Tutorials and Reviews CHUA'S PERIODIC TABLE - Semantic Scholar

Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1451–1464 c World Scientific Publishing Company

CHUA’S PERIODIC TABLE R. TONELLI and F. MELONI INFM, Department of Physics, University of Cagliari, Italy Received May 10, 2001; Revised February 20, 2002 An interdisciplinary analysis is presented in this paper on the relationship between the bifurcation behavior of chaotic systems, specifically Chua’s circuit, and the quantized energy level of atoms. We show that it is possible to associate special capacitance values in Chua’s circuits with the electronic energy levels En of atoms calculated from Bohr’s Law. In particular, these particular capacitance values correspond to the bifurcation points αn of Chua’s circuits. We found a functional relation associating a specific Chua’s circuit to the Hydrogen atom and obtained a map of “Atom” to “Chua’s Circuit”. This map can be extended to other elements in the Periodic Table, thereby demonstrating an almost universal relation between the two different physical systems. With this map we can calculate the energy levels of different atoms from bifurcation diagrams (or vice-versa) and express the analytical relations in terms of the two variables αn and En Keywords: Chua’s circuit; periodic table; Chua’s atom; spectroscopy.

1. Introduction Nonlinear systems are ubiquitous among almost all aspects of real life. Nonlinearity is found in entirely different fields of scientific investigation. Particularly interesting for the purpose of this paper is the possibility that nonlinear systems may change their behavior from a simple, periodic motion to a very complicated or chaotic one. Some general features of these systems include extreme sensitivity to initial conditions [Parker & Chua, 1989] (even if we allow the imposition of control schemes [Tonelli et al., 2000]), bifurcations [Devaney, 1986], and chaotic attractors [Eckmann & Ruelle, 1985]. Among all nonlinear systems Chua’s circuit [Madan, 1993] plays a special role in view of its simplicity and rich dynamics, including the possibility to exhibit an immense variety of chaotic attractors. In a recent paper [Romano et al., 1999], Chua’s circuit is used to introduce a chaotic model of electrons in atoms and to give a physical analogy between the synchronization of two Chua’s circuits and the covalent bond in molecules. We review and extend their conjecture into another method, which enable

us to exploit the similarities existing between bifurcation diagrams in Chua’s circuit and the spectra of atoms. In this paper, we have focused on the period-doubling bifurcation cascade of Chua’s circuit. The sequence of its bifurcation points is in fact quite similar to the series of lines in atomic spectra. In order to construct a map between them, we use the atomic Bohr’s model to calculate the electronic energy levels and associate them to the Chua’s bifurcation diagram obtained by the Poincar´e Section Technique [Ott, 1993]. We have constructed this map for all elements in the first row of the Periodic Table. It is possible to apply a nonlinear transformation to the bifurcation diagram so that most of the bifurcation points are mapped accurately to corresponding energy levels. This nonlinear transformation allows us to calculate the electron energy levels from the bifurcation diagrams, and vice-versa. The paper is organized as follows. In Sec. 2, we introduce some basic notions of spectroscopy. Examples of atomic spectra are shown to be similar to bifurcation diagrams. The Hydrogen spectrum is discussed in Sec. 3 and the formula giving the

1451

1452 R. Tonelli & F. Meloni

position of spectral lines are studied. In order to derive a theoretical scheme and to apply it to heavier atoms the Bohr’s model is described in Sec. 4. This enables us to calculate the electron energy levels for Hydrogen-like atoms. In Secs. 5 and 6, we introduce the concept of bifurcation diagram of chaotic systems and apply it specifically to Chua’s circuit. By comparing the bifurcation diagrams and the spectroscopy series of atoms, we construct an “Atomsto-Chua’s Circuit” Map in Sec. 7. In Sec. 8, this map is extended to other elements in the Periodic Table. Some concluding remarks are given in Sec. 9.

