q$-value ($q$ is a running parameter), cannot do it abruptly.  Thus, $q$ is preserved for at least a few cycles, obeying the so-called *adiabatic* constraint [@4].  If the dynamics are not chaotic, the attractors are closed curves as in the case of conservative systems.  We can use this similarity to describe the dynamics in terms of generalized action-angle variables [@4], where angle $\varphi$ is a cyclic or ignorable coordinate, and the generalized momentum or reduced action (or simply action), $J =
S/2\pi$ ($S$ is the total action accumulated during the entire cycle), is a conserved variable.  The latter can also be used to parameterize the attractors.  Having selected $q$ as a dynamical parameter, we would like to connect the $J$-parameterization with the $q$-parameterization.  This canonical transformation can be accomplished via dimensional analysis if we define $J$ as: $$\begin{aligned}
J = \eta{}q^2
\label{eq:one},\end{aligned}$$ where $\eta$ is a conversion constant that has the physical dimensions of electromagnetic impedance.  To be more specific, we assign $\eta$ to be the vacuum impedance, $\eta = \sqrt{\mu_0/\varepsilon_0}$, where $\varepsilon_0$ is vacuum permittivity and $\mu_0$ is vacuum permeability. 

To further simplify our analysis, we replace each continuous trajectory in the state space with a set of points $\vec{\psi}_i$ (one point per loop) that are selected using a procedure called the Poincaré section [@5] (Fig.\[fig:Fig3\]). 

![ \[fig:Fig3\] Selection of map points $\vec{\psi_i}$ using the Poincaré section.](Fig3.pdf){width="2.5in"}

After applying the Poincaré section, we obtain a one-dimensional recurrent map called a Poincaré map: $$\begin{aligned}
\vec{\psi}_k = \vec{F}\left(\vec{\psi}_{k-1}\right)
\label{eq:two},\end{aligned}$$ where $\vec{F}\left(\vec{\psi}\right)$ is a recursive function.  According to our assumptions, this function can be parameterized by charge $q$ and denoted as $\vec{F}_q\left(\vec{\psi}\right)$, or by action $J$ and denoted as $\vec{F}_J\left(\vec{\psi}\right)$.

Based on similarities between the particle phylogenetic tree diagram (Fig.\[fig:Fig1\]) and the period-doubling bifurcation diagram (Fig.\[fig:Fig2\]), we assume that the dynamical electron is similar to systems in which the dynamics possess a period-doubling bifurcation structure.  The class of period-doubling bifurcation systems is wide [@6].  It includes all maps with smooth unimodal (having a single extremum) recursive functions (see examples in Fig.\[fig:Fig4\]a-c), and many known SOSs such as mechanical and electronic oscillators, Bénard cells, Belousov-Zhabotinsky chemical reactions, Couette-Taylor flow, etc. 

![ image](Fig4.pdf){width="6.5in"}

A bifurcation diagram maps limit points $x_\infty$ (vertical axis) against control parameter values (horizontal axis).  It represents a hierarchy of period-doubling bifurcations.  Each bifurcation is a phase transition that is accompanied by a doubling in the number of limit points (attractor loops).  The distance between bifurcations progressively shrinks with parameter, and the scaling factor quickly converges to a number called the Feigenbaum constant.  Our three example maps (Fig.\[fig:Fig4\]a-c) have different recursive functions, yet their bifurcation diagrams still look very similar.  They are called quadratic maps because the extrema of their recursive functions can be approximated by quadratic parabolas.  According to the Feigenbaum universality [@3a; @3b; @3c], the scaling factor for these maps, as for all unimodal quadratic maps, is independent of all other details in their recursive functions, and always converges to the Feigenbaum number $\delta = 4.669\ldots$. We assume that the electron map represented by equation (\[eq:two\]) also belongs to the class of quadratic maps.  The corresponding hypothetical bifurcation diagram of the particle recursive function $\vec{F}_J\left(\vec{\psi}\right)$ is shown in Fig.\[fig:Fig4\]d.  We select $J$ (not $q$) as the parameter for the following reason.  In all quadratic maps, parameter $A$ controls the strength of the feedback or self-interaction.  In electromagnetic theory, feedback is a two-step process: 1) the excitation of the electromagnetic field by the charge distribution, and 2) the formation of the charge distribution by this field.  The intensity of each step is proportional to $q$, and the intensity of the entire feedback is proportional to $q^2$, or assuming equation (\[eq:one\]), to $J$. Thus, $J$ plays the same feedback role in electron self-interaction as parameter $A$ does in quadratic maps.  For convenience, we place the origin of the $J$-axis, $J = 0$, at the Feigenbaum point [@5], and the origin of the $\psi$-axis, $\psi =
0$, at the point where the electron recursive function $\vec{F}\left(\vec{\psi}\right)$ reaches its hypothetical extremum. 

