Dynamics Control of the Complex Systems via Nondifferentiability

Abstract

A new topic in the analyses of complex systems dynamics, considering that the movements of complex system entities take place on continuum but nondifferentiable curves, is proposed. In this way, some properties of complex systems (barotropic-type behaviour, self-similarity behaviour, chaoticity through turbulence and stochasticization, etc.) are controlled through nondifferentiability of motion curves. These behaviours can simulate the standard properties of the complex systems (emergence, self-organization, adaptability, etc.).

1. Introduction

Complex systems are very large interdisciplinary research topics that have been intensively studied, particularly since the 1980s, by means of a combination of basic theory, derived especially from physics and computer simulation. Such kind of systems is composed of many interacting elemental units that were called “agents.” Examples of complex systems can be found in human societies, the brain, internet, ecosystems, biological evolution, stock markets, economies, and many others [1–4].

The way in which such a system manifests cannot be predicted only by the behaviour of individual elements or by adding their behavior but is determined by the manner in which the elements relate to influence global behaviour. Among the most significant properties of complex systems are emergence, self-organization, adaptability, and so forth [5–9].

The emergence of a complex system can be represented by that state of the whole which does not include the sum of its elements.

Self-organization is another characteristic of complex systems. Examples of organization that manifests in nature are found from cells to organisms, ecosystems, and also planets, stars, and galaxies [5–9].

Another characteristic of complex systems is adaptability. This occurs when the system changes in response to some information. Continuously oscillating between equilibrium and disorder, complex systems are not “rigid.” Any modification induced at microlevel generates a series of fluctuations, exploring new states of self-organization [5–9].

An example of a complex system is represented by polymers. Their forms present a multitude of organizations starting from simple, linear chains of identical structural units to very complex chains consisting of sequences of amino acids that form the building blocks of living things. Probably one of the most intriguing complex systems in nature is DNA, which creates cells by means of a simple but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems like organs, and these organs form even more complex systems like organisms. The study of complex systems can offer a glimpse about the realistic dynamics of polymers, solving difficult problems as protein folding [6–10].

Correspondingly, the theoretical models that describe the complex systems dynamics become sophisticated and ambiguous too [6–10]. However the situation can be standardized taking into account that the complexity of interaction process imposes various temporal resolution scales, and the pattern evolution imposes different degrees of freedom.

In order to develop a new theoretical model we must admit that the complex systems that display chaotic behaviour are known to acquire self-similarity (space-time structures seem to appear) in association with strong fluctuations at all possible space-time scales [6–8]. Then, for temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a collection of potential trajectories and the concept of definite positions by that of probability density [11–14]. One of the most interesting examples is the collisions processes in plasma discharge as a complex system, where the dynamics of the particles can be described by Levi-type movements [15]: between two successive collisions, the particle trajectory is a straight line, with the trajectory becoming nondifferentiable in the impact point, which implies that there are left and right derivatives in this point. We note that the Brownian-type motion is a particular case of Levi-type motion [16].

Since the nondifferentiability appears as a universal property of the complex systems, it is necessary to construct a nondifferentiable physics. In such conjecture, by considering that the complexity of the interactions processes is replaced by nondifferentiability, it is no longer necessary to use the whole classical “arsenal” of quantities from the standard physics (differentiable physics) [12–14, 17, 18].

This topic was developed in [12–14, 19–23] using the scale relativity theory (SRT) [12–14]. In the framework of SRT we assume that the movements of complex system entities take place on continuous but nondifferentiable curves (fractal curves) so that all physical phenomena involved in the dynamics depend not only on the space-time coordinates but also on the space-time scales resolution. From such a perspective, the physical quantities that describe the dynamics of complex systems may be considered fractal functions [12–14]. Moreover, the entities of the complex system may be reduced to and identified with their own trajectories, so that the complex system will behave as a special interactionless “fluid” by means of its geodesics in a nondifferentiable (fractal) space.

In the present paper, we propose a new topic to analyse the complex systems dynamics using SRT. Considering that the entities of the complex system are moving on continuous but nondifferentiable curves, we show that the control of different behaviours of these systems implies nondifferentiability.

2. Geodesics Equations

Considering that the dynamics of the complex system entities take place on continuous but nondifferentiable curves, that is, fractal curves (e.g., the Koch curve, the Peano curve, or the Weierstrass curve [11–14, 17, 18]), they are given by the fractal operator 𝑑/𝑑𝑡 (for details see Appendices A and B): 𝑑=𝜕𝑑𝑡+𝜆𝜕𝑡𝐕⋅∇−𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ, (1) where 𝐕=𝐕_𝐷−𝑖𝐕_𝐹 (2) is the complex velocity, 𝐕_𝐷 is the differentiable and resolution scale independent velocity, 𝐕_𝐹 is the nondifferentiable and resolution scale dependent velocity, 𝐕⋅∇ is the convective term, 𝜆²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝜆Δ=²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}×𝜕²𝜕𝑥²+𝜕²𝜕𝑦²+𝜕²𝜕𝑧² (3) is the dissipative term, 𝛿𝑡/𝜏 is the scale resolution identified here with 𝑑𝑡/𝜏 by substitution principle [12–14], 𝐷_𝐹 is the fractal dimension of the movement curves, 𝜆 is the reference length scale, 𝜏 is the reference time scale, and 𝜆²/𝜏≡𝐷 is the Nottale coefficient specific to fractal-nonfractal transition [12–14]. In the case of fractal dimension 𝐷_𝐹, we can use any definition (the Hausdorff-Besicovitch fractal dimension, the Kolmogorov fractal dimension, etc. [11]), but once such definition is accepted, it has to be constant over the entire complex system dynamics analysis.