2. Elementary Notions of Spectroscopy Optical spectra are the most important source of information concerning the structure of atoms, molecules and solids. They provide indications on the electronic structure and on the composition of

emitting sources. There are continuous, band, and line spectra [Hindmarsh, 1967; Haken et al., 1984]. The continuous spectra are emitted by solids or dense gases. The band spectra which consist of groups of extremely close spectral lines (most of the times appearing to form a continuum) are emitted generally by molecules. The line spectra consist of isolated lines, which can be ordered in a characteristic series, indicating the types of atoms (examples shown in Fig. 1). Normally, the wavelength (or frequency) of the element spectra can be categorized into different regions corresponding to the visible, infrared, ultraviolet, X-ray, microwave, and radio bands. They are usually expressed in several different units for historical reasons. For example, the light wavelength λair measured in air is different from the wavelength λvac measured in vacuum; namely, λvac (1) λair = η

He

H Fig. 1. Table 1.

The line spectra of Hydrogen and Helium.

Spectroscopic quantities and fundamental constants.

Quantity

Units −10

1 ˚ A = 10

Wavelength λ Wave number ν =

1 λ

Energy E

m = 0.1 nm

1 cm−1 = 29.979 GHz 1 eV (electron-volt) = 1.6 × 10−19 J = 8066 cm−1 E = hν =

hc = hcν λ

Velocity of light

c = 2.99792458 1010 cm/s

Rydberg constant

RH = 13.595 eV = 1.097 107 m−1

Rest Electron Mass m

m = m0 = 0.911 × 10−27 g = 511 KeV/c2

Electron Charge e

e = 1.602 × 10−19 C

Planck constant h

h = 6.62618 × 10−34 J s = 4.14 × 10−15 eV s

Reduced Planck constant

~=

h = 1.05 × 10−34 J s 2π

Chua’s Periodic Table 1453

where η is the refractive index of air. Independent of the propagation medium, light may be characterized by its frequency: ν=

c

(2)

λvac

where c is the velocity of light in vacuum. In addition, the wave number ν=

1 ν = c λvac

(3)

represents another characteristic quantity of light. The frequency ν is related to the light energy E = hν

3. The Hydrogen Spectrum Hydrogen is the simplest element which consists of only one proton and one electron. Its spectrum shows three characteristic lines in the visible region, referred to as Hα , Hβ and Hγ , respectively corresponding to 656.3 nm, 486.1 nm and 434.1 nm (shown as in Fig. 2) [Series, 1957]. At the near ultraviolet region, a series of other lines appear and their spacing decreases in a regular manner. These lines approach a limiting wavelength (H∞ ) where an infinite number of lines converge. Balmer [Haken &

434.05

Hγ 400 nm Fig. 2.

3 4 5 6 7 8 9 10

15233.2 20564.8 23032.5 24373.0 25181.3 25705.8 26065.5 26322.8

15233.0 20564.5 23032.3 24372.8 25181.1 25705.7 26065.4 26322.6

(4)

where h is the Planck constant. The most common units and conversion factors are listed in Table 1.

410.17

Table 2. Comparison between observed and predicted wave numbers from the Balmer formula.   1 1 n ν vac RH − 22 n2

486.13

Hβ

Wolf, 1984] generated the wavelength sequence with the formula: n2 (5) λ=k× 2 n −4 where n > 2 is an integer. The same equation can be written in terms of frequencies in the following more general form 

ν = RH

1 1 − n22 n21



(6)

where RH is the Rydberg constant (listed in Table 1), n1 and n2 are both integers called principle quantum numbers, with n1 ≥ n2 . This simple formula is in extraordinary agreement with experimental observations (see Table 2). By substituting

656.29

Hα 700 nm

Experimental and schematic representation of the Hydrogen spectrum showing the most important lines.