Bifurcation points divide maps into segments of stability [@5].  Each map has a continuous interval where it converges to a single fixed point that corresponds to a single loop attractor (we do not account for the degenerated case of thermodynamic equilibrium where the attractor is just a point).  In the next parameter interval, the original map looses stability but its second iteration $\vec{\psi}_k =
\vec{F}_A\left(\vec{F}_A\left(\vec{\psi}_{k-1}\right)\right)$ still converges, implying the existence of a period-2 attractor.  Now we have two limit points in the bifurcation diagram $\left(\vec{\psi}^1_\infty, \vec{\psi}^2_\infty\right)$, which means that the attractor crosses the Poincaré section two times, and represents a closed double-loop trajectory (Fig.\[fig:Fig5\]).  Period-doubling bifurcation manifests a phase transition that changes the structure of attractors [@7].  It can be considered as the emergence of a second periodic motion with a doubled period (schematically shown in Fig.\[fig:Fig5\] by small circles) whose dynamical mode is orthogonal to the original mode.  Describing the new dynamical state requires an additional degree of freedom.  The description of a new trajectory requires two complex numbers, and taking into account period-doubling, can be represented by a two-component *spinor*.  This can be considered as a doubling of the state space dimensionality by the bifurcation. 

As the control parameter reaches its second critical value, another period-doubling bifurcation occurs.  The stability shifts from a period-2 attractor to a period-4 attractor.  The state space experiences a similar metamorphosis and its dimensionality doubles again.  The projection of the attractor onto the original state space now represents four loops and needs a four-component spinor for its description.  At the next critical value a period-8 attractor replaces the period-4 attractor and so on.  This progression ends when the number of branches, the dimensionality of the dynamical space, and the number of spinor components reaches infinity, the so-called Feigenbaum point. 

![ \[fig:Fig5\] Double-loop attractor can be represented by a two-component spinor.](Fig5.pdf){width="2.25in"}

One quadratic map that has been extensively explored is now considered to be the paradigm for the entire class.  It is called the logistic map (Fig.\[fig:Fig4\]a).  We will use it in our numerical explorations. 

The logistic map is given by the equation: $$\begin{aligned}
x_k = Ax_{k-1}\left(1-x_{k-1}\right)
\label{eq:three},\end{aligned}$$ where $A$ is the control parameter. 

Origin of Quantization - An Inherent Feature of SOS Dynamics
============================================================

Now let us examine how the dissipation rate varies with the control parameter in the stability intervals, i.e. between bifurcations. 

The dissipation rate $D$ can be defined as the inverse number of steps $k$ (iterations), $D = 1/k$, required for the system to go from a randomly selected initial point $x_0$ to the vicinity of the corresponding fixed point, i.e. within the interval $\left(x_{k=\infty} - \varepsilon, x_{k=\infty} + \varepsilon\right)$ for any sufficiently small $\varepsilon$. Fig.\[fig:Fig6\] demonstrates the dissipation rate of a map given by equation (\[eq:three\]) for the first three stability intervals when $x_0 =
0.367$, $k_\infty = k_{1000}$, and $\varepsilon = 10^{-14}$.

