# Dynamics Control of the Complex Systems via Nondifferentiability

Authors: Carmen Nejneru; Anca Nicuţă; Boris Constantin; Liliana Rozemarie Manea; Mirela Teodorescu; Maricel Agop
Journal: Journal of Applied Mathematics
 (2013)

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## 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.).


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## Body

## 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].Theemergence 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 isadaptability. 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):
(1)d^dt=∂∂t+V^·∇-iλ2τ(dtτ)(2/DF)-1Δ,
where
(2)V^=VD-iVF
is the complex velocity, VD is the differentiable and resolution scale independent velocity, VF is the nondifferentiable and resolution scale dependent velocity, V^·∇ is the convective term,
(3)(λ2τ)(dtτ)(2/DF)-1Δ=(λ2τ)(dtτ)(2/DF)-1×[(∂2∂x2)+(∂2∂y2)+(∂2∂z2)]
is the dissipative term, δt/τ is the scale resolution identified here with dt/τ by substitution principle [12–14], DF 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 DF, 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:
(4)d^V^dt=∂V^∂t+(V^∇)V^-iλ2τ(dtτ)(2/DF)-1ΔV^=0.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)-1ΔV^, 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(5a)∇×V^=0,(5b)∇×VD=0,(5c)∇×VF=0we can choose V^ of the form
(6)V^=-2iλ2τ(dtτ)(2/DF)-1∇lnψ
or explicitly, with ψ=ρexp(iS),(7a)V^=2λ2τ(dtτ)(2/DF)-1∇S-iλ2τ(dtτ)(2/DF)-1∇lnρ,(7b)VD=2λ2τ(dtτ)(2/DF)-1∇S,(7c)VF=λ2τ(dtτ)(2/DF)-1∇lnρ,where ρ is an amplitude and S a phase. The function ψ by means of ln defines the velocity scalar potential
(8)Φ=2λ2τ(dtτ)(2/DF)-1S-iλ2τ(dtτ)(2/DF)-1lnρ.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
(9)m0[∂VD∂t+(VD·∇)VD]=-∇Q,(10)∂ρ∂t+∇(ρVD)=0
with Q being  the fractal potential
(11)Q=-2meλ4τ2(dtτ)(4/DF)-2Δρρ=-m0VF22-m0λ2τ(dtτ)(2/DF)-1∇VF
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 DF=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]
(12)σ^il=m0λ4τ2(dtτ)(4/DF)-2[∇iρ∇lρ-(∇iρ∇lρρ)]=η(∂VFi∂xl+∂VFl∂xi),η=12m0ρλ2τ(dtτ)(2/DF)-1
of which divergence is equal to the usual force density associated with Q(13)∇iσ^il=-ρ∇lQ.