1454 R. Tonelli & F. Meloni

Fig. 3. Different spectral series of Hydrogen. The colors do not correspond to the true spectral colors (for example, the Lyman series is in the infrared region) but are useful for distinguishing the overlap between different series.

n2 with various integers, all series in the Hydrogen spectrum can be obtained. The first series (n2 = 1) is the Lyman series, in the far ultraviolet region. The second series (n2 = 2) is the Balmer series. Other well-known series in the infrared region are the Paschen (n2 = 3), Brackett (n2 = 4) and Pfund (n2 = 5) series (shown in Fig. 3). Equation (6) can be expressed by two terms: ν=

RH RH − 2 = T2 − T1 2 n2 n1

terms in Eq. (7) is known as the Ritz combination principle. It is valid, subject to certain limitations, not only for Hydrogen-like atoms but also for other elements in the periodic table. Moreover, the two terms of Eq. (7) can be regarded as the energy levels of electrons in the Hydrogen atom, and the spectral lines can be graphically represented as transitions between two energy levels in a Grotrian diagram (Fig. 4).

(7)

We see that the frequency approaches a limit ν∞ = RH /n22 as n1 → ∞, and an infinite number of spectral lines converge to this value. The fact that the frequency of each spectral line can be obtained directly from the difference between two

4. Bohr’s Model According to the Maxwell’s electromagnetic field theory, the emission of light radiation is due to the acceleration of electric charges. It predicts that electrons in accelerated motion around the nucleus would emit radiation of some fundamental frequency together with their harmonics, and the atomic spectra should consist of a simple line corresponding to the fundamental frequency plus the lines related to the harmonics. However, this was not the case in experimental observations. Furthermore, the electrons moving in orbits around the nucleus must lose energy due to radiation and hence spiral into the nucleus while emitting a continuous spectrum of light. The Balmer formula could be used to predict the line positions but did not provide a theoretical explanation. Hence, the classical theory of electromagnetism cannot explain the behavior of microscopic entities like electrons or atoms. In modern quantum theory [Gasiorowitz, 1974], the concept of quantized energy was introduced successfully to explain light radiations, such as the blackbody radiation, and the photoelectric effect. The discrete energy quanta is given by ∆E = hν

Fig. 4. Grotrian scheme relating the lines in the Hydrogen spectrum to the atomic energy levels.

where h is the Planck constant.

(8)

Chua’s Periodic Table 1455

Fig. 5.

Bohr’s atom showing the quantum transitions in the Hydrogen spectral series.

By including both the energy quanta and the Balmer formula, Bohr developed a model to explain the connections between spectral lines and atomic structure [Haken & Wolf, 1984]. Bohr’s model assumed that there is a heavy nucleus of positive charge Ze in each atom and Z electrons are continuously rotating around the nucleus. Contrary to Maxwell’s equation, Bohr postulates that the electrons do not emit radiation even though they are in accelerated motion. The model is based on an ad hoc definition of “stationary orbits” (shown in Fig. 5). The atom is assumed to radiate energy only when an electron jumps from an orbit to another one with a quantum transition (depicted in Fig. 6). The amount of radiation emitted is assumed to be equal exactly to the difference between the energy levels of the two quantized orbits involved in the jump: (9) En1 − En2 = hν where the energy levels En1 and En2 are respectively proportional to the terms T1 and T2 of Eq. (7). An energy value of E = 0 implies that the electron is

completely removed from the atom. All other energy levels are assumed as a set of discrete values, and this model can be used to explain why the spectrum appears as a set of discrete lines. Normally, electrons stay in the ground state and occupy the low energy levels of an atom. When atoms absorb energy their electrons reach an excited state. They release the absorbed energy spontaneously by emitting radiation and returning to the ground state. The Bohr quantization rule applies also to angular momentum. Considering the Coulomb’s attraction between the nucleus charge Ze and each electron, we have Ze2 mv2 Ze2 ⇒ mvr = = r2 r v

(10)

where r is the orbit radius and mvr is the angular momentum of electron. Bohr postulates that the angular momentum is given by mvr = n

h 2π

(11)

1456 R. Tonelli & F. Meloni

Fig. 6.

