Chaotic discrete map of pulse oscillator dynamics with threshold nonlinear rate coding

The study presents 1D discrete map (DM) to describe the dynamics of the oscillator with chaotic pulse position modulation (PPM). The model circuit has pulse voltage-controlled oscillator (PVCO) and feedback (FB) loop with a threshold of pulse rate coding, which performs non-retriggerable monostable multivibrator (MMV). DM is based on the analysis of this circuit using a simple approximation of the frequency modulation, which includes a threshold condition on the pulse period and sigmoid function of rate coding. The model circuit and DM demonstrate dynamic chaos in a wide range of control parameters. The transition to the chaos occurs by a jump either from a fixed point (tangent bifurcation), or from a limit cycle. An experimental (digital-analog) circuit of the chaotic pulse oscillator, in which the FB unit is MMV with a microcontroller (MC), is implemented. The relationship between the presented DM and the well-known sawtooth (Bernoulli) map (STM), widely used in engineering, is discussed.


Introduction
Chaotic systems have been the subject of intense research in various fields of science and technology in recent decades.New systems and models with chaotic dynamics are constantly appearing, and their fields of application are increasingly expanding [1][2][3][4][5][6][7].
Electronic circuits with chaotic dynamics are implemented both in analog and digital versions.The advantages of the digital implementation are undoubtedly the use of fully integrated technology, low noise (or even its actual absence), easy processing and storage of data.In addition, digital chaotic systems based on DMs are easier to analyze than analog (continuous) systems, which are described by differential equations.DM are easily implemented in electronic circuits [8,9], and are applied in communication systems [10][11][12], generation of pseudo-random sequences [4,13], cryptography [5,14,15], neural networks [16][17][18] and neuromorphic systems [19][20][21].
One of the first methods of DM construction, also used nowadays [22,23], consists in replacing the differential equations describing the initial continuous dynamical system by finite-difference schemes.The second, widespread method is the construction of Poincaré sections [24], in which the dynamics of the system are recorded only at moments of intersection of the phase trajectories with some transversal hyperplane.The DM function is constructed by point approximations of these intersections, which still must be found by solving differential equations.This makes the Poincaré section method quite complicated.But in some cases DM are obtained from the continuous dynamics of the system virtually unambiguously and naturally, without calculating the original model, only on the basis of simple analytical or empirical approximations.
Recently, in [7], a model scheme of a chaotic LIF oscillator is proposed based on a switching element with an S-shaped I-V characteristic (S-switch).This oscillator has a frequency (rate) instability of pulse generation due to the resistive FB including a second-order filter.Modeling of the circuit has shown that when the filter is set to self-excited (oscillatory) mode, chaotic PPM is realized, and the transition to chaos occurs according to the scenario of a period-doubling bifurcation cascades.The experimental (analog) circuit of this model [25], built on operational amplifiers and field-effect transistors, also demonstrates PPM chaos in certain parameter regions.However, the dynamics of the circuit has a strong stochastic blurring due to random switching initiated by element noise, primarily transistors.It is also worth noting the complexity of the circuit and the mathematical model of the LIF oscillator, which consists of a system of five differential equations.
In this study, a simple digital-analog circuit of pulse oscillator with unstable rate coding is proposed.The digital control unit is an Aurduino MC with a digitalto-analog converter (DAC), which are included in the FB loop of the oscillator.In contrast to the analog circuit [25], the digital circuit does not contain a secondorder filter, but the control unit (FB) must have a pulse shaper -non-retriggerable MMV to implement PPM chaos.The dynamics of the chaotic oscillator is described by a one-dimensional DM based on a simple approximation of the frequency modulation and includes a threshold condition on the pulse period, which is analogous to the MMV action.Modeling of oscillator circuits is performed in Matlab/Simulink soft; DM analysis is performed using Python.The pulse train of PVCO Posc come to the input of the MMV (Fg.1) and pulses increase in width up to a certain value Tth.If the period of input pulses coming to the MMV becomes less than Tth, then MMV keeps a logic 1 (5 V) at the output for the time Tth and does not react to the input pulses (see Fig. 3).This state of the MMV I called "1-hold".Only after the MMV is reset to state 0, it will be ready to generate the next pulse when a new input pulse arrives.