We notice that $D(A)$-curves have a sharp maximum close to the center of the corresponding stability interval.  These maxima occur when attractors have limit points that coincide with points where the recursive function reaches its extremum: $x = 0.5$ for the logistic map and $\psi = 0$ for the electron map.  Here the stability parameter, called the Lyapunov exponent and determined as $\lambda =
\left(1/n\right)\ln\left(\left|dF\left(x_0\right)/dx\right|\left|dF\left(x_1\right)/dx\right|\ldots\left|dF\left(x_n\right)/dx\right|\right)$ [@5], has a singularity, $\lambda\rightarrow-\infty$. The corresponding attractors are called supercycles or superattractors.  Because superattractors play an important role in our model, we denote their parameter values with subscripts representing the stability interval number.  For example, $J_i$ and $q_i$ are the action and charge corresponding to the superattractor of the $i$-th stability interval. 

![ \[fig:Fig6\] Dissipation rate as a function of the control parameter for three intervals of stability (bottom) and the corresponding segments of the bifurcation diagram (top).](Fig6.pdf){width="3.25in"}

Let us conditionally divide each converging trajectory into two parts, the first being the transient spiral, and the second being the vicinity of the attractor.  For each selected trajectory, the higher the convergence rate, the shorter the transient time, the more time the system spends near the attractor, and thus the higher the probability of finding the system in the vicinity of the attractor. 

Despite the adiabatic constraints, $J$ is still a free running parameter.  Vacuum fluctuations or other external noise can kick the electron from one state trajectory to another and from one $J$-value to another.  However, due to the profound differences in dissipation rates for different attractors (Fig.\[fig:Fig6\]), the average time spent near the superattractor is much longer than the time spent near other attractors, i.e. the superattractors are by far the most attractive attractors.  Hence, the probability of finding the system near a superattractor is much higher than the probability of finding it in other regions of the state space.  Thus, we come to the important conclusion that in our self-organized system there exists *a set of special, discrete, most probable, and super-stable closed trajectories*.  This can be interpreted as quantization, a phenomenon similar to the one postulated in QT.  The difference is that we did not introduce this quantization a priori.  It is an inherent feature of SOS dynamics.  The quantization of attractors imposes a quantization of their parameters.  In reference to our model, this means that both the action $J$ and the charge $q$ have preferred discrete values, i.e. they are also *quantized*. 

It is interesting that in order to find superattractors we simply need to look for trajectories where the fixed point $\vec{\psi}_\infty$ corresponds to the extremum of the recursive function $F\left(\vec{\psi}_\infty\right) = F_{extremum}$, which can be understood as a sort of *variational principle*.  This variational principle can be extended to the situation when a particle interacts with an external field and the interaction deforms the recursive function, shifting its extremum and the extrema of its iteratives to other fixed points $\vec{\psi}_\infty$ (eigenstates).  The deformation may also change the values of $J_n$ and $q_n$. In the case of stronger interactions, the system jumps to the next stability interval with a different state space that has a doubled dimensionality and a different topology. 

Numerical Surprises
===================

According to our model an electron is a self-organized system for which the internal dynamics can be represented by a discrete set of orbits in a state space or by levels $q_i$ and $J_i$ in the charge/action scale.  The specific values of charge and action depend on the level of excitation.  The Feigenbaum universality makes it possible to obtain some quantitative relations between these parameters at different excitation levels.  The parameter values $J_k$ obey the same scaling law as the entire bifurcation treethey have the same asymptotic behavior and converge to the same Feigenbaum delta [@3a; @3b; @3c]: $$\begin{aligned}
\lim_{k\rightarrow\infty}\left(J_{k-1}/J_k\right) = \delta
\label{eq:four}.\end{aligned}$$ From equations (\[eq:one\]) and (\[eq:four\]), the ratio between two charge values at two adjacent levels converges as: $$\begin{aligned}
\lim_{k\rightarrow\infty}\left(q_{k-1}/q_k\right) = \delta^{1/2}
\label{eq:five}.\end{aligned}$$ The convergence rate is usually high, and even for relatively small numbers $k$ we can substitute equations (\[eq:four\]) and (\[eq:five\]) with their approximations: $$\begin{aligned}
J_{k-1}/J_k = \delta
\label{eq:six}\end{aligned}$$ and $$\begin{aligned}
q_{k-1}/q_k = \delta^{1/2}
\label{eq:seven}.\end{aligned}$$