Mechanism of light emission from an excited atom resulting in a quantum jump to a lower energy orbit.

where n is an integer called the principal quantum number. The quantized energy of each orbit can be written as: E=−

2π 2 me4 Z 2 Ze2 1 + mv2 ⇒ En = − r 2 h2 n2

(12)

Z2 n2

and rn ∼

n2 Z

(13)

2π 2 me4 2 1 1 Z − 2 2 3 h n2 n2

1

N

(15)

dt

and its vector form

1 (En1 − En2 ) h 

=

2

dt

Applying these conditions, together with the Bohr rule for frequency emission, we obtain ν=

 dx(t)    = F1 (x1 , . . . , xN )   dt        dx2 (t) = F (x , . . . , x )   ..   .      dx (t)    N = FN (x1 , . . . , xN )

Consequently, En ∼

differential equations with N variables xi :



(14)

It follows from Bohr’s Model that the value of the Rydberg constant is equal to RH = 2π 2 me4 /h3 , in excellent agreement with measurements. According to this model the spectra of all atoms with a single electron should be obtained by replacing the one positive charge of the Hydrogen nucleus with the corresponding Z number for each atomic element. The Hydrogen-like atomic spectra, such as those from He+ , Li2+ , Be3+ ions, or even heavier ionized atoms, can be explained by Bohr’s model and have been experimentally confirmed.

5. Chaotic Systems Consider a dynamical system defined by a set of N

dx(t) = F(x(t)) dt

(16)

where x(t) is an N -dimensional vector function of time t. For any initial condition x(0) one can in principle solve the equations to obtain its evolution. The space spanned by the variables x1 , . . . , xn is referred to as the phase space and the curve described by the variables in this space is called an orbit or trajectory [Ott, 1993]. The main feature of a chaotic dynamical system is its sensitive dependence on initial conditions. A dynamical system is usually called chaotic if it exhibits this sensitive dependence phenomenon, i.e. if the system starts from two different and infinitely close initial conditions, the corresponding orbits x1 (t) and x2 (t) will diverge from each other exponentially with time [Eckmann et al., 1985]; namely, k∆x(t)k = kx2 (t) − x1 (t)k ' exp(λt)

(17)

where λ > 0 is a positive parameter called the Liapunov exponent. One useful tool to analyze chaotic systems is the method of Poincar´e Sections [Abarbanel

Chua’s Periodic Table 1457

the (N −1)-dimensional cross-section. This dynamics no longer has a continuous time evolution but the subsequent points can be identified by a discrete index n and their dynamics is described by a mapping: (18) xn+1 = f (xn ) where the quantity xn+1 is uniquely determined from xn . The Poincar´e Section technique is particularly useful in a bifurcation analysis. A bifurcation is a change in qualitative behavior of a system depending on the variation of a system parameter [Devaney, 1986]. In order to examine a typical sequence of bifurcations leading to a chaotic state (called a period-doubling bifurcation route to chaos), let us recast the dynamical system of Eq. (16) into the more explicit form Fig. 7. Example of a two-dimensional Poincar´e section for a three-dimensional dynamical system.

dx(t) = F(x(t), λ) dt

et al., 1993]. This technique reduces a continuous system to a discrete iterative map by choosing an appropriate (N − 1)-dimensional surface in the N -dimensional phase space and looking for the intersections of the orbits. As shown in Fig. 7, any two successive intersections are completely determined by the orbit flow in the phase space and so a discrete evolution dynamics is established onto

where λ = (λ1 , . . . , λn ) denotes some external parameters. A change in the parameters can lead the system to bifurcate from one periodic orbit to another periodic orbit with a different period, or to a chaotic orbit. The universal feature of the perioddoubling route to chaos is that as a parameter is changed the system passes through a sequence of transitions from a period-one periodic orbit to a

(19)

2 1.8 ·

1.6 ²

µ

´

µ

1.4 ·

1.2

³

¶

²

1 ´

µ

0.8 ³

²

0.6 ®¯

°

0.4 0.2 0 7

œ

7.5



ž

Ÿ

¡

¢

£

¤



8

¥

§

¨

£

ª

¢

£

¬

8.5

­

Fig. 8. Bifurcation diagram of Chua’s circuit. The upper and lower values of the variable y on the Poincar´e section are plotted versus the parameter α.