Applying the fractal operator (1) to the complex velocity (2) and accepting a generalized inertial principle (a generalization of Nottale’s principle of scale covariance [12–14]), we obtain the geodesics in the form of the Navier-Stokes-type equations: 𝑑𝐕=𝜕𝐕𝑑𝑡+𝜆𝜕𝑡𝐕∇𝐕−𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ𝐕=0. (4)

Equation (4) shows that at any point of a fractal path, local acceleration, 𝜕_𝑡𝐕, convection, (𝐕∇)𝐕, and dissipation, (𝜆²/𝜏)(𝑑𝑡/𝜏)^{(2/𝐷𝐹)−1}Δ𝐕, is in equilibrium. According to [2–6], the complex system can be assimilated to a “rheological” fractal fluid. The “rheology” of the fractal fluid gives hysteretic properties to the complex system (the complex system has a hysteresis cycle, memory, etc.).

Since the movement of the complex system’s entities lacks interaction, we practically make use of self-convection and self-dissipation type mechanisms.

3. Fractal Hydrodynamics Model

For irrotational motion∇×𝐕=0,∇×𝐕_𝐷=0,∇×𝐕_𝐹=0 (5a)(5b)(5c)we can choose 𝐕 of the form 𝜆𝐕=−2𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}∇ln𝜓 (6) or explicitly, with 𝜓=√𝜌exp(𝑖𝑆),𝜆𝐕=2²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝜆∇𝑆−𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝐕∇ln𝜌,_𝐷𝜆=2²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝐕∇𝑆,_𝐹=𝜆²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}∇ln𝜌, (7a)(7b)(7c)where 𝜌 is an amplitude and 𝑆 a phase. The function 𝜓 by means of ln defines the velocity scalar potential 𝜆Φ=2²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝜆𝑆−𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}ln𝜌. (8)

By substituting ((7a)–(7c)) in (4) and separating the real and imaginary parts, up to an arbitrary phase factor which may be at zero by a suitable choice of the phase 𝜓, we obtain 𝑚₀𝜕𝐕_𝐷+𝐕𝜕𝑡^𝐷⋅∇𝐕_𝐷=−∇𝑄,𝜕𝜌𝜕𝑡+∇𝜌𝐕_𝐷=0 (9)(10) with 𝑄 being the fractal potential 𝑄=−2𝑚_𝑒𝜆⁴𝜏²𝑑𝑡𝜏^{(4/𝐷𝐹)−2}Δ√𝜌𝜌𝑚=−⁰𝐕²_𝐹2−𝑚₀𝜆²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}∇𝐕_𝐹 (11) and 𝑚₀ the rest mass of the complex system’s entity. The first equation (9) is the momentum conservation law, the second equation (10) is the density conservation law, and they define together the fractal hydrodynamics (FH) model. We note that, for Peano-type movements in fractal dimension 𝐷_𝐹=2 and supposing that fractal potential is assimilated with the pressure, the fractal hydrodynamic model becomes the standard hydrodynamic (for details see [19–23]).

The fractal potential (11) comes from the nondifferentiability and must be considered as a kinetic term and not as a potential one. Moreover, the fractal potential (11) can generate a viscosity stress tensor type [22, 23] 𝜎_𝑖𝑙=𝑚₀𝜆⁴𝜏²𝑑𝑡𝜏^{(4/𝐷𝐹)−2}∇_𝑖𝜌∇_𝑙∇𝜌−^𝑖𝜌∇_𝑙𝜌𝜌=𝜂𝜕𝑉_𝐹𝑖𝜕𝑥_𝑙+𝜕𝑉_𝐹𝑙𝜕𝑥_𝑖1,𝜂=2𝑚₀𝜌𝜆²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1} (12) of which divergence is equal to the usual force density associated with 𝑄∇_𝑖𝜎_𝑖𝑙=−𝜌∇_𝑙𝑄. (13)

4. Barotropic-Type Behaviours of the Complex Systems via Nondifferentiability

For a barotropic-type behaviour of the complex system [24], 𝜎_𝑖𝑙=𝜎𝛿_𝑖𝑙=𝜌𝑐²𝛿_𝑖𝑙 (14) with 𝑐 being the critical velocity (for details see [24]); the FH equations become𝜕𝐕_𝐷+𝐕𝜕𝑡^𝐷⋅∇𝐕_𝐷𝑐=−²𝜌∇𝜌,𝜕𝜌𝜕𝑡+∇𝜌𝐕_𝐷=0. (15a)(15b)

In the following, using (15a) and (15b) in a plane symmetry, we analyze the complex fluid dynamics. The presence of an external field is specified by adequate initial and boundary conditions (e.g., spatial-temporal Gaussian). In this situation, let us introduce the normalized coordinates𝜔𝑡=𝜏₀,𝑉𝑘𝑥=𝜉,𝑘𝑦=𝜂,_𝑥𝑘𝜔=𝑉_𝜉,𝑉_𝑦𝑘𝜔=𝑉_𝜂,𝜌𝜌₀=𝑁, (16a)(16b)(16c)(16d)(16e)(16f)where 𝜔, 𝑘, and 𝜌₀ are critical parameters of the complex fluid (in [19, 25–27] the complex fluid is identified with a laser produced plasma; in such a context 𝜔 is the plasma pulsation, 𝜌₀ is the density of plasma at thermodynamic equilibrium, 𝑘 is the inverse of the Debye length, and 𝑐 is the ion acoustic speed).