There is a special $q_i$ value that we refer to as $q_e$. It corresponds to the experimentally measured electron charge $e$: $$\begin{aligned}
q_e = e = 1.6021\ldots\times10^{-19}C
\label{eq:eight}.\end{aligned}$$ Let us explore the partice dynamics at different excitation levels around $q_e$. As part of the exploration we will compare the dynamical variables $J_i$ and $q_i$ at these levels. 

First we compare the parameters at the level where $q_e = e$ with the parameters two levels above where $q_{e-2} = \delta{}e$. Using equations (\[eq:one\]), (\[eq:six\]), (\[eq:seven\]), and (\[eq:eight\]) we find that: $$\begin{aligned}
J_{e-2} = \delta^2J_e = \eta\delta^2e^2 = 2.1083\ldots\times10^{-34}J\cdot{}s
\label{eq:nine},\end{aligned}$$ which equals twice the value of the Planck constant, $2\hbar =
2\left(1.05457\ldots\times10^{-34}J\cdot{}s\right)$, accurate to 0.04%.  This implies that there is a direct connection between the two fundamental quanta $e$ and $\hbar$: $$\begin{aligned}
\hbar = \sqrt{\mu_0/\varepsilon_0}\left(\delta{}e\right)^2/2
\label{eq:ten}.\end{aligned}$$ Essentially ignored by QT, this relation plays a crucial role in our proposed model.  The relation described by equation (\[eq:ten\]) carries important implications:

1.   It indicates that quantization of charge and quantization of action have the same origin. 

2.   The presence of the Feigenbaum delta implies the relevance of period-doubling bifurcation dynamics. 

3.   The value of delta, $\delta = 4.669\ldots$, suggests the involvement of dissipative dynamics (the period-doubling transition to chaos has also been found in Hamiltonian systems but in this case $\delta_H
    \approx 8.721\ldots$ [@5]). 

4.   And finally, despite the fact that $\delta$ is a fundamental mathematical constant, it has only been used in one physical context, that of dynamical systems which experience transitions from smooth dynamics to turbulent dynamics, i.e. $\delta$ belongs exclusively to *chaos theory*.  The latter is essentially a non-quantum theory.  For several fundamental constraints (like superposition principle, uncertainty principle, and quantization itself), QT is incapable of describing truly-chaotic systems. ( The confusing term quantum chaos relates not to real chaotic systems, but rather to non-chaotic quantum systems whose classical counterparts are chaotic [@5; @8].)  Therefore, it is unlikely that one will be able to understand equation (\[eq:ten\]) (and the roots of charge and action quantization) within the framework of quantum theory. 

Equation (\[eq:ten\]) leads us to several other striking results.  The fine structure constant, $\alpha = e^2/4\pi\varepsilon_0c\hbar$, can be expressed exclusively via the mathematical constants $\pi$ and $\delta$: $$\begin{aligned}
\alpha = \left(2\pi\delta^2\right)^{-1} \cong \frac{1}{137}
\label{eq:eleven}.\end{aligned}$$ By examining the next level, where charge is $q_{e-1} =
\delta^{1/2}e$, we find that the fine structure constant is $\alpha_{e-1} =
\left(2\pi\delta\right)^{-1} \approx 1/29$ [@9], and corresponds to the experimental value of the weak coupling constant. 

One more level above, where charge $q_{e-2} = \delta{}e$, we find that the fine structure constant is $\alpha_{e-2} = \left(2\pi\right)^{-1}\cong 0.16$, and corresponds to the experimental values of the low-energy ($\sim$15 GeV) strong coupling constant $\alpha_s$ obtained through JADE data [@10; @11]. 

Our findings suggest that electromagnetic, weak, and strong interactions obey a hierarchy that can be explained by period-doubling bifuration dynamics as shown in Fig.\[fig:Fig7\].  The $\delta$-relations between coupling constants (but not their explicit values) were previously observed by Goldfain [@12b]. 

