Double-Period Gravitational Dynamics from a Multifractal Perspective of Motion

Abstract

Assimilating complex systems to multifractal-type objects reveals continuous and non-differentiable curve dynamics, aligning with the Multifractal Theory of Motion. Two scenarios, a Schrödinger-type and a Madelung-type multifractal scenario, are possible in this setting. If the Madelung scenario employs maximized information entropy for a distribution density, then Newtonian and oscillator-type forces can be determined. In the presence of these forces and a matter background, we analyze the two-body problem. The obtained results are as follows: a generalized Hubble-type law, a dependence of Newton’s constant on the epoch and background density, a generalization of Lorentz transform (involving the Hubble constant, Newton’s constant, the speed of light, and cosmic matter density), etc. Moreover, in the same scenario, the functionality of a diffusion-type equation implies instabilities, such as period doubling, through an SL(2R) invariance. Thus, multiple infragalactic and extragalactic instabilities are exemplified.

1. Introduction

Since non-differentiability is a fundamental trait of complex systems, it is imperative to establish a non-differentiable physics model. This hypothesis proposes that the complexity of interaction processes may be substituted with non-differentiability, rendering the whole classical repertory of variables from conventional physics (differentiable physics) superfluous.

This subject was formulated within the context of the Scale Relativity Theory (SRT) [1] and the Multifractal Theory of Motion (MTM), specifically incorporating an arbitrary constant fractal dimension into the Scale Relativity Theory [2]. In the framework of SRT or MTM, we posit that the trajectories of entities within complex systems unfold along continuous yet non-differentiable curves (multifractal curves), whereby all physical phenomena related to the dynamics depend not only on the space–time coordinates but also on the resolution of space–time scales. From this perspective, the physical attributes delineating complex system behaviors may be seen as multifractal functions [1,3]. Moreover, the elements of the intricate system may be distilled and associated with their individual trajectories, enabling the system to operate as a distinct interaction-less “fluid” via its geodesics in a non-differentiable (multifractal) space (Schrödinger or hydrodynamic representations).

This research introduces an innovative way to analyze the dynamics of complex systems through MTM. We begin by equating any intricate system to a multifractal mathematical entity. The characterization of complex system dynamics is accomplished by continuous and non-differentiable curves (multifractal curves), leading to two distinct scenarios. The initial scenario is a Schrödinger-type multifractal scenario, whereby the laws of motion may be associated with the SL(2R) algebra invariant functions. The second scenario is a Madelung-type multifractal scenario, whereby the non-differentiable elements of the velocity field appear as a gradient of a scalar function. The two situations are interrelated, emphasizing SL(2R) invariances (i.e., symmetries produced by the SL(2R) groups) and Riccati-type gauges in these dynamics. Several examples are given.

We must specify that our approach is not singular.

Firstly, we must mention the work of Ershkov et al. [4], which introduced a new restricted two-body problem (R2BP) solution approach to calculating the precise coordinates of a mass point in orbit around a much more massive main body. This method uses a continuing fraction potential different from Kepler’s R2BP formulation. Additionally, the authors examined a system of equations of motion in order to find an analytical way to describe the answer in polar coordinates.

Secondly, in their work, Christianto and Smarandache [5] argue that the Schrödinger representation of the Navier–Stokes equation may be written using the Riccati equation. The suggested methodology varies from others, but offers an advantage by allowing for the extension of the Navier–Stokes equation to quaternionic and biquaternionic versions, such as via Kravchenko and Gibbon’s routes.

Lastly, Christiano analyzed several ways to predict the quantization of planetary orbits [6], including the Cantorian Superfluid Vortex concept. The Quantum Cosmology theory suggests a relationship between both approaches, maybe owing to gravitation-related boson condensation processes.

Moreover, several dynamics related to the previous works were analyzed in [7,8].

2. Two Scenarios for Analyzing Dynamics in the Universe as a Complex System

Let us assume that the Universe, as a complex system, can be assimilated to a multifractal-type mathematical object. The hypothesis that the Universe can be likened to a multifractal model is supported by several arguments based on observations, theory, and advanced mathematical models. Here are the main arguments in favor of this hypothesis:

(a)

Distribution of matter and energy in the Universe [9]:

•

Large-scale structures: the Universe exhibits a hierarchical distribution of matter (galactic filaments, cosmic voids, and galaxy clusters), suggesting self-similarity on different scales. This is a fundamental feature of multifractals.

•

Density heterogeneity: analysis of the distribution of galaxies and their clusters suggests non-uniform variations that fit a multifractal model better than a uniform or simple fractal model.

(b)

Multifractal theory and gravity [10]:

•

Nonlinear gravitational interaction: as gravity forms cosmic structures, their evolution is influenced by nonlinear processes that generate multifractal distributions of matter.

•

Cosmological perturbations: fluctuations in the density of dark matter or cosmic background radiation can be efficiently described by a multifractal dimension spectrum.