Then (15a) and (15b) become𝜕𝜕𝜏₀𝑁𝑉_𝜉+𝜕𝜕𝜉𝑁𝑉²_𝜉+𝜕𝜕𝜂𝑁𝑉_𝜉𝑉_𝜂=−𝑁⁻¹𝜕𝑁,𝜕𝜕𝜉𝜕𝜏₀𝑁𝑉_𝜂+𝜕𝜕𝜉𝑁𝑉_𝜉𝑉_𝜂+𝜕𝜕𝜂𝑁𝑉²_𝜂=−𝑁⁻¹𝜕𝑁,𝜕𝜂𝜕𝑁𝜕𝜏₀+𝜕𝜕𝜉𝑁𝑉_𝜉+𝜕𝜕𝜂𝑁𝑉_𝜂=0. (17a)(17b)(17c)

For the numerical integration we will impose the initial conditions𝑉_𝜉𝑉0,𝜉,𝜂=0,_𝜂𝑁=10,𝜉,𝜂=0,0,𝜉,𝜂5,1≤𝜉≤2,0≤𝜂≤1 (18a)(18b)(18c)(18d)(18e)as well as the boundary conditions𝑉_𝜉𝜏₀,1,𝜂=𝑉_𝜉𝜏₀𝑉,2,𝜂=0,_𝜂𝜏₀,1,𝜂=𝑉_𝜂𝜏₀𝑉,2,𝜂=0,_𝜉𝜏₀,𝜉,0=𝑉_𝜉𝜏₀𝑉,𝜉,1=0,_𝜂𝜏₀,𝜉,0=𝑉_𝜂𝜏₀𝑁𝜏,𝜉,1=0,₀,1,𝜂=𝑁𝜏₀=1,2,𝜂5,𝑁𝜏₀=1,𝜉,0−𝜏10exp⁰−1/51/5²exp−𝜉−3/21/5²,𝑁𝜏₀=1,𝜉,15. (19a)(19b)(19c)(19d)(19e)(19f)(19g)

By using the finite differences method [28], the system ((17a)–(17c)) with the initial conditions ((18a)–(18e)) and the boundary ones ((19a)–(19g)) was numerically resolved.

In Figures 1(a), 1(b), and 1(c), three-dimensional dependences of the normalized density 𝑁, normalized velocities, 𝑉_𝜉 and 𝑉_𝜂, on the normalized coordinates, 𝜉 and 𝜂, are given for the normalized time 𝜏₀=0.54. Also in Figures 1(d), 1(e), and 1(f) the two-dimensional contours of the normalized density 𝑁, and normalized velocities, 𝑉_𝜉 and 𝑉_𝜂, are given for the same normalized time. The followings result in (i) the generation of two complex fluid structures; (ii) the symmetry of the normalized velocity, 𝑉_𝜉, with respect to symmetry axis of the spatial-temporal Gaussian; and (iii) vertices at the complex fluid periphery for the normalized velocity field, 𝑉_𝜂. These results are in agreement with the experimental data on the behaviours of a laser produced aluminium plasma [19, 26, 27, 29–33].

Figure 1 

Three-dimensional dependences of the normalized density 𝑁 and normalized velocities, 𝑉𝜉 and 𝑉𝜂, on the normalized coordinates, 𝜉 and 𝜂, for the normalized time 𝜏0=0.54 ((a), (b), and (c)); two-dimensional contour of the normalized density 𝑁 and normalized velocities, 𝑉𝜉 and 𝑉𝜂, for the same normalized time ((d), (e), and (f)).

Starting from these numerical solutions we can build the normalized current density 𝑗𝜉,𝜂,𝜏₀=𝑁𝜉,𝜂,𝜏₀𝑉²_𝜉𝜉,𝜂,𝜏₀+𝑉²_𝜉𝜉,𝜂,𝜏₀^1/2. (20) See Figures 2(a)–2(c).

(a)

(b)

(c)

Figure 2 

Three-dimensional dependences of the normalized density current 𝑗, on the normalized coordinates, 𝜉 and 𝜂, for the normalized time 𝜏0=0.54 (a); two-dimensional contour of the normalized current density 𝑗 for the same normalized time (b); time evolution of the normalized density current for 𝜉=0.5 and 𝜂=0.4 (c).

It results in the splitting of the complex fluid into two structures. By assimilating the complex fluid with a laser produced aluminium plasma, the previous theoretical results are in agreement with the experimental data [32, 33].