Here are some additional examples suggesting that the Feigenbaum delta may have a profound relevance to particle physics. 

According to the Standard Model, the photon and Z-boson are described as mixtures of the electromagnetic B-boson and the weak W$^3$-boson.  The ratios of their relative inputs are characterized by the weak mixing angle $\theta_W$. Experiments show that $\sin^2{\left(\theta_W\right)} \approx 0.23$ [@9], a value close to $\delta^{-1}$ (see also[@12b]). 

Another example is the mixing of quarks from different families.  Here, the Cabibbo angle $\theta_C$ plays the role of the mixing angle.  According to experiment, $\sin\left(\theta_C\right) \approx
0.23$, which is close to $\delta^{-1}$ (see also [@12b]). 

![ \[fig:Fig7\] Transformation of the strong force into the weak force into the electromagnetic force via bifurcations.](Fig7.pdf){width="2.25in"}

Finally, a proton and a neutron can be considered as two states of the same particle, the nucleon.  One parameter that characterizes the two particles is a dimensionless physical constant called the $g$-factor.  Experimental values of the $g$-factor for a proton and a neutron are $+5.585\ldots$ and $-3.826\ldots$ respectively, which are two asymmetric numbers.  We can restore the symmetry if we represent them as $g_{p,n} \approx 1\pm\delta$, which is consistent to an accuracy of a few percent. 

Spin - Another Possible Consequence of the SOS Model
====================================================

The physical dimension of the reduced action $J$ is the same as the dimension of angular momentum, and at the level $q_{e-2}$ its value is $2\hbar$, suggesting that $J$ may be related to particle spin.  Even more suggestive is the $SU\left(2\right)$ symmetry, which is immanent of spin-1/2 particles and plays a profound role in period-doubling bifurcation dynamics.  We propose the following construction in order to incorporate spin into our model. 

According to our model particle dynamics can be described by a $2^n$-component spinor, where $n$ is the excitation level.  To some approximation, these $2^n$ degrees of freedom can be viewed as independent.  Similar to a gas system where the average energy is distributed among all degrees of freedom, we conjecture that in our particle system the average action is shared by all degrees of freedom.  We call this democracy the equal action distribution rule.  We also assume that particles can be located at different excitation levels and have different numbers of loops depending on the level they inhabit. 

For example, at the lowest excitation level, an electron possesses a one-loop superattractor.  This level corresponds to small perturbations where we can neglect the probability of positron generation and spin flipping.  At this level we assign the charge as $q_e = e$ and the action as $J_e = \eta{}e^2$. At the highest level of electron excitation there are four quasi-independent electron statesspin-up electron, spin-down electron, spin-up positron, and spin-down positron.  They can be represented by a four-component spinor (recall the Dirac spinor) or a four-loop attractor (Fig.\[fig:Fig8\]a).  These dynamics emerge two levels above $q_e = e$, where $q_{e-2} = \delta{}e$ and $J_{e-2} =
\delta^2\eta{}e^2 = 2\hbar$. Applying the equal action distribution rule we find that each degree of freedom in the four-loop superattractor has action $1/4\left(2\hbar\right) = \hbar/2$, which corresponds to the electron spin. 

![ \[fig:Fig8\] Schematic attractor topology for (a) the electron, (b) the photon, and possibly (c) the graviton. ( d) Electron branch of the phylogenetic tree.](Fig8.pdf){width="3in"}

Photons come in two distinct polarizations and possess two degrees of freedom that can be associated with two-loop attractors (Fig.\[fig:Fig8\]b).  To find the lowest photon level, we recall that a photon can be viewed as the fusion (annihilation) of two particlesthe electron and the positron.  In the particle phylogenetic tree, the electron and positron branches merge just one level above the lowest electron level (Fig.\[fig:Fig8\]d).  Here the charge is $q_{e-1} =
\delta^{1/2}e$. The two-loop photon superattractor emerges at the next level where $q_{e-2} = \delta{}e$ and $J_{e-2} =
\eta\delta{}J_{e-1} = \eta\delta\left(\delta^{1/2}e\right)^2 =
2\hbar$. At this level the photon action per degree of freedom is $\hbar$, and corresponds to the photon spin. 