(c)

Cosmic background radiation (CMB) [11]:

•

Observed anisotropies: the anisotropic structures in the cosmic microwave background (CMB) demonstrate a complex distribution that can be analyzed by multifractal techniques.

•

Studies of the power spectra of temperature fluctuations in the CMB indicate the existence of phenomena involving self-similar dynamics on multiple scales.

(d)

Cosmic fluid dynamics [12]:

•

The Universe as a multifractal fluid: theoretical models describing the Universe in terms of a non-differentiable (multifractal) fluid provide explanations for phenomena, such as gravitational instabilities and the turbulent behavior of dark matter.

•

Navier–Stokes-type equations: some studies have suggested that the equations governing the evolution of the Universe can be associated with transformations to Schrödinger-type equations using a multifractal model.

(e)

Compatibility with the cosmological principle [13]:

•

The generalized cosmological principle: although the Universe is homogeneous and isotropic on the largest scales, at lower levels, its structure can be described by a multifractal spectrum of dimensions.

•

This hypothesis does not contradict current observations, but extends the understanding of structure on intermediate scales.

(f)

Mathematical and physical applications of multifractals [14]:

•

Modeling complex processes: multifractals are already used in physics to describe systems with complex dynamical processes, such as turbulence. The same mathematical tools are also applicable in cosmology.

•

Fractal dimension spectrum: multifractal models allow for the use of a full-dimension spectrum to explain the diversity of structures in the Universe.

(g)

Philosophical and epistemological implications [15]:

•

The complexity of the nature of the Universe: the assimilation of the Universe to a multifractal model reflects a more realistic description of the complex and interconnected nature of cosmic structures.

•

Unification of theories: multifractals can provide a framework for connecting models from quantum physics, chaos theory, and general relativity, suggesting a unified theory of structures.

(h)

Recent observations and empirical data [16]:

•

Galaxy mapping: studies such as the Sloan Digital Sky Survey and other cosmic mapping support the hypothesis that the distribution of galaxies exhibits multifractal features.

•

Large-scale anomalies: phenomena such as the polarization of the cosmic background radiation or the distribution of dark matter can be explained in terms of multifractal properties.

According to our previous hypothesis, let us describe any gravitational system dynamics by means of MTM. Subsequently, in accordance with MTM [2], two scenarios become compatible in the description of such dynamics:

(i)

A Schrödinger multifractal scenario described by the following differential equation (for details, see [2]):

$${{\lambda }^2\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}{\partial }^l{\partial }_l\psi +i{\lambda \left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}-1\right]}{\partial }_t\psi =0$$

(1)

where

$${\partial }_t=\frac{\partial }{\partial t}, {\partial }_l=\frac{\partial }{\partial x^l}, {\partial }^l{\partial }_l=\frac{{\partial }^2}{\partial x_l^2}$$

(2)

In Equations (1) and (2), the quantities have the following meanings: $x_l$ is the multifractal (non-differentiable) spatial coordinate, t is the non-multifractal (differentiable) temporal coordinate, $\lambda$ is a multifractal–non-multifractal transition coefficient, $f\left(\alpha \right)$ is the singularity spectrum of order $\alpha$, where $\alpha \equiv \alpha \left(D_F\right)$ and $D_F$ is the motion curves’ fractal dimension, dt is the scale resolution, and $\psi$ is the state function [2,3].

We note that for $\lambda =\hslash /2m_0$ (where $\hslash$ is the reduced Planck constant and $m_0$ is the microparticle’s rest mass) and $D_F=2$ [2], the monofractal dynamics of complex systems can be characterized by Peano-type curves. Then, Equation (1) is reduced to Schrödinger’s differential equation from Quantum Mechanics.

(ii)

A Madelung multifractal scenario described by a multifractal hydrodynamics differential equations system, as follows [2]:

$$\begin{gathered} {\partial }_t{V}_D^i+\left(V_D^l{\partial }_l\right)V_D^i=-{\partial }^{i}Q \\ {\partial }_t\rho +{\partial }^i\left(\rho V_D^i\right)=0 \end{gathered}$$

(3)

with

$$\begin{array}{c}{\widehat{V}}^l=V_D^l-i{V}_F^l\\ Q=-2{{\lambda }^2\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}\frac{{\partial }^l{\partial }_l\sqrt{\rho }}{\sqrt{\rho }}=-\frac{V_F^l{V}_{Fl}}{2}-{\lambda \left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}-1\right]}{\partial }_i{V}_F^i\\ V_D^i=2\lambda {\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}{\partial }^i\varphi , V_F^i={\lambda \left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}-1\right]}{\partial }^i\varphi \\ \psi =\sqrt{\rho }{e}^{i\varphi },\overline{\psi }=\sqrt{\rho }{e}^{-i\varphi }, \rho =\psi \overline{\psi },\end{array}$$

(4)

where ${\widehat{V}}^l$ is the global velocity, $V_D^l$ is the differentiable velocity, $V_F^l$ is the non-differentiable velocity, $Q$ is the specific multifractal potential, $\sqrt{\rho }$ is the amplitude, and $\varphi$ is the phase of the state function. This equations system corresponds to two conservation laws: a specific multifractal momentum conservation law—(Equation (3)), and a multifractal states density conservation law—(Equation (3)). In our opinion, the multifractal potential in the Madelung scenario could have a significant influence on force generation, both in the context of Newtonian (gravitational) potentials and oscillatory potentials (such as harmonic ones). This influence derives from the complex and nonlinear nature of the multifractal potential and the way it interacts with the dynamical structures associated with a quantum system described by the Madelung formalism.