Model circuit of chaotic pulse oscillator
After MMV, the voltage Upvc is generated using PVC proportional to the current periods Tmv of monovibrator pulses: Upvc =a1Tmv.The output voltage Ufb in the last (NM) module is sigmoid function of the input voltage Upvc with a shift (inflection point) Uo: ( ) where Umin and U define the minimum and maximum variation of the sigmoid.
Consider the circuit of PVCO in the form a simple pulse oscillator (Fig. 1b) based on two NOT logic elements with RC-coupling, in which the period and the width of the output pulses change proportionally to the resistance R varies by the voltage Ufb.The dynamics of the Fig. 1b briefly presented in Apendix A. If the variable resistance is proportional to the control voltage (R = a2Ufb), then R will also be the sigmoid of the pulse period Tmv: where β = α/a1, R = a2U and Ro = a2Umin are the exponent, maximum change, and resistance reference value, respectively, To = α/a1•Uo.
The dynamics of the pulse oscillator circuit (Fig. 1a) based on the Fg.1b circuit and R(T) function (2) was analyzed by means of a Matlab-based simulation.
Simulink modules are used as MMV, PVC and NM units in FB loop, as well as linear variable resistance element R in PVCO.Note that MMV and R modules were developed and used earlier in the impulse Hopfield network [26].They are presented in Apendix B. The parameters of all calculations in this study are summarized in Table 1.Model circuit, Fig. 1 С =100 nF Logic NOT (U s = 5V)  The modeling shows that the periods Tosc and Tmv are always multiples of each other.Indeed, as can be seen from the oscillograms of Fig. 3, the current values of periods are either equal, if there is no 1-hold, or Tmv is twice Tosc, if there is 1-hold.
In the second case the periods Tosc become less than the width Tth.As a result MMV will not switch to zero with the arrival of the next PVCO pulse and its period Tmv remains equal to the previous value (until the arrival of the next pulse).
Thus, the PVCO output voltage Upvc and the controlled resistance R do not change, and MMV responds with one pulse to two PVCO pulses, i.e., for new values of periods Tmv =2Tosc.1.
The modeling also shows that the multiplicity n (Tmv=nTosc) in 1-hold states increases as Tth and the average pulse frequency (rate) growing, and one MMV pulse can correspond to 3, 4 or more PVCO pulses.Moreover, in the chaotic regime the multiplicity n is an irregular sequence of integers, that is, in different 1-hold states Tmv will be a multiple of Tosc in different number of times.For example, Fig. 4 shows a variant of chaotic PPM, where a random sequence of MMV and PVCO pulse multiples with n = 1, 2 and 3 is generated.The same dynamics with an irregular sequence of pulse multiplicities is implemented in the circuit experiment (Sect.4).
Further, in the construction of the map, we restrict ourselves to the case of n = 2, as in the Fig. 3 dynamics, in which only the double multiplicity is observed.
Once again, it should be emphasized that the multiplicity of PVCO and MMV periods is a property of both chaotic and regular circuit dynamics.