5. Self-Similarity Behaviour of the Complex Systems via Nondifferentiability

Neglecting both convection 𝐕⋅∇𝐕 and dispersion (𝜆³/𝜏)(𝑑𝑡/𝜏)^{(3/𝐷𝐹)−1}∇³𝐕, (4) takes the form 𝑑𝐕=𝜕𝐕𝑑𝑡𝜆𝜕𝑡−𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ𝐕=0 (21) or, by separating the resolution scales, 𝜕𝐕_𝐷−𝜆𝜕𝑡𝜏²𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ𝐕_𝐹=0 (22) for the differentiable scale and 𝜕𝐕_𝐹−𝜆𝜕𝑡𝜏²𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ𝐕_𝐷=0 (23) for fractal scale. The velocity fields are totally separated, first applying (22) and (23) to the Δ operator, that is, 𝜕𝜕𝑡Δ𝐕_𝐷−𝜆²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ²𝐕_𝐹𝜕=0,𝜕𝑡Δ𝐕_𝐹+𝜆²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ²𝐕_𝐷=0, (24) then, substituting in the transformed equations the dissipative terms by using (22) and (23). It results in the Kirchhoff-type equations 𝜕²𝜕𝑡²+𝜆⁴𝜏²𝑑𝑡𝜏^{(4/𝐷𝐹)−2}Δ²𝐕_𝐷𝐕_𝐹=00. (25)

For one-dimensional case, the previous equations with the substitutions𝑥𝐿=𝑡𝜉,𝑇=𝐿𝜏,₄𝑇²=𝜆⁴𝜏²𝑑𝑡𝜏^{(4/𝐷𝐹)−2},𝐕_𝐷;𝐕_𝐹≡𝐾_𝑖𝜉,𝜏,𝑖=1,2 (26a)(26b)(26c)(26d)take the unitary form 𝐿⁴𝜕⁴𝐾_𝑖𝜕𝜉⁴+𝑇²𝜕²𝐾_𝑖𝜕𝜏²=0. (27)

On (27), we impose for “clamping” conditions at 𝜉=1𝜕²𝐾_𝑖(1,𝜏)𝜕𝜉²𝜕=0,₃𝐾_𝑖(1,𝜏)𝜕𝜉³=0 (28a)(28b)

and for boundary conditions at 𝜉=0𝐾_𝑖(0,𝜏)𝜕=0,𝐾_𝑖(0,𝜏)𝜕𝜉=0. (29a)(29b)

These four boundary conditions in 𝜉 associated with the two initial conditions𝐾_𝑖𝜉,0=𝐾_𝑖0,𝜕𝐾_𝑖𝜉,0𝜕𝜏=0 (30a)(30b)imply a unique solution 𝐾_𝑖(𝜉,𝜏) to (27); see Figure 3.

Figure 3 

Numerical solution of the Kirchhoff equation (27) with “clamped-free” unitary conditions, for a uniform initial 𝐾𝑖0. 𝐾𝑖(0,𝜏) relaxes to zero within the first few time steps.

Owing to the scaling 𝑥∼𝐿𝑡/𝑇, we seek a solution of (27) in the form 𝐾_𝑖𝜉,𝜏=𝐾_𝑖0𝑢_𝑖𝜂, (31) where the self-similarity variable is 𝜂=𝜉√𝜏=(𝑥/𝐿)=𝑥𝑡/𝑇.𝛾𝑡 (32)

The boundary conditions for 𝑢(𝜂) are derived from those for 𝐾_𝑖:𝑢_𝑖(0)𝑢=0,_′𝑖(0)𝑢=0,_𝑖(+∞)⟶1. (33a)(33b)(33c)

Substituting this self-similar form of 𝐾_𝑖(𝜉,𝜏) into (27) yields the following equation for the self-similar solution 𝑢_𝑖(𝜂): 4𝑑⁴𝑢_𝑖𝜂𝑑𝜂⁴+𝜂²𝑑²𝑢_𝑖𝜂𝑑𝜂²+3𝜂𝑑𝑢_𝑖𝜂𝑑𝜂=0. (34)

Imposing that 𝑢_𝑖(𝜂) matches the initial condition for large 𝜂 implies that 𝑑²𝑢_𝑖(0)𝑑𝜂²=0, (35) as shown with the help of an integral of motion. This last condition, combined with the previous ones, yields a unique self-similar solution to (34): 2𝐾_𝑖𝜉,𝜏=2𝐾_𝑖0𝜂√,2𝜋 (36) where we have introduced the Fresnel sine integral, 𝑆(𝑥)=∫^𝑥₀sin(𝜋𝑦²/2)𝑑𝑦, also arising in diffraction theory.

Equation (36) describes a self-similar solution. This reflects the dispersive nature of (27), see Figure 4.

Figure 4 

The self-similar solution (36) is a function of 𝜂. At later times, 𝜏∼1, reflections are generated from the “clamped” end 𝜉=1.

In these conditions the phase 𝑆 associated with the velocity scalar potential is presented in Figure 5.

Figure 5 

The phase dependence versus normalized space-time coordinates.

The states density associated with the same velocity scalar potential has the same behaviour as the one presented in Figure 5. It results in a self-similarity behaviour.