Continuing with the speculations, we come to the one-loop superattractor (Fig.\[fig:Fig8\]c) at the level $q_{e-2} =
\delta{}e$. Here action is $J_{e-2} = \eta\left(\delta{}e\right)^2 =
2\hbar$ and corresponds to a spin-2 particle.  The only elementary particle (albeit a hypothetical one) that is known to have spin $2\hbar$ is the graviton. 

The bifurcation tree branch describing these three types of particles is shown in Fig.\[fig:Fig8\]d.  We refer to it as the electron branch. 

Particle Zoo
============

Smashing particles in accelerators or other strong perturbations may excite the system to levels beyond $q_{e-2}$. Such excitations may result in the emergence of more branches in the bifurcation tree (Fig.\[fig:Fig9\]).  Each new bifurcation doubles the number of degrees of freedom.  As the system relaxes to its original level it is faced with more choices for particle self-organization.  For example, after being excited to level $q_{e-3}$ (Fig.\[fig:Fig9\]), the system can relax to a lower level either through the electron branch described in the previous section, or through a new branch (the neutrino branch), transforming into particles from a different group.  The set of all options includes one more spin-2 particle, two more spin-1 particles, and four more spin-1/2 particles.  Altogether we now have four spin-1 bosons that can be associated with electroweak vector bosons, and eight fermions that can be associated with spin-up/spin-down electrons/positrons and neutrinos/anti-neutrinos. 

![ \[fig:Fig9\] Family I branch of the phylogenetic tree.](Fig9.pdf){width="2.8in"}

The number of particles doubles again after an excitation to the level $q_{e-4}$. Now we have four spin-2 particles, eight spin-1 bosons, and sixteen fermions.  The bosons can be associated with gluons, and the eight extra fermions with spin-up/spin-down u-quarks/d-quarks and their anti-particles.  This particle zoo has a lot of similarities with the Standard Model taxonomy.  However, the analogy is not complete.  For instance, some of the gluons coincide with electroweak bosons and photons. 

To accommodate particles from the second and third fermion families of the Standard Model we need to add more branches to our bifurcation diagram.  If particle dynamics belong to the period-3 window, this will be the end of the story.  Otherwise, future experiments will discover new particles belonging to a fourth family and so on.  Adding new branches to the first family of particles assumes the existence of more bosons than is currently known from experiment.  This oddity may be explained by the fact that it is difficult to distinguish among bosons of different families.  Whereas fermions can be distinguished exclusively by their mass (e.g. electrons, muons, and tau-particles have different masses), bosons are either massless or too heavy to be generated by our current particle accelerators. 

Relation to the Standard Model
==============================

In our efforts we strive to be as close to the Standard Model as possible.  Therefore, it is not surprising that the two models have a lot in common: $SU\left(2\right)$ symmetry, the same types of forces, similar values of their coupling constants, the same number of fermions, electro-weak bosons, and gluons, etc.  However, despite these similarities the two models have principal differences. 

The major distinction is that they are built on fundamentally different premises.  The Standard Model views conservation laws and the corresponding symmetries (or rather broken symmetries), both local and global, as the basic principles for particle existence.  Our proposed model is necessarily based on dissipation, which inherently implies time-arrow asymmetry (the violation of T-invariance) and the existence of directional flows.  These flows may also be responsible for violations of other global symmetries (like P-invariance, C-invariance, and CP-invariance).  In addition, local symmetries and gauge invariance do not play the same fundamental and mystic role as they do in the Standard Model.  The $SU\left(2\right)$ symmetry simply reflects the relations among solutions of a special class of nonlinear equations (the recursive function and its iterations) that describe the evolution of particle dynamics under perturbations.  Our model is formulated in a space-time independent framework.  Both dynamical variables, action and charge, are Lorentz invariant.  Thus, space-time symmetries are irrelevant.  Some other profound differences between the two models are listed in the table in Fig.\[tbl:Tbl1\]. 