Irrespective of the scenario considered, using $f\left(\alpha \right)$ has the following advantages:

(i)

The existence of a dominant fractal dimension in any complex system dynamics facilitates the recognition of a universal pattern. The elucidation of this pattern aligns with global structural and functional characteristics unique to monofractal dynamics.

(ii)

The existence of a fractal dimension “set” in any complex system dynamics facilitates the recognition of zonal patterns. The elucidation of these patterns aligns with local structural and functional characteristics unique to multifractal dynamics.

(iii)

The singularity spectrum of order $\alpha$ enables the identification of universality classes within any complex system dynamics, even when the associated attractors exhibit distinct characteristics.

3. Types of Forces Through Informational Entropy

In this study, we employ the Madelung scenario as a framework to investigate the multifractal nature of complex systems. The Madelung formalism is particularly suited to this analysis, as it bridges not only quantum mechanics and hydrodynamics but also multifractal mechanics and hydrodynamics, allowing for the incorporation of non-differentiable dynamics and nonlinear interactions. However, we acknowledge that other approaches, such as numerical simulations, perturbative techniques, or methods inspired by chaos theory and non-equilibrium thermodynamics, may provide additional insights. These methodologies could complement or expand upon the findings derived from the Madelung scenario and are valuable avenues for future exploration.

However, in the Madelung multifractal model, the probability density represents the probability of the particle distribution at any scale resolution. Information entropy maximization is closely related to the determination of this probability density in the presence of constraints imposed by the Schrödinger or Madelung equations [1].

The principle of entropy maximization can be used to infer the probability density if we know certain macroscopic variables (such as average energy or average momentum).

The informational entropy of a distribution can be defined as [17,18,19]:

$$H=-\int \rho \ln \rho dx$$

(5)

where the distribution density is given by $\rho \left(x\right)$ and we denote with x the random variables of the problem, with dx being the elementary measure of their domain.

This function quantifies the level of uncertainty in the definition of probabilities. The function is positive, demonstrating an increase with rising uncertainty, characterized by the broadening of the distribution, and is additive when considering independent uncertainty sources. To admit the maximum of informational entropy in probability inference [17,18,19] when only partial information is available is tantamount to admitting that we cannot know more. The distributions thus obtained must, therefore, be the least displaced distributions with respect to the real ones, since no restrictive hypothesis has been made on the missing information. In other words, such a distribution is realizable in the largest possible number of ways. The partial information that is most often at hand is given as the average of a function $f\left(x\right)$, or of several functions, as follows:

$$\overline{f}=\int \rho \left(x\right)f\left(x\right)dx$$

(6)

Here, we assume that $\overline{f}$ is known by measurement.

Equation (6), together with the density distribution summation relation

$$\int \rho \left(x\right)dx=1$$

(7)

are now the constraints for Equation (5), used in order to provide the distribution density corresponding to the maximum informational entropy. In this concrete case, the method of Lagrange’s undetermined multipliers leads to the following [20]:

$$\rho \left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left[-\mu -\nu f\left(x\right)\right]$$

(8)

which can be multivariate if we are dealing with several constraints of type (7). If, in addition to this type of constraint, we also specify the variance

$${\left(∆f\right)}^2=\int \rho \left(x\right){\left[f\left(x\right)-\overline{f}\left(x\right)\right]}^{2}dx$$

(9)

then the nature of the usual distribution changes. It becomes the Gaussian

$$\rho \left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left[-\mu -\nu f\left(x\right)-\epsilon f^2\left(x\right)\right]$$

(10)

In this context, let us consider the specific distribution densities (with a radial symmetry):

$$\rho \left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\nu r\right)$$

(11)

and

$$\rho \left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\epsilon r^2\right)$$

(12)

obtained from Equation (10) by a convenient choice of $\mu , \nu ,$ and $\epsilon$. Then, the specific multifractal potential, Q, explicated in the form

$$Q\left(r\right)=-2{\lambda }^2{\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}\frac{1}{\sqrt{\rho }}\left(\frac{d^2\sqrt{\rho }}{d{r}^2}+\frac{2}{r}\frac{d\sqrt{\rho }}{dr}\right)$$

(13)

will induce the following specific multifractal forces:

$$\overrightarrow{F}\left(r\right)=-\nu {\lambda }^2{\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}\frac{\overrightarrow{r}}{r^3}$$

(14)

corresponding to Equation (11), and

$$\overrightarrow{F}\left(r\right)=2\epsilon {\lambda }^2{\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}\overrightarrow{r}$$

(15)

corresponding to Equation (12).