1458 R. Tonelli & F. Meloni

period-two, then to a period-four, period-eight, periodic orbits and so on, until the parameter reaches the critical value for the transition to a chaotic state. A useful scheme to represent this behavior is the bifurcation diagram, in which the long-time behavior of the system is plotted versus the value of the parameter λ (as shown in Fig. 8). A period-doubling route to chaos may be characterized by certain universal numbers that do not depend on the particular system. For example, in one-dimensional maps, there is a universal constant δ, called the Feigenbaum Number [Feigenbaum, 1978] which is the ratio of the spacing between consecutive values of the parameter λ, defined as: δ = lim

n→∞

λn − λn−1 = 4.6692 λn+1 − λn

(a)

(20)

where λn−1 , λn , λn+1 are the consecutive bifurcation points of the parameter λ.

(b) Fig. 9. Chua’s circuit. (b) Voltage–current characteristic of Chua’s diode.

6. Chua’s Circuit Chua’s circuit is a nonlinear circuit that for its simplicity has played a crucial role in the study of chaotic systems [Madan, 1993]. Despite its simplicity it exhibits a great variety of phenomena typical of chaotic systems; e.g. strange attractors, stochastic resonance, chaotic orbits, basin bifurcations, etc. Even the possibility to achieve secure communication via chaotic synchronization has been experimentally demonstrated [Grassberger & Procaccia, 1983; Anishchenko et al., 1992; Kocarev et al., 1993]. Chua’s circuit [shown in Fig. 9(a)] is composed of two capacitors, an inductor, a resistor and a Chua’s diode. Chua’s diode is the active nonlinear element in the circuit and is responsible for the chaotic behavior. Chua’s diode is actually a nonlinear diode [Cruz & Chua, 1992] whose i–v characteristic is shown in Fig. 9(b). The nonlinear differential equations of the Chua’s circuit are given by

1 dv2 = (v1 − v2 ) + i3 dt R

      di   L 3 = −v2 − R0 i3

dt

1 f (v) = Gb v+ (Ga −Gb )(|v+Bp |−|v−Bp |) 2

(21)

(22)

describing the Chua’s diode characteristic: The same kind of piecewise-linear function has been used to implement the nonlinear behavior in Cellular Nonlinear Networks [Chua, 1998]. This is a thirdorder dynamical system whose equations are usually written in the following dimensionless form:    x˙ = α(y − x − f (x))  

y˙ = x − y + z z˙ = −βy − γz

(23)

where the dimensionless variables are defined as follows:  v   x=    Bp    

 dv1 1   C1 = (v2 − v1 ) − f (v1 )    dt R    

C  2

where v1 and v2 are the voltages across the capacitors and i3 is the current in the inductor. R0 is the internal resistance of the inductor and f (v) is a piecewise-linear function

τ=

t RC2

      C   α = 2

C1

y=

v2 Bp

z=

a = RGa β=

R2 C2 L

Ri3 Bp

b = RGb γ=

C2 RR0 L

(24)

Chua’s Periodic Table 1459

with x˙ = dx/dτ and f (x) = bx +

1 (a − b)(|x + 1| − |x − 1|) 2

(25)

For certain values of the circuit parameters, the two voltages across the capacitors and the current in the inductor have a periodic time variation. While, for some other values of these circuit parameters, the previous variables may exhibit a very complicated time evolution with a broadband Fourier spectrum in which a continuum of frequencies are present. In this case the regime is chaotic. Sequentially, with increasing values of α, the circuit undergoes a sequence of transitions between different regular states until a chaotic regime is reached. These transitions constitute bifurcations in the phase space and occur at a fixed value of αn . The difference αn+1 − αn between two subsequent bifurcation values decreases at a geometric rate and all bifurcation points accumulate to one single value α∞ where the transition to a chaotic regime occurs (as shown in Fig. 8).