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

For a barotropic-type behaviour of the complex system [24],
(14)σ^il=σδil=ρc2δil
with c  being  the critical velocity (for details see [24]); the FH equations become(15a)[∂VD∂t+(VD·∇)VD]=-c2ρ∇ρ,(15b)∂ρ∂t+∇(ρVD)=0.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 coordinates(16a)ωt=τ0,(16b)kx=ξ,(16c)ky=η,(16d)Vxkω=Vξ,(16e)Vykω=Vη,(16f)ρρ0=N,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(17a)∂∂τ0(NVξ)+∂∂ξ(NVξ2)+∂∂η(NVξVη)=-N-1∂N∂ξ,(17b)∂∂τ0(NVη)+∂∂ξ(NVξVη)+∂∂η(NVη2)=-N-1∂N∂η,(17c)∂N∂τ0+∂∂ξ(NVξ)+∂∂η(NVη)=0.For the numerical integration we will impose the initial conditions(18a)Vξ(0,ξ,η)=0,(18b)Vη(0,ξ,η)=0,(18c)N(0,ξ,η)=15,(18d)1≤ξ≤2,(18e)0≤η≤1as well as the boundary conditions(19a)Vξ(τ0,1,η)=Vξ(τ0,2,η)=0,(19b)Vη(τ0,1,η)=Vη(τ0,2,η)=0,(19c)Vξ(τ0,ξ,0)=Vξ(τ0,ξ,1)=0,(19d)Vη(τ0,ξ,0)=Vη(τ0,ξ,1)=0,(19e)N(τ0,1,η)=N(τ0,2,η)=15,(19f)N(τ0,ξ,0)=110exp[-(τ0-1/51/5)2]exp[-(ξ-3/21/5)2],(19g)N(τ0,ξ,1)=15.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 Figures1(a), 1(b), and 1(c), three-dimensional dependences of the normalized density N, normalized velocities, Vξ and Vη, 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, Vξ and Vη, 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, Vξ, with respect to symmetry axis of the spatial-temporal  Gaussian;  and (iii) vertices at the complex fluid periphery for the normalized velocity field, Vη. 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 densityN 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(20)j(ξ,η,τ0)=N(ξ,η,τ0)[Vξ2(ξ,η,τ0)+Vξ2(ξ,η,τ0)]1/2.
See Figures 2(a)–2(c).Three-dimensional dependences of the normalized density currentj, 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).
(a)
(b)
(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 convectionV^·∇V^ and dispersion (λ3/τ)(dt/τ)(3/DF)-1∇3V,^ (4) takes the form
(21)dV^dt=∂V^∂t-iλ2τ(dtτ)(2/DF)-1ΔV^=0
or, by separating the resolution scales,
(22)∂VD∂t-λ2τ(dtτ)(2/DF)-1ΔVF=0
for the differentiable scale and
(23)∂VF∂t-λ2τ(dtτ)(2/DF)-1ΔVD=0
for fractal scale.  The velocity fields are totally separated,  first applying (22) and (23) to the Δ operator, that is,
(24)∂∂t(ΔVD)-λ2τ(dtτ)(2/DF)-1Δ2VF=0,∂∂t(ΔVF)+λ2τ(dtτ)(2/DF)-1Δ2VD=0,
then, substituting in the transformed equations the dissipative terms by using (22) and (23). It results in the Kirchhoff-type equations
(25)(∂2∂t2+λ4τ2(dtτ)(4/DF)-2Δ2)(VDVF)=(00).For one-dimensional case, the previous equations with the substitutions(26a)xL=ξ-,(26b)tT=τ-,(26c)L4T2=λ4τ2(dtτ)(4/DF)-2,(26d)(VD;VF)≡K-i(ξ-,τ-),i=1,2take the unitary form
(27)L4∂4K-i∂ξ-4+T2∂2K-i∂τ-2=0.On (27), we impose  for  “clamping” conditions at ξ-=1(28a)∂2K-i(1,τ-)∂ξ-2=0,(28b)∂3K-i(1,τ-)∂ξ-3=0and for boundary conditions atξ-=0(29a)K-i(0,τ-)=0,(29b)∂K-i(0,τ-)∂ξ-=0.These four boundary conditions inξ- associated with the two initial conditions(30a)K-i(ξ-,0)=K-i0,(30b)∂K-i(ξ-,0)∂τ-=0imply 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 K-i0. K-i(0,τ) relaxes to zero within the first few time steps.Owing to the scalingx~Lt/T, we seek a solution of (27) in the form
(31)K-i(ξ-,τ-)=K-i0ui(η-),
where the self-similarity variable is
(32)η-=(ξ-τ-)=(x/L)t/T=xγt.The boundary conditions foru(η-) are derived from those for K-i:(33a)ui(0)=0,(33b)ui′(0)=0,(33c)ui(+∞)⟶1.Substituting this self-similar form ofK-i(ξ-,τ-) into (27) yields the following equation for the self-similar solution ui(η-):
(34)4d4ui(η-)dη-4+η-2d2ui(η-)dη-2+3η-dui(η-)dη-=0.Imposing thatui(η-) matches the initial condition for large η- implies that
(35)d2ui(0)dη-2=0,
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):
(36)2K-i(ξ-,τ-)=2K-i0(η-2π),
where we have introduced the Fresnel sine integral, S(x)=∫0xsin(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 phaseS 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 Figure5. 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,VF, 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
(37)En=2m0𝔇2(nπa)2,𝔇=λ2τ(dtτ)(2/DF)-1
and the eigenfunctions
(38)ϕn={(2a)1/2sin(nπxa),neven|x|≤a2,(2a)1/2cos(nπxa),nodd|x|≤a2,
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 levelsEn around the main energy En- (see method from [34])
(39)En=En-+4πm0𝔇[n-n-Tα+(n-n-)2Tβ+⋯],
where often the zero of energy is shifted to remove the En- term. Regrouping the infinite square well energies (17a)–(17c) in this form gives
(40)En=E1n2=E1n-2+2E1n-(n-n-)+E1(n-n-)2.
And comparing (39) and (40) we relate
(41)Tα=2πm0𝔇n-E1,Tβ=4πm0𝔇E1.We note that the time scaleTβ 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 timet=0 speed scalar potential in the infinite square well as
(42)ψ(x,t=0)=ψi(x).We expand this speed scalar potential using the energy eigenstate basis(43)ψi(x)=∑n=1∞cnϕn(x)
with
(44)cn=∫-∞+∞ϕn(x)ψi(x)dx.Using the time scaleTβ, the time evolution in the energy eigenstate basis was found from Schrödinger-type equation to be
(45)ψ(x,t)=∑nexp[-2πi(tTβ)n2]cnϕn(x).


#### 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
(46)ψ(x,t=t0+2kTβ)=ψ(x,t=t0)
or
(47)ψ(x,t0+pqTβ)=ψ(x,t=t0)
for any time t0 and k, p, and q integers. In any of the situations above, either for t=TF=2kTβ or for t=TSH=(p/q)Tβ, we can introduce Reynolds-type criterions
(48)ReF=VFLFνF=2k,ReSH=VSHLSHνSH=pq,
where(49a)E1≡EF/SH=12m0V2F/SH,(49b)LF/SH=VF/SHTF/SH,(49c)νF/SH=8πλ2τ(dtτ)(2/DF)-1have the usual signification from fluid mechanics [24]. Up to the values ReF/SHc the fractal fluids become turbulent. Then through TF/Tβ=2k and ReF=VFLF/νF=2k it is formally simulated  through  the criterion of evolution to chaos via Feigenbaum scenario (cascade of period doubling bifurcations), while through Tβ/TSH=ωSH/ωβ=q/p with p>q and ReSH=VSHLSH/νSH=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 DF≤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.