Now, a possible correspondence with physical reality, according to [21], implies the following functionalities in the case of monofractal dynamics $f\left(\alpha \right)\to D_F$ in the fractal dimension $D_F\to 2$ (dynamics on fractal curves of the Peano type) [1,3]:

$$\nu {\lambda }^2=GM$$

(16)

where G is Newton’s constant, M is the rest mass of the gravitational source, and

$$2\epsilon {\lambda }^2=H^2-\frac{4\pi }{3}G{\rho }_0\equiv \frac{1}{T^2}$$

(17)

where H is the Hubble constant, and ${\rho }_0$ is the density of the matter background [21,22]. In this context, T corresponds to the inverse of an effective Hubble constant. In our opinion, the previous results can have the following implications:

(i)

Equation (14) can correspond to Newtonian-type forces, while Equation (15) can correspond to oscillator-type forces.

(ii)

A solution to the “delicate” problem of the Hubble tension (the dependence of the Hubble constant on the degree of self-structuring of gravitational systems in the Universe) can be found. We must mention that in the standard cosmological model, known as the Lambda Cold Dark Matter model, the Hubble constant is related to the total density of energy and matter in the Universe [23]. Moreover, some relevant observational studies seem to support our previous affirmations, including Hubble constant measurements using gravitational waves [24] and data obtained by the James Webb Spatial Telescope [25].

(iii)

If the repulsive forces (introduced together with the constant H) are exactly compensated by the attractive forces exerted by the rest of the matter in the Universe (metagalaxy), then

$$H^2-\frac{4\pi }{3}G{\rho }_0=0$$

(18)

In such a context, the motion equations in the two-body problem can be reduced to their Newtonian form. Moreover, if, in this case, we choose for H an expression such as

$$H=\frac{c}{R_0}$$

(19)

where c is the speed of light, and $R_0$ is the Euclidian radius of the Universe (considered spherical), the following relation is obtained:

$$\frac{G{M}_0}{c^2{R}_0}=1$$

(20)

where $M_0$ corresponds to the mass of the metagalaxy. This relation is analogous to Milne’s relation [26] and is very close to the similar relation in Einstein’s static model [27].

(iv)

The gravitational interaction energy between two punctiform bodies will have two components: an attractive and a repulsive one. In such a context, one can arrive at a classical solution of Seelinger’s gravitational paradox, based on repulsive forces [27,28].

(v)

If $H^2-\frac{4\pi }{3}G{\rho }_0\equiv h^2>0$, then the two-body problem implies Hubble’s law (of the proportionality between radial velocity and distance) as an asymptotic relation [29].

(vi)

In the general case (the Universe assimilated to a multifractal), the interaction constants from Equations (14) and (15) are dependent on the scale resolution, so that

$$G=G_0{\left(dt\right)}^{\left[\frac{4}{f\left(\alpha \right)}-2\right]}$$

and

$$T=T_0{\left(dt\right)}^{\left[1-\frac{2}{f\left(\alpha \right)}\right]}$$

where $G_0$ and $T_0$ are the values of $G$ and $T$ for monofractal dynamics $f\left(\alpha \right)\to D_F$ in the fractal dimension $D_F\to 2$ (dynamics on fractal curves of the Peano type).

4. On a Class of Multifractal Cosmological Models

We start from the observation that many cosmological models, although using the equations of general relativity, which include a non-zero matter tensor, still conclude that the expansion is of inertial origin [30,31]. On the other hand, Hubble’s empirical law predicts accelerated ( $\overrightarrow{v}\propto \overrightarrow{r}$) rather than inertial motion. In a broader sense, accelerated motion can be attributed to an “inertial nature”, if the equations of motion arise from geodesic motion in a non-Minkowskian metric, for which the curvature tensor, $R_{\mu \nu }$, is identically zero, but the affine connections, ${\Gamma }_{\mu \nu }^{\lambda }$, are non-zero. A transformation group more general than the Lorentz group will correspond to such a metric, that latter of which contains physical constants that are scale-resolution-dependent, such as the speed of light and the Hubble constant. From considerations related to the distribution of the masses and velocities of matter in the Universe, as well as the fact that the Universe is seen to be spherically symmetric with respect to the origin of the coordinate frame, no matter which galaxy nucleus the origin is placed, the following equations result:

$$m_{01}\left(\frac{d^2{\overrightarrow{r}}_1}{d{t}^2}-\frac{{\overrightarrow{r}}_1}{T^2}\right)=-\frac{\partial \Phi }{\partial {\overrightarrow{r}}_1}, m_{02}\left(\frac{d^2{\overrightarrow{r}}_2}{d{t}^2}-\frac{{\overrightarrow{r}}_2}{T^2}\right)=-\frac{\partial \Phi }{\partial {\overrightarrow{r}}_2}, \Phi \equiv \Phi \left(\left|{\overrightarrow{r}}_2-{\overrightarrow{r}}_1\right|\right)$$