The theoretical oscillations (obtained by means of self-elimination of the dissipative terms between differentiable and fractal scales) can explain the experimental ionic oscillations of the current density that take place in a laser produced aluminium plasma [19, 26, 27, 29]. In Figure 6 we give by comparison the experimental curve (red curve, which reflects the ionic oscillations of the current density with time that take place in a laser produced aluminium plasma at a laser power of 40 mJ/pulse) and the theoretical curve (blue curve). We note that in Figure 6 we used the normalized coordinates.

Figure 6 

The theoretical (blue) and experimental (red) curves of the ionic oscillations of the normalized current density, 𝐽, with normalized time, 𝜏, in a laser produced aluminium plasma at a laser power of 40 mJ/pulse.

It results in a good agreement (correlation factor 0.81) between theoretical model and the experimental data.

6. Chaoticity through Turbulence and Stochasticization via Nondifferentiability

Through the fractal velocities field, 𝐕_𝐹, the specific fractal potential 𝑄 is a measure of nondifferentiability of the complex system particle trajectories, that is, of their chaoticity.

Since the position vector of the particle is assimilated with a stochastic process of Wiener type (for details see [11–13], 𝜓 is not only the scalar potential of a complex speed (through ln𝜓) in the frame of fractal hydrodynamics but also density of probability (through |𝜓|²) in the frame of a Schrödinger-type theory.

It results in the equivalence between the formalism of the fractal hydrodynamics and the one of the Schrödinger-type equation. Moreover, the chaoticity, either through turbulence in the fractal hydrodynamics approach or through stochasticization in the Schrödinger-type approach, is generated only by the nondifferentiability of the movement trajectories in a fractal space. In this way the nondifferentiability becomes a “control parameter” of the complex system dynamics.

6.1. Full and Fractional Speed Scalar Potential Revivals in the Infinite Square Well: Various Criteria of Evolution to Chaos

6.1.1. Infinite Square Well System

Let us consider that the external perturbation applied to complex system simulates, in our opinion, one-dimension square well system. After solving the time-dependent Schrödinger-type equation according to the method described in [23] we obtain the discrete eigenvalues 𝐸_𝑛=2𝑚₀𝔇²𝑛𝜋𝑎²𝜆,𝔇=²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1} (37) and the eigenfunctions 𝜙_𝑛=2𝑎^1/2sin𝑛𝜋𝑥𝑎|𝑥|≤𝑎,𝑛even2,2𝑎^1/2cos𝑛𝜋𝑥𝑎|𝑥|≤𝑎,𝑛odd2, (38) where 𝑎 is the well’s width and 𝑚₀ is the rest mass of the fractal fluid particle.

6.1.2. Time Scales

Some time scales of a speed potential evolution are contained in the coefficients of the Taylor series of the quantized energy levels 𝐸_𝑛 around the main energy 𝐸_𝑛 (see method from [34]) 𝐸_𝑛=𝐸_𝑛+4𝜋𝑚₀𝔇𝑛−𝑛𝑇_𝛼+(𝑛−𝑛)²𝑇_𝛽+⋯, (39) where often the zero of energy is shifted to remove the 𝐸_𝑛 term. Regrouping the infinite square well energies (17a)–(17c) in this form gives 𝐸_𝑛=𝐸₁𝑛²=𝐸₁𝑛²+2𝐸₁𝑛(𝑛−𝑛)+𝐸₁(𝑛−𝑛)². (40) And comparing (39) and (40) we relate 𝑇_𝛼=2𝜋𝑚₀𝔇𝑛𝐸₁,𝑇_𝛽=4𝜋𝑚₀𝔇𝐸₁. (41)

We note that the time scale 𝑇_𝛽 does not depend on the mean energy level 𝑛. This will provide us with a “universal” time scale for describing speed potential evolution that does not depend on the particle average energy.

6.1.3. Time Evolution

We write the particle’s time 𝑡=0 speed scalar potential in the infinite square well as 𝜓(𝑥,𝑡=0)=𝜓_𝑖(𝑥). (42)

We expand this speed scalar potential using the energy eigenstate basis 𝜓_𝑖(𝑥)=^∞_𝑛=1𝑐_𝑛𝜙_𝑛(𝑥) (43) with 𝑐_𝑛=^+∞_−∞𝜙_𝑛(𝑥)𝜓_𝑖(𝑥)𝑑𝑥. (44)

Using the time scale 𝑇_𝛽, the time evolution in the energy eigenstate basis was found from Schrödinger-type equation to be 𝜓(𝑥,𝑡)=_𝑛𝑡exp−2𝜋𝑖𝑇_𝛽𝑛²𝑐_𝑛𝜙_𝑛(𝑥). (45)