![ \[tbl:Tbl1\] Key differences between the proposed model and the Standard Model.](Table1.pdf){width="3.4in"}

Conclusions
===========

We proposed a new framework for studying elementary particles.  This framework seems promising for penetrating into the previously inaccessible territory of the internal structure and dynamics of elementary particles.  Particle self-organization is described via *feedback* that is controlled by running coupling constants (charges and actions).  We explored the simplest system that mimics quadratic maps.  We found sensible answers to a number of questions that have never been explained by quantum physics.  We demonstrated that our model is capable of explaining the phenomenon and has the potential to provide a basis for particle taxonomy and the relations among strong, weak and electromagnetic forces accepted by the Standard Model. 

Our key result is the discovery of the deep connection between two fundamental constants of quantum physics, the Planck constant $\hbar$ and elementary charge $e$. The relation $\hbar =
\sqrt{\mu_0/\varepsilon_0}\left(\delta{}e\right)^2/2$ is experimentally verifiable and should be considered as an experimental fact irrespective of the correctness of our proposed model.  Ironically, the two most fundamental *quantum constants*, $\hbar$ and $e$, are linked through the Feigenbaum $\delta$, a constant that belongs to the physics of deterministic chaos and is thus exclusively *non-quantum*. 

Our results are assonant with t Hoofts proposal that the theory underlying quantum mechanics may be dissipative [@15].  They also suggest that quantum theory, albeit being both powerful and beautiful, may be just a quasi-linear approximation to a deeper theory describing the non-linear world of elementary particles.  As one of the founders of quantum theory, Werner Heisenberg once stated, …it may be that…the actual treatment of nonlinear equations can be replaced by the study of infinite processes concerning systems of linear differential equations with an arbitrary number of variables, and the solution of the nonlinear equation can be obtained by a limiting process from the solutions of linear equations.  This situation resembles the other one…where by an infinite process one can approach the nonlinear three-body problem in classical mechanics from the linear three-body problem of quantum mechanics. [ @14]

Acknowledgements
================

I want to express my gratitude to the authors of numerous books and papers that inspired my research.  The most influential ones are listed below.  I am also grateful to the people who read my manuscript at different stages and made critical comments, or simply encouraged me to continue this exploration: Vladimir Litvinov, Ryszard Gajewski, Tomasz Jannson, Lev Sadovnik, and Vitaly Dugaev.  I also want to thank my children, Julia and Alexander, for numerous discussions and corrections that made this paper more readable.\

**Nonlinear dynamics, complexity, and chaos theory**\
R.H. Abraham and C. D. Shaw in *Dynamics: The Geometry of Behavior*.\
A. A. Andronov, A. A. Vitt, and S. E. Khaikin in *Theory of Oscillators*.\
R. X. Hilborn in *Chaos and Nonlinear Dynamics*.\
K. Mainzer in *Symmetry and Complexity: The Spirit and Beauty of Nonlinear Science*.\
G. Nicolis and I. Prigogine in *Exploring Complexity*.\
I. Prigogine in *From being to becoming: time and complexity in physical sciences*.\
P. O. Peitgen, H. Jürgens, and D. Saupe in *Chaos and Fractals: New Frontiers of Science*.\
I. Stewart in *Does God Play Dice?  The New Mathematics of Chaos*.\
S. H. Strogatz in *Nonlinear Dynamics and Chaos*.\

**The Standard Model and above**\
G. D. Goughlan and J. E. Dodd in *The Ideas of Particle Physics*.\
B. Green in *The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory*.\
G. t Hooft in *Determinism and Dissipation in Quantum Gravity*.\
I. D. Lawrie in *A Unified Grand Tour of Theoretical Physics*.\
R. Penrose in *The Road to Reality: A Complete Guide to the Laws of the Universe*.\
B. A. Schumm in *Deep Down Things: The Breathtaking Beauty of Particle Physics*.\
L. Smolin in *Three Roads to Quantum Gravity*.\
A. Zee in *Quantum Field Theory in a Nutshell*.\