(21)

where $m_{01}$ and $m_{02}$ are the rest masses of the bodies assimilated to material points, $T=T_0{\left(dt\right)}^{\left[1-\frac{2}{f\left(\alpha \right)}\right]}$, and $\Phi$ is the gravitational potential. They generalize the equations of Newtonian analytical mechanics in the two-body problem, based on the multifractal hydrodynamics model, and are invariant to the following set of transformations:

$${\overrightarrow{r}}^{\prime }=\overrightarrow{r}+{\overrightarrow{V}}_{o}T\mathrm{sh}\left(\frac{t-t_0}{T}\right), t^{\prime }=t$$

(22)

The supplementary accelerations, proportionate to the distances in Equation (21), represent the influence of the ordered component of cosmic matter motion on the two-body aggregate. The group of transformations (22) passes into a Galilei-type group of transformations, when $T\to \infty$. In the absence of “local interaction”, $\Phi =0$, and the motion of a point body will be described by the equation

$$\frac{d^2\overrightarrow{R}}{d{t}^2}-\frac{\overrightarrow{R}}{T^2}=0$$

(23)

whose general solution is

$$\overrightarrow{R}={\overrightarrow{R}}_0\mathrm{ch}\left(\frac{t-t_0}{T}\right)+{\overrightarrow{V}}_{0}T\mathrm{sh}\left(\frac{t-t_0}{T}\right)$$

(24)

If the duration of motion is much smaller than the constant T, then $\frac{t-t_0}{T}\ll 1$ and Equation (24) describes inertial motion in Galileo’s formulation:

$$\overrightarrow{R}\approx {\overrightarrow{R}}_0+{\overrightarrow{V}}_0\left(t-t_0\right)$$

(25)

If $\frac{t-t_0}{T}\gg 1$, Equation (24) has another limit, which can be found as follows. We first derive the equation with respect to time and eliminate ${\overrightarrow{R}}_0$ between the original equation and the derived equation. We obtain

$$\overrightarrow{V}=\frac{{\overrightarrow{V}}_0}{\mathrm{ch}\left(\frac{t-t_0}{T}\right)}+\frac{\overrightarrow{R}}{T}\mathrm{th}\left(\frac{t-t_0}{T}\right)$$

(26)

The limit sought is, therefore, a multifractal Hubble law:

$$\overrightarrow{V}=\frac{\overrightarrow{R}}{T}=\frac{\overrightarrow{R}}{T_0{\left(dt\right)}^{\left[1-\frac{2}{f\left(\alpha \right)}\right]}}$$

(27)

The standard Hubble law corresponds to the case in which the density of background matter is zero for monofractal dynamics $f\left(\alpha \right)\to D_F$ in the fractal dimension $D_F\to 2$ (dynamics on fractal curves of the Peano type).

Let us now perform the coordinate and time transformation

$$\tau =T\mathrm{t}\mathrm{h}\frac{t-t_0}{T}, \overrightarrow{\rho }=\frac{\overrightarrow{R}}{\mathrm{ch}\frac{t-t_0}{T}}$$

(28)

under the action of which the transformation group (22) becomes the multifractal Galilei transformation group:

$${\overrightarrow{\rho }}^{\prime }=\overrightarrow{\rho }+{\overrightarrow{V}}_0\tau , {\tau }^{\prime }=\tau$$

(29)

The standard Galilei group corresponds to the case of monofractal dynamics, where $f\left(\alpha \right)\to D_F$ in the fractal dimension and $D_F\to 2$ (dynamics on fractal curves of the Peano type).

At the same time, we have the identity

$$\frac{d^2\overrightarrow{\rho }}{d{\tau }^2}\equiv \left(\frac{d^2\overrightarrow{R}}{d{t}^2}-\frac{\overrightarrow{R}}{T^2}\right){\mathrm{ch}}^3\left(\frac{t-t_0}{T}\right)$$

(30)

Now, at any scale resolution, from Equations (23) and (30), it follows that in the measure $\left(\overrightarrow{\rho },\tau \right),$ a body on which only the (smoothly distributed) matter of the Universe acts moves rectilinearly and uniformly, according to the Newtonian prescription. In other words, at any scale resolution, the effect of the matter of the Universe on the sample body, free from local actions, is exactly compensated, which means that the motion of matter has only an ergodic component. Therefore, at any scale resolution, by switching from measure $\left(\overrightarrow{R},t\right)$ to measure $\left(\overrightarrow{\rho },\tau \right)$, the ordered (hydrodynamic) component of the motion of matter is canceled. The transition $\left(\overrightarrow{R},t\right)\to \left(\overrightarrow{\rho },\tau \right)$ is thus equivalent to the transition from the observable frame to the frame which is operated in conventional cosmology. Going back to Equation (21), let us specify the form of function $\Phi$, corresponding to the gravitational interaction:

$$\Phi =-\mathrm{G}\frac{m_{01}{m}_{02}}{\left|{\overrightarrow{r}}_2-{\overrightarrow{r}}_1\right|}$$