7. Chua’s Atom Let us now consider the similarities between the above two physical systems. Our idea is to construct a map associating atoms from the periodic table to Chua’s Circuits by comparing the bifurcation cascade of Chua’s Circuit to the atomic spectra via Bohr’s model. The first common feature is that both atomic spectra and bifurcation diagrams have a series of critical points exhibiting the same behavior. The distance between critical points decreases monotonically, and tends to a geometric ratio. There exists a single value at which the sequence of points accumulates and converges to. Before this critical point the systems can be in only one of an infinitely many discrete states. Beyond this critical point the “spectrum” becomes continuous and the system exists in an infinite continuum of states. The characteristic distances in the spectra are not fixed but have a parametric dependence in both systems. The determining parameter is the atomic number Z in the case of atomic spectra and the circuit capacitance (related to the parameter β) in the case of Chua’s circuit. The positions of the critical points undergo a translation and there is a contraction (resp. expansion) when the atomic number Z or the circuit parameter α decreases (resp. increases).

In the case of atoms, it is more convenient to derive the energy levels by theoretical calculations from Bohr’s formula [i.e. Eq. (13)] instead of by experimental measurements from atomic spectral lines. Similarly, in the case of Chua’s circuit, we can let αn denote the value of the parameter α for the nth bifurcation point. Our goal is to establish a one-to-one correspondence between the sequence of bifurcation values αn in a specific Chua’s circuit and the series of energy levels En in a fixed atom. Applying the fourth-order Runge–Kutta algorithm to the dimensionless Chua’s equations of Eq. (23), we have performed numerical integrations over a range of parameter values to determine when the system’s asymptotic behavior is periodic. We then calculate the bifurcation points αn using a Poincar´e twodimensional cross-section in the three-dimensional phase space. In our numerical calculations, we have chosen a = −8/7 and b = −5/7 in the piecewise-linear function of Eq. (25) and set the parameter γ = 0 in Eq. (23). The bifurcation points of Chua’s circuit are then determined by varying the two dimensionless parameters α and β [Chua et al., 1986]. We identify the parameter β with the atomic number Z. For a specific atom with atomic number Z, we can keep β fixed and use α as the bifurcation parameter to calculate the discrete values αn . These αn correspond to the energy levels of the atom, and the differences in two successive bifurcation values, denoted by ∆αn = αn+1 − αn , correspond to the differences in two consecutive atomic energies. We have observed that a different β-value will produce a translation and contraction (resp. expansion) of the bifurcation values αn , in the same way as a variation along the Periodic Table will produce a translation and contraction (resp. expansion) of the energy levels. It is natural therefore to choose the parameter β in correspondence to the atomic number Z in the Periodic Table. It is possible to rescale the bifurcation diagram of Chua’s circuit to obtain a closer correspondence with the energy diagram of the atom via a nonlinear transformation given in the next section. Figure 10 shows a map between the bifurcation diagram of Chua’s circuit and the energy diagram of the Hydrogen atom. Here, only the first five bifurcation points are calculated to map the first five energy levels of the Hydrogen atom. It follows that an atom in each energy state is related to a Chua’s circuit in one of its stable periodic orbits. An energy level transition of the atom

1460 R. Tonelli & F. Meloni

Fig. 10. A comparison between bifurcation diagrams and energy levels in atoms. The arrows show the correspondence. It is evident that the nonlinear transformation has rescaled the bifurcation diagram (right) so that its bifurcation points are closer to the energy levels than the ones obtained from the unscaled diagram (left).

has a corresponding state transition in Chua’s circuit between two stable periodic orbits in Chua’s circuit. The “order-to-chaos” transition in Chua’s circuit corresponds to an ionization of the related atom, namely to a “bounded-to-not bounded” state transition. The correspondence αn versus En is shown in Fig. 11 where we used Z = 1 in the Bohr’s formula and β = 10.246 to calculate the Chua’s bifurcation diagram. From this “Atom-toChua’s Circuit” map it is possible to find the atomic energy levels from the knowledge of Chua’s bifurcation diagram and vice-versa.

physical systems is maintained for large values of Z, we have confirmed that this phenomenon is consistently preserved, namely, up to Z = 14, corresponding to the Silicon atom. For each parameter β, a nonlinear transformation can be introduced to rescale the bifurcation diagram so that it resembles better the energy level diagram (shown in Fig. 10). In order to figure out the nonlinear transformation, we integrate Bohr’s formula with the above map in the following way. We set first the final asymptotic state of the bifurcation diagram via the following function: α0n = C − log(α∞ − αn )