6.1.4. Simulation of the Evolution to Chaos Criteria

Now, the full and fractional revivals formalism may be applied. Full and fractional revivals of a speed scalar potential in the infinite square well occur when a speed scalar potential evolves in time to a state that can be described as a collection of spatially distributed subspeed scalar potentials that each closely reproduces the initial speed scalar potential shape; see for details [34]. Therefore, the full and fractional revivals of a speed scalar potential in the infinite square well impliy either 𝜓𝑥,𝑡=𝑡₀+2^𝑘𝑇_𝛽=𝜓𝑥,𝑡=𝑡₀ (46) or 𝜓𝑥,𝑡₀+𝑝𝑞𝑇_𝛽=𝜓𝑥,𝑡=𝑡₀ (47) for any time 𝑡₀ and 𝑘, 𝑝, and 𝑞 integers. In any of the situations above, either for 𝑡=𝑇_𝐹=2^𝑘𝑇_𝛽 or for 𝑡=𝑇_SH=(𝑝/𝑞)𝑇_𝛽, we can introduce Reynolds-type criterions Re_𝐹=𝑉_𝐹𝐿_𝐹𝜈_𝐹=2^𝑘,Re_SH=𝑉_SH𝐿_SH𝜈_SH=𝑝𝑞, (48) where𝐸₁≡𝐸_𝐹/SH=12𝑚₀𝑉²_𝐹/SH,𝐿_𝐹/SH=𝑉_𝐹/SH𝑇_𝐹/SH,𝜈_𝐹/SH𝜆=8𝜋²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1} (49a)(49b)(49c)have the usual signification from fluid mechanics [24]. Up to the values Re^𝑐_𝐹/SH the fractal fluids become turbulent. Then through 𝑇_𝐹/𝑇_𝛽=2^𝑘 and Re_𝐹=𝑉_𝐹𝐿_𝐹/𝜈_𝐹=2^𝑘 it is formally simulated through the criterion of evolution to chaos via Feigenbaum scenario (cascade of period doubling bifurcations), while through 𝑇_𝛽/𝑇_SH=𝜔_SH/𝜔_𝛽=𝑞/𝑝 with 𝑝>𝑞 and Re_SH=𝑉_SH𝐿_SH/𝜈_SH=𝑝/𝑞 the criterion of evolution to chaos via a cascade of subharmonic bifurcations.

We admit that in any of these two situations mentioned above the fractal velocity (7c) is null, since 𝜌=|𝜓|²=const; meanwhile the differential velocity (7b) is not zero, since the phase 𝑆 is not constant, with the increase of the systems phase incoherence being associated with the increase in turbulence of fractal fluid.

We note that in the standard model (Landau’s scenario [24, 35]) the Fourier spectrum is always discrete and cannot approximate a continuum spectrum than that in case of a large number of frequencies that will generate an unlimited number of spectral components as a result of their beats which appear thanks to the presence of nonlinearities in the system. Yet, considering standard model, the flow can never be truly chaotic, because, in case of multiple periodic functions, correlations tend to be not null but having an oscillating character. Therefore, Landau’s scenario can describe transition towards chaotic behaviour only in a system with an infinite number of degrees of freedom, such as a fluid. In our case, because the Reynolds numbers (48) present scale dependencies (49c), when 𝑑𝑡→0 for 𝐷_𝐹≤2, the fractals physical values that describe the dynamics of the system are no longer defined. So, in this approximation, a simulation of a system with an infinite number of degrees of freedom is used.

We note that the results from the present paragraph permit the chaoticity and self-structuring control by means of nondifferentiability. Thus the experimental results on the formation, dynamics, and evolution towards chaos of complex space charge structures that emerge in front of a positively biased electrode immersed in a quiescent plasma from [20] are in agreement with our previous theoretical results.

7. Conclusions

In the present paper, we propose a new topic in the control of complex systems dynamics using the nondifferentiability of complex system movement curves. This topic was developed through the scale relativity approach. Considering the dynamics of complex system entities that take place on fractal curves, we show that the control of different behaviours of these systems implies nondifferentiability. The main conclusions of the present paper are as follows: (i) the geodesics equations in the form of a Navier-Stokes-type equation are obtained. It results in the rheologic properties of the complex system which implies memory and so forth; (ii) considering that the fractal fluid flows are irrotational, fractal hydrodynamics equations are obtained. These equations are formed from the density and momentum conservations laws, wih the presence of nondifferentiability being induced by the fractal potential; (iii) a barotropic-type behaviour of the complex system is numerically simulated using FH equations. In this way we highlight the self-multiplication mechanism of the substructures that constitute the complex system; (iv) the implications of self-similarity in the dynamics of the complex system are presented; (v) the chaoticity through turbulence and stochaticization via nondifferentiability are obtained; (vi) these previous behaviours can simulate the standard properties of the complex system.

In this way, standard properties of complex systems such as emergence, self-organisation, and adaptability are controlled through nondifferentiability of motion curves of the sub-systems that compose the complex system. We note that general aspects of the dynamics control of the complex system are described in [36], while the concrete cases of this control are presented in [37–39].

Appendices

A. Consequences of Nondifferentiability

The nondifferentiability implies the following [12–14, 19–22].

(i) A continuous and a nondifferentiable curve (or almost nowhere differentiable) is explicitly scale dependent. This means that its length tends to infinity, when the scale interval tends to zero. Therefore, a continuous and nondifferentiable space is fractal, in the general meaning given by Mandelbrot to this concept [11].