(31)

Then, in the equations obtained by inserting Equation (31) into Equation (21), we perform the transformation

$$\mathrm{t}=t_0+T\ln \sqrt{\frac{T+\tau }{T-\tau }}, \overrightarrow{r}=\frac{\overrightarrow{\rho }}{\sqrt{1-\frac{{\tau }^2}{T^2}}}$$

(32)

which is the inverse of transformation (28). This leads to the following motion equations:

$$m_{01}\frac{d^2{\overrightarrow{\rho }}_1}{d{\tau }^2}=\frac{G}{\sqrt{1-\frac{{\tau }^2}{T^2}}}\frac{m_{01}{m}_{02}}{{\rho }^3} \overrightarrow{\rho }, { m}_{02}\frac{d^2{\overrightarrow{\rho }}_2}{d{\tau }^2}=-\frac{G}{\sqrt{1-\frac{{\tau }^2}{T^2}}}\frac{m_{01}{m}_{02}}{{\rho }^3} \overrightarrow{\rho }$$

(33)

Because, at any scale resolution, in measure $\left(\overrightarrow{\rho },\tau \right),$ Galileo’s inertia principle is functional, it follows that Equation (33) must coincide with Newton’s equations. As a consequence, the G constant, from Equations (31) and (33), must depend, at any scale resolution, on the epoch according to the following law:

$$G=\overline{G_0}\sqrt{1-\frac{{\tau }^2}{T^2}}=\frac{\overline{G_0}}{\mathrm{ch}\frac{t-t_0}{T}}$$

(34)

Although the law (33) corresponds, at any scale resolution, to a decrease in the constant G with the epoch, the difference with the cosmologies of Dirac and Dicke is considerable, due to the fact that $\left(\dot{G}/G\right)=0$ for $\mathrm{t}=t_0$. So far, we have developed an unconventional cosmological model at the non-relativistic level, dependent on the scale resolution. The insertion of the relativity amendments must, therefore, be carried out in such a way that Equation (27), which prescribes the extended inertial character of the motion of a body (free from local actions), remains unchanged. The way forward is then obvious: transforms (29) must be replaced by Lorentz transforms of measure $\left(\overrightarrow{\rho },\tau \right)$, at any scale resolution. Then, switching from measure $\left(\overrightarrow{\rho },\tau \right)$ to measure $\left(\overrightarrow{R},t\right)$, a transformation group generalizing Lorentz’s transforms is obtained, for any scale resolution, i.e.,

$$\begin{array}{c}{\overrightarrow{R}}^{\prime }={\left(1-{\epsilon }^2\right)}^{-\frac{1}{2}}\left[\frac{\overrightarrow{R}}{\mathrm{ch}\alpha }+\gamma {\overrightarrow{V}}_{0}T\mathrm{th}\alpha +\frac{{\gamma }^2}{\gamma +1}\frac{\left(\overrightarrow{R}{\overrightarrow{V}}_0\right){\overrightarrow{V}}_0}{c^2\mathrm{ch}\alpha }\right], t^{\prime }=t_0+T\ln \sqrt{\frac{1+\epsilon }{1-\epsilon }}, \gamma \equiv \\ {\left(1-\frac{{\overrightarrow{V}}_0^2}{c^2}\right)}^{-\frac{1}{2}}, \alpha \equiv \frac{t-t_0}{T}, \epsilon \equiv \gamma \left[\mathrm{th}\alpha +\frac{1}{c^2}\frac{\left(\overrightarrow{R}{\overrightarrow{V}}_0\right)}{T\mathrm{ch}\alpha }\right]\end{array}$$

(35)

These transformations preserve the invariance of the following metric:

$$\left(dS\right)=\frac{dt}{{\mathrm{ch}}^2\alpha }\sqrt{1-\frac{1}{c^2}{\left(\overrightarrow{V}\mathrm{ch}\alpha -\frac{\overrightarrow{R}}{T}\mathrm{sh}\alpha \right)}^2}$$

(36)

and the motion in Equation (23) corresponds to the geodetic motion in this metric at any resolution scale. Finally, in the particular case of monofractal dynamics $f\left(\alpha \right)\to D_F$ in the fractal dimension $D_F\to 2$, the size of the Universe is estimated, in this model, assuming that, in the measure $\left(\overrightarrow{\rho },\tau \right)$, the aggregation of matter occurs until the gravitational charge of the Universe becomes zero, i.e.,

$$q_G=\frac{\sqrt{G_0}}{c^2}\left(M_0{c}^2-\frac{3}{5}{G}_0\frac{M_0^2}{cT}\right)=0, \frac{1}{T}=H$$

(37)

This gives a value for the inertial mass, $M_0$, comparable to that estimated by Arthur Eddington and Edward Milne [32,33,34].