8. Chua’s Periodic Table The next step in our model is to choose which atom corresponds to a Chua’s circuit defined by a given set of parameters. With different values of the parameter β and the atomic number Z, we can obtain a set of correspondences between the bifurcation values αn and the energy values En of atoms. We found that the correspondences were kept in the same functional shape (shown in Figs. 11–16) if the parameter β was chosen to begin from β1 = 10 246 and then increasing it by one unit each time the atomic number Z is increased by one in the Periodic Table. To check if the map between the two

(26)

The spacing between two successive α0n will be constant since the spacing between two successive αn tends to a geometric ratio as in Eq. (20). Here C is a constant value depending on α∞ , and hence depending also on β. We can write Eq. (26) for the mapping of β and Z as: α0n = C(β) − log(α∞ − αn ) = f (αn ) ' k(α∞ ) × n (27) where k(α∞ ) is a constant depending on the infinite bifurcation point, which is basically β. The function C(β) is chosen such that the first rescaled bifurcation point α01 matches k(α∞ ). On the other

Chua’s Periodic Table 1461

Fig. 11.

Fig. 12.

Functionality form of the relationship between αn and En for Hydrogen (Z = 1).

Functionality form of the relationship between αn and En for Helium (Z = 2).

1462 R. Tonelli & F. Meloni

7.6 Lithium Z = 3; β = 12.286

Bifurcation Parameter α

7.4 7.2 7 6.8 6.6 6.4 6.2 −140 Fig. 13.

−120

−100 −80 −60 Bohr’s Energy E

−40

−20

0

Functionality form of the relationship between αn and En for Lithium (Z = 3).

Bifurcation Parameter α

8.2 8

Berillium Z = 4; β = 13.286

7.8 7.6 7.4 7.2 7 6.8 6.6 6.4 −250

Fig. 14.

−200

−150 −100 Bohr’s Energy E

−50

0

Functionality form of the relationship between αn and En for Beryllium (Z = 4).

Chua’s Periodic Table 1463

Fig. 15.

Functionality form of the relationship between αn and En for Boron (Z = 5).

9

Bifurcation Parameter α

Carbon Z = 6; β = 15.286 8.5

8

7.5

7 −500 Fig. 16.

−400

−300 −200 Bohr’s Energy E

−100

0

Functionality form of the relationship between αn and En for Carbon (Z = 6).

1464 R. Tonelli & F. Meloni

hand, we can recast Bohr’s formula into the form −En = k0 (Z) ×

1 1 ⇒√ = k00 (Z) × n (28) n2 −En

Comparing Eqs. (27) and (28) we obtain the following analytical expression for the nonlinear transformation: 1 (29) C(β) − log(α∞ − αn ) = K 0 (Z, β) × √ −En where K 0 (Z, β) is a constant depending on β and Z such that the two spacing k(α∞ ) in Eq. (27) and k00 (Z) in Eq. (28) are the same. Obviously, Eq. (29) generalizes the mapping of “Atom-to-Chua’s Circuit” to all elements in the Periodic Table. It provides us with a simple way to derive the energy levels of an atom.

9. Concluding Remarks We have investigated two different fields guided by a similarity existing between the structure of the period-doubling bifurcation route to chaos in Chua’s circuit, and the structure of the spectral series emitted by atoms. Starting from Bohr’s model and analyzing Chua’s circuit bifurcation diagram we construct a mapping relating the energy of electronic levels in atoms to the parameter value α at which bifurcations occur. The map reveals a common feature on the shape of the connecting function as the circuit parameters and atomic elements are changed. We also presented an analytical formula which allows one to go back and forth from Bohr’s energy levels to bifurcation values.

References Anishchenko, V. S., Safonova, M. A. & Chua, L. O. [1992] “Stochastic resonance in Chua’s circuit,” Int. J. Bifurcation and Chaos 2(2), 397–401.

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