## 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
(37)En=2m0𝔇2(nπa)2,𝔇=λ2τ(dtτ)(2/DF)-1
and the eigenfunctions
(38)ϕn={(2a)1/2sin(nπxa),neven|x|≤a2,(2a)1/2cos(nπxa),nodd|x|≤a2,
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 levelsEn around the main energy En- (see method from [34])
(39)En=En-+4πm0𝔇[n-n-Tα+(n-n-)2Tβ+⋯],
where often the zero of energy is shifted to remove the En- term. Regrouping the infinite square well energies (17a)–(17c) in this form gives
(40)En=E1n2=E1n-2+2E1n-(n-n-)+E1(n-n-)2.
And comparing (39) and (40) we relate
(41)Tα=2πm0𝔇n-E1,Tβ=4πm0𝔇E1.We note that the time scaleTβ 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 timet=0 speed scalar potential in the infinite square well as
(42)ψ(x,t=0)=ψi(x).We expand this speed scalar potential using the energy eigenstate basis(43)ψi(x)=∑n=1∞cnϕn(x)
with
(44)cn=∫-∞+∞ϕn(x)ψi(x)dx.Using the time scaleTβ, the time evolution in the energy eigenstate basis was found from Schrödinger-type equation to be
(45)ψ(x,t)=∑nexp[-2πi(tTβ)n2]cnϕn(x).


### 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
(46)ψ(x,t=t0+2kTβ)=ψ(x,t=t0)
or
(47)ψ(x,t0+pqTβ)=ψ(x,t=t0)
for any time t0 and k, p, and q integers. In any of the situations above, either for t=TF=2kTβ or for t=TSH=(p/q)Tβ, we can introduce Reynolds-type criterions
(48)ReF=VFLFνF=2k,ReSH=VSHLSHνSH=pq,
where(49a)E1≡EF/SH=12m0V2F/SH,(49b)LF/SH=VF/SHTF/SH,(49c)νF/SH=8πλ2τ(dtτ)(2/DF)-1have the usual signification from fluid mechanics [24]. Up to the values ReF/SHc the fractal fluids become turbulent. Then through TF/Tβ=2k and ReF=VFLF/νF=2k it is formally simulated  through  the criterion of evolution to chaos via Feigenbaum scenario (cascade of period doubling bifurcations), while through Tβ/TSH=ωSH/ωβ=q/p with p>q and ReSH=VSHLSH/νSH=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 DF≤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.


## 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
(37)En=2m0𝔇2(nπa)2,𝔇=λ2τ(dtτ)(2/DF)-1
and the eigenfunctions
(38)ϕn={(2a)1/2sin(nπxa),neven|x|≤a2,(2a)1/2cos(nπxa),nodd|x|≤a2,
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 levelsEn around the main energy En- (see method from [34])
(39)En=En-+4πm0𝔇[n-n-Tα+(n-n-)2Tβ+⋯],
where often the zero of energy is shifted to remove the En- term. Regrouping the infinite square well energies (17a)–(17c) in this form gives
(40)En=E1n2=E1n-2+2E1n-(n-n-)+E1(n-n-)2.
And comparing (39) and (40) we relate
(41)Tα=2πm0𝔇n-E1,Tβ=4πm0𝔇E1.We note that the time scaleTβ 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 timet=0 speed scalar potential in the infinite square well as
(42)ψ(x,t=0)=ψi(x).We expand this speed scalar potential using the energy eigenstate basis(43)ψi(x)=∑n=1∞cnϕn(x)
with
(44)cn=∫-∞+∞ϕn(x)ψi(x)dx.Using the time scaleTβ, the time evolution in the energy eigenstate basis was found from Schrödinger-type equation to be
(45)ψ(x,t)=∑nexp[-2πi(tTβ)n2]cnϕn(x).


## 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
(46)ψ(x,t=t0+2kTβ)=ψ(x,t=t0)
or
(47)ψ(x,t0+pqTβ)=ψ(x,t=t0)
for any time t0 and k, p, and q integers. In any of the situations above, either for t=TF=2kTβ or for t=TSH=(p/q)Tβ, we can introduce Reynolds-type criterions
(48)ReF=VFLFνF=2k,ReSH=VSHLSHνSH=pq,
where(49a)E1≡EF/SH=12m0V2F/SH,(49b)LF/SH=VF/SHTF/SH,(49c)νF/SH=8πλ2τ(dtτ)(2/DF)-1have the usual signification from fluid mechanics [24]. Up to the values ReF/SHc the fractal fluids become turbulent. Then through TF/Tβ=2k and ReF=VFLF/νF=2k it is formally simulated  through  the criterion of evolution to chaos via Feigenbaum scenario (cascade of period doubling bifurcations), while through Tβ/TSH=ωSH/ωβ=q/p with p>q and ReSH=VSHLSH/νSH=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 DF≤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].



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*Source: 137056-2013-09-24.xml*