5. Synchronized Dynamics in Gravitational Systems

Let us reconsider the states density conservation law (3b). If the differential and nondifferential scale dynamics satisfy the constraint (a momentum conservation law)

$$v_D^i=2{\lambda \left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}-1\right]}{\partial }^i\Phi =-v_F^i={\lambda \left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}-1\right]} {\partial }^i\mathrm{l}\mathrm{n}\rho$$

(38)

then the states density conservation law takes the form of the multifractal hydrodynamics equation

$${\partial }_t\rho ={\lambda \left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}-1\right]}∆\rho$$

(39)

This differential equation, in the case of one-dimensional dynamics, remains invariant with respect to the transforms group

$${\mathrm{t}}^{\prime }=\frac{\alpha t+\beta }{\gamma t+\delta }, r^{\prime }=\frac{r}{\gamma t+\delta }, \alpha ,\beta ,\gamma ,\delta \in \mathbb{R}$$

(40)

Now, let us reconsider the first Equation (40), which represents the homographic action of the generic matrix:

$$\widehat{M}=\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)$$

(41)

The matter that requires attention is as follows: a relation must be established between the ensemble of matrices $\widehat{M}$ and the ensemble of values corresponding to $t$, for which $t^{\prime }$ remains constant.

From a geometric standpoint, this entails identifying the collection of points $\left(\alpha ,\beta ,\gamma ,\delta \right)$, univocally corresponding to the values of the parameter $t$. By using the first Equation (40), along with the methods from [21,22] (i.e., a Riccati-type gauge), we obtain

$$dt+{\omega }_1{t}^2+{\omega }_{2}t+{\omega }_3=0$$

(42)

where the following notations are used [2,35]:

$${\omega }_1=\frac{\gamma d\alpha -\alpha d\gamma }{\alpha \delta -\beta \gamma }, {\omega }_2=\frac{\delta d\alpha -\alpha d\delta +\gamma d\beta -\beta d\gamma }{\alpha \delta -\beta \gamma }, {\omega }_3=\frac{\delta d\beta -\beta d\delta }{\alpha \delta -\beta \gamma }$$

(43)

It is readily apparent that the metric

$$d{s}^2=\frac{{\left(\alpha d\delta +\delta d\alpha -\beta d\gamma -\gamma d\beta \right)}^2}{4{\left(\alpha \delta -\beta \gamma \right)}^2}-\frac{d\alpha d\delta -d\beta d\gamma }{\alpha \delta -\beta \gamma }$$

(44)

is directly related to the discriminant of the quadratic polynomial from Equation (42)

$$d{s}^2=\frac{1}{4}\left({{\omega }_2}^2-4{\omega }_1{\omega }_2\right)$$

(45)

We must point out the fact that, for a particular case of 1-forms ${\omega }_1,{\omega }_2,{\omega }_3$ [2], the metric (45) also assumes the SL(2R) invariance.

The three differential forms of Equation (43) can provide a coframe [2] in any point of the absolute space. This facilitates the conversion of the geometric characteristics of absolute space into algebraic qualities associated with differential Equation (42).

In this case, the 1-forms ${\omega }_1, {\omega }_2, {\omega }_3$ are differentiated exactly in the same parameter $\tau$ of the geodesic. Along these geodesics, Equation (46) is transformed into a Riccati-type differential equation (a Riccati-type gauge):

$$\frac{dt}{d\tau }=P\left(t\right), P\left(t\right)=c_1{t}^2+2c_{2}t+c_3$$

(46)

We will now discuss the in-phase correlative dynamics of complex systems.

Now, the parameters $c_1$, $c_2$, and $c_3$ are constants that characterize a certain geodesic of the family. In this case, by employing the method from [2,21], Equation (46) highlights dynamics correlated by a Stoler transform [36,37,38,39]:

$$z=-\frac{c_2}{c_1}+\frac{\omega }{c_1}\left[\frac{2r\sin \left[2\omega \left(\tau -{\tau }_0\right)\right]}{1+r^2+2r\cos \left[2\omega \left(\tau -{\tau }_0\right)\right]}+i\frac{1-r^2}{1+r^2+2r\cos \left[2\omega \left(\tau -{\tau }_0\right)\right]}\right]$$

(47)

where $r$ and ${\tau }_0$ are two real constants specific to the solution.

We present, in Figure 1, the explication of double-period dynamics in complex systems of a gravitational type, based on Equation (47), through $F\left(\omega ,\tau =\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\right)$, for ${\omega }_{max}=6$, $r=1,$ and ${\tau }_0=0$.

Period doubling is a widespread instability in gravitational dynamics, both on infragalactic and extragalactic scales. We give some examples below.