(ii) Physical quantities will be expressed through fractal functions, namely, through functions that are dependent both on spatial and temporal coordinates and on resolution scale. The invariance of the physical quantities in relation with the resolution scale generates special types of transformations, called resolution scale transformations. Particularly, the differentiality of the generalized spatial coordinates, 𝐗, takes the form 𝑑𝐗=𝑑𝐱+𝑑𝝃, (A.1) where 𝑑𝐱 is the classical differential element and 𝑑𝝃 is a differential fractal one.

(iii) There is infinity of fractal curves (geodesics) relating to any couple of points (or starting from any point) and applied for any scale. The phenomenon can be easily understood at the level of fractal surfaces, which, in their turn, can be described in terms of fractal distribution of conic points of positive and negative infinite curvatures. As a consequence, we have replaced velocity on a particular geodesic by fractal velocity field of the whole infinite ensemble of geodesics. This representation is similar to that of fluid mechanics where the motion of the fluid is described in terms of its velocity field, density, and pressure. We will, indeed, recover the fundamental equations of fluid mechanics (Euler and continuity equations), but we will write them in terms of a density of probability (as defined by the set of geodesics) instead of a density of matter and adding an additional term of quantum pressure (the expression of fractal geometry).

(iv) The local differential time invariance is broken, so the time-derivative of the fractal field 𝑄 can be written twofold: 𝑑₊𝑄𝑑𝑡=lim_Δ𝑡⟶0+𝑄(𝑡+Δ𝑡)−𝑄(𝑡),𝑑Δ𝑡₋𝑄𝑑𝑡=lim_{Δ𝑡⟶0−}𝑄(𝑡)−𝑄(𝑡−Δ𝑡).Δ𝑡 (A.2)

Both definitions are equivalent in the differentiable case 𝑑𝑡→−𝑑𝑡. In the nondifferentiable situation, these definitions are no longer valid, since limits are not defined anymore. Fractal theory defines physics in relationship with the function behaviour during the “zoom” operation on the time resolution 𝛿𝑡, which is here identified with the differential element 𝑑𝑡 (substitution principle), which is considered an independent variable. The standard field 𝑄(𝑡) is therefore replaced by fractal field 𝑄(𝑡,𝑑𝑡), explicitly dependent on time resolution interval, whose derivative is not defined at the unnoticeable limit 𝑑𝑡→0. As a consequence, this leads to the two derivatives of the fractal field 𝑄 as explicit functions of the two variables 𝑡 and 𝑑𝑡𝑑₊𝑄𝑑𝑡=lim_Δ𝑡⟶0+𝑄(𝑡+Δ𝑡,Δ𝑡)−𝑄(𝑡,Δ𝑡),𝑑Δ𝑡₋𝑄𝑑𝑡=lim_{Δ𝑡⟶0−}𝑄(𝑡,Δ𝑡)−𝑄(𝑡−Δ𝑡,Δ𝑡).Δ𝑡 (A.3)

Notation “+” corresponds to the forward process, while “−” to the backward one.

Let us particularize (A.3) for the spatial coordinates. 𝑡 results in 𝑑_±𝐗=𝑑_±𝐱+𝑑_±𝝃. (A.4)

Since, according to [12–14, 19–22], we can write 𝑑_±𝑋^𝑖=𝑑_±𝑥^𝑖 (A.5) from (A.4) by averaging, it results in 𝑑𝜉^𝑖_±=0. (A.6)

(v) The differential fractal part satisfies the fractal equation: 𝑑_±𝜉^𝑖=𝜆^𝑖_±𝑑𝑡𝜏^1/𝐷𝐹, (A.7) where 𝜆^𝑖_± are some constant coefficients, 𝑑𝑡 is the time differential, 𝜏 is the reference time scale, and 𝐷_𝐹 is a constant fractal dimension. We note that the use of any Kolmogorov or Hausdorff definition [12–14, 19–22] can be accepted for fractal dimension, but once a certain definition is admitted, it should be used until the end of analyzed dynamics.

(vi) The local differential time reflection invariance is recovered by combining the two derivatives, 𝑑₊/𝑑𝑡 and 𝑑₋/𝑑𝑡, in the complex operator: 𝑑=1𝑑𝑡2𝑑₊+𝑑₋−𝑖𝑑𝑡2𝑑₊−𝑑₋.𝑑𝑡 (A.8)

Applying this operator to the “position vector,” a complex velocity yields 𝐕=𝑑𝐗=1𝑑𝑡2𝑑₊𝐗+𝑑₋𝐗−𝑖𝑑𝑡2𝑑₊𝐗−𝑑₋𝐗=𝐕𝑑𝑡₊+𝐕₋2𝐕−𝑖⁺−𝐕₋2=𝐕_𝐷−𝑖𝐕_𝐹 (A.9) with 𝐕_𝐷=𝐕₊+𝐕₋2,𝐕_𝐹=𝐕₊−𝐕₋2. (A.10)

The real part, 𝐕_𝐷, of the complex velocity 𝐕 represents the standard classical velocity, which does not depend on resolution, while the imaginary part, 𝐕_𝐹, is a new quantity coming from fractality.