A galaxy system’s rhythmic nature can double the period between cycles, generally due to nonlinear dynamics and chaotic behavior. In several galaxy systems, period doubling and similar characteristics have been observed or postulated. Galactic systems may experience period doubling in several cases [40,41,42,43]:

(i)

Galactic disk nonlinear oscillations: Nonlinear dynamics are used to model spiral galaxies’ density waves in the galactic disk. As they interact with galactic stars and gas, density waves can oscillate. Nonlinear effects, such as those in spiral arms, can double the density wave period in some models. The spiral arms’ oscillation period doubles as the disk’s stars interact with the spiral structure, causing chaotic behavior. This period doubling may indicate nonlinear dynamics that cause spiral arm creation and evolution in galaxies.

(ii)

Stellar orbit nonlinear dynamics: A core supermassive black hole may impact star orbits in galactic centers, especially nucleus-dominated galaxies. Star orbits in these regions can oscillate periodically or quasi-periodically. Gravitational interactions between stars and the central black hole can cause period doubling in these stellar orbits, especially in systems with large gravitational potential nonlinearities. In star clusters or in the galactic center, the black hole’s influence on star dynamics can cause bifurcations, where stars’ orbital periods quadruple, signaling chaotic behavior in their trajectories.

(iii)

Galaxy merger tidal interactions: Tidal interactions between galaxies during mergers can cause complex gravitational dynamics. Merging can cause gas and stellar oscillations in galaxies. The dynamic reaction of the system might double the oscillation of the merging galaxies’ gas disks or star components if the interaction is large or intense. Nonlinear feedback mechanisms in the galaxy’s response to tidal forces may lead certain properties, like gas or star orbits, to exhibit periodicity that doubles as the galaxies approach a more stable configuration.

(iv)

Precessing galactic binary orbits: Gravitational interactions can cause periodic behavior in binary galaxy systems or systems with interacting dwarf galaxies, notably in star and gas cloud orbits. Nonlinear gravitational interactions can cause orbits to bifurcate, increasing the period of star or gas cloud oscillations. In a galaxy pair undergoing close gravitational contact, stars and gas clouds may oscillate regularly, but their oscillations may abruptly quadruple, indicating a chaotic or complicated gravitational regime.

(v)

Bar and halo dynamics in disk galaxies: The gravitational interaction between the bar and the galactic disk can cause stars in bar–disk galaxies to oscillate. When the galactic bar grows or undergoes internal dynamical changes, these oscillations might double in a period. Nonlinearities in the potential that drives star motion within the bar can cause the system to bifurcate, doubling the bar’s oscillation time as it transitions between states. This is commonly linked to the secular development of bars, where galaxy dynamics change over time.

6. Conclusions

This paper presents the following main conclusions:

(i)

By equating any complex system to a multifractal-type object, it is demonstrated that, in alignment with the Multifractal Theory of Motion, its dynamics can be characterized by continuous and non-differentiable curves.

(ii)

In such a context, two scenarios (a Schrödinger-type and a Madelung-type multifractal scenario) become operable.

(iii)

In the Madelung scenario, using the principle of maximizing the informational entropy for a distribution density, the types of forces (Newtonian and oscillator-type forces) are determined by means of the specific multifractal potential. Such a procedure involves multifractalization by stochasticization.

(iv)

The two-body problem is analyzed both in the context of their interactions through the forces determined above and also in the presence of a matter background.

(v)

A generalized Hubble-type law is revealed.

(vi)

The dependence of Newton’s constant on the epoch and cosmic density is highlighted.

(vii)

A generalization of Lorentz transformations (transformations that depend on the Hubble constant, the speed of light, and the density of matter in the Universe) is obtained.

(viii)

The size of the Universe is estimated, assuming that, in a specific measure, the aggregation of matter occurs until the gravitational charge of the Universe becomes zero.

(ix)

Our analysis, rooted in the Madelung scenario, offers a semi-analytical perspective on complex systems, emphasizing multifractal structures and non-differentiable dynamics. While this approach has been effective in capturing key characteristics of the systems under investigation, it is not exhaustive. Alternative methods, such as direct numerical simulations or chaos-theory-based approaches, could provide complementary perspectives. For instance, numerical simulations may better capture high-dimensional interactions in certain complex systems, while perturbative methods could refine our understanding of small-scale phenomena. Future studies should aim to integrate these alternative approaches to validate and extend the findings presented here.

(x)

Assuming the functionality of a diffusion-type equation in the Madelung scenario, it is shown that the presence of a SL(2R) invariant induces various instabilities, such as period doubling. In this sense, various such instabilities are specified, both on the infragalactic and extragalactic scales.

(xi)

Future work will focus on comparing the findings obtained through the Madelung scenario with those derived from alternative techniques, such as machine learning, chaos theory and direct numerical simulations. This comparative analysis will help establish a more comprehensive framework for understanding complex systems and their multifractal behaviors.