B. Covariant Derivative

Let us now assume that curves describing movements of complex systems (continuous but nondifferentiable) are immersed in a 3-dimensional space and that 𝐗 of components 𝑋^𝑖(𝑖=1,3) is the position vector of a point on the curve. Let us also consider the fractal field 𝑄(𝐗,𝑡) and expand its total differential up to the second order: 𝑑_±𝑄=𝜕𝑄𝜕𝑡𝑑𝑡+∇𝑄⋅𝑑_±1𝐗+2𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗𝑑_±𝑋^𝑖𝑑_±𝑋^𝑗. (B.1)

Relations (B.1) are valid in any point both for the spatial manifold and for the points 𝐗 on the fractal curve selected in relations (B.1). Hence, the forward and backward average values of these relations take the form 𝑑_±𝑄=𝜕𝑄+𝜕𝑡𝑑𝑡∇𝑄⋅𝑑_±𝐗+12𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗𝑑_±𝑋^𝑖𝑑_±𝑋^𝑗. (B.2)

The following aspects should be mentioned: the mean value of function 𝑄 and its derivatives coincide with themselves, and the differentials 𝑑_±𝑋^𝑖 and 𝑑𝑡 are independent; therefore, the average of their products coincides with the product of averages. Consequently, (B.2) becomes 𝑑_±𝑄=𝜕𝑄𝑑𝜕𝑡𝑑𝑡+∇𝑄_±𝐗+12𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗𝑑_±𝑋^𝑖𝑑_±𝑋^𝑗 (B.3) or more, using (A.4) and (A.6), 𝑑_±𝑄=𝜕𝑄𝜕𝑡𝑑𝑡+∇𝑄⋅𝑑_±1𝐗+2𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗×𝑑_±𝑋^𝑖𝑑_±𝑋^𝑗+𝑑_±𝜉^𝑖𝑑_±𝜉^𝑗. (B.4)

Even if the average value of the fractal coordinate 𝑑_±𝜉^𝑖 is null (see (A.6)), for higher order of fractal coordinate average, the situation can still be different. Let us focus on the averages ⟨𝑑_±𝜉^𝑖𝑑_±𝜉^𝑗⟩. If 𝑖≠𝑗, these averages are zero due to the independence of 𝑑_±𝜉^𝑖 and 𝑑_±𝜉^𝑗. So, using (A.7), we can write 𝑑_±𝜉^𝑖𝑑_±𝜉^𝑗=𝜆^𝑖_±𝜆^𝑗_±𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝑑𝑡𝜏. (B.5)

Then, (B.4) may be written as follows: 𝑑_±𝑄=𝜕𝑄𝜕𝑡𝑑𝑡+𝑑_±1𝐱⋅∇𝑄+2𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗𝑑_±𝑥^𝑖𝑑_±𝑥^𝑗+12𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗𝜆^𝑖₊𝜆^𝑗₊𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝑑𝑡𝜏. (B.6)

If we divide by 𝑑𝑡 and neglect the terms containing differential factors, (B.6) is reduced to 𝑑_±𝑄=𝑑𝑡𝜕𝑄𝜕𝑡+𝐕_±1∇𝑄+2𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗𝜆^𝑖₋𝜆^𝑗₋𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}. (B.7)

These relations also allow us to define the operators: 𝑑_±=𝜕𝑑𝑡𝜕𝑡+𝐕_±𝜆⋅∇+^𝑖+𝜆^𝑗₊2𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}𝜕²𝜕𝑋^𝑖𝜕𝑋^𝑗. (B.8)

Under these circumstances, let us calculate 𝑑𝑄/𝑑𝑡. Taking into account (B.8), (A.8), and (A.9), we will obtain 𝑑𝑄=1𝑑𝑡2𝑑₊𝑄+𝑑𝑑𝑡⁻𝑄𝑑𝑑𝑡−𝑖⁺𝑄−𝑑𝑑𝑡⁻𝑄=𝑑𝑡𝜕𝑄+()𝜕𝑡𝐕⋅∇+𝑑𝑡/𝜏^{(2/𝐷𝐹)−1}⋅𝜆4𝜏_𝑖+𝜆^𝑗₊+𝜆^𝑖₋𝜆^𝑗₋−𝑖𝜆^𝑖₊𝜆^𝑗₊−𝜆^𝑖₋𝜆^𝑗₋𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗. (B.9)

This relation also allows us to define the fractal operator: 𝑑=𝜕𝑑𝑡+()𝜕𝑡𝐕⋅∇+𝑑𝑡/𝜏^{(2/𝐷𝐹)−1}⋅𝜆4𝜏_𝑖+𝜆^𝑗₊+𝜆^𝑖₋𝜆^𝑗₋−𝑖𝜆^𝑖₊𝜆^𝑗₊−𝜆^𝑖₋𝜆^𝑗₋𝜕²𝑄𝜕𝑋^𝑖𝜕𝑋^𝑗. (B.10)

Particularly, by choosing 𝜆^𝑖₊𝜆^𝑗₊=−𝜆^𝑖₋𝜆^𝑗₋=2𝜆²𝛿^𝑖𝑗, (B.11) where 𝜆 is the reference length scale, the fractal operator (B.10) (covariant derivative) takes the usual form: 𝑑=𝜕𝑑𝑡+𝜆𝜕𝑡𝐕⋅∇−𝑖²𝜏𝑑𝑡𝜏^{(2/𝐷𝐹)−1}Δ. (B.12)