Discrete map of pulse oscillator dynamics
The voltage (current) in relaxation RC oscillators based on threshold switching elements changes according to the exponential law with time constant  = RC.The period of relaxation oscillations in simple circuits is estimated by linear dependence on resistance and capacitance: ( ) where the proportionality factor k is a linear combination of logarithms of the threshold switching parameters (see, for example, formula (4) in [27]).In Appendix A an estimation of the factor k is given for the pulse oscillator (PVCO, Fig. 1b) based on logic elements, in which the threshold parameters are low (Ulow) and high (Uhigh) voltage levels (see Table 1).Thus, if we use variable elements ( ) where A1=kCRo and A2=kCR.The rate function (4) must be supplemented with the condition corresponding to the 1-hold MMV state, when the current pulse period Tj-1 becomes less than Tth and it is necessary to set the multiplicity of the PVCO and MMV periods.In our case (Fig. 3) n=2, the period Tj must be doubled, after which rate (4) is applied: Thus, the DM (4,5) is finally defined by five parameters (A1, A2, β, T o and Tth).Fig. 6a shows the DM function (4,5), as well as its analytical approximation Fdm(T), constructed using the sigmoid S(T, Tth) with a large exponent γ, which model step functions: ( ) ( ) ) and its approximations (6).Black solid curve 1 is DM function, short dot curve is DM approximation (6): red curve 2 with γ =1000 and blue curve 3 with γ =50.Black dash lines are functions T j =2T j-1 and T j =2T j-1 −1, which correspond to STM (9).(b)

F T rate T ×S T T rate T ×S T T S T T TT 
The Lameray diagram of DM with the starting point T 0 .The green solid line is the bisector T j =T j-1 .
The parameters of DM in Table 1.
As can be seen from Fig. 6a, the DM function is divided by the vertical line T=Tth into two branches: rate (5) at T<Tth and rate (4) at T>Tth.The analytical approximation Fdm(T) (6) with large exponent γ =1000 (curve 2) almost coincides with DM (curve 1).
The Lameray diagram exhibits a chaotic attractor (Fig. 6b) bounded by the rate (4) and rate (5) branches.If DM iterations tend to infinity, the attractor will fill the solid region crossed by the vertical line T=Tth.A similar situation is observed with variations in the other DM parameters.In all cases, the region of continuous filling (chaos) is single with adjacent windows of discrete (regular) iterations, and the transition from chaos to a stable stationary point (dashed arrows, Fig. 7a and c) is always sharp.There are differences in the first transition "regular regime -chaos", which can be from a stationary point (Fig. 7b) or from a limiting multi-period cycle (Fig. 7d).Also, the region of chaos may be continuous or there may be small windows of regular iterations within it.
The Lyapunov exponent () of DM (4,5) is calculated using analytical approximation (6) with γ >10 12 and the known limit [24]:  and Tth (b).The calculation parameters and vertical (short dot) lines are the same as in Fig. 6.
As can be seen from Fig. 9, the dependences of Lyapunov exponent on DM parameters Tth and A2 correspond exactly to the bifurcation diagrams of Fig. 7: in the windows of discrete filling (regular iterations)  < 0, in the region of continuous filling (chaos)  > 0, the vertical short dot lines cross the level  = 0 in the transition point "discrete iterations -continuous iterations".Note Fig. 9b, where we can see the self-similarity of the function (Tth) at  < 0 before the "regular regime -chaos" transition.The maximum value (above 0.5) of the Lyapunov exponent is reached when the chaotic regime is exited.

Circuit experiment
The model circuit based on PVCO in Fig. 1b is inconvenient to implement, as it uses resistive control.However, the general concept of the chaotic pulsed oscillator circuit presented in Fig. 1a assumes any PVCO circuit in which the pulse periods is linearly controlled by the voltage.Therefore, I use standard IC circuits of PVCO and MMV with direct voltage control.In addition, an MC is part of the FB loop, with which the functions of the PVC and NM modules are easily digitally implemented.
The experimental (hybrid digital-analog) circuit of the chaotic pulse oscillator includes in series PVCO and MMV based on NE5555 IC timer (Fig. 10) and Arduino Uno MC with an output 12-bit DAC (not shown in Fig. 1a).The MMV receives (TRIG, Fig. 10b) the pulse train Posc from the PVCO output (OUT, Fig. 10a) and sends the output pulse train Pmv (OUT, Fig. 10b) to the digital input of the Arduino MC.The DAC output is connected to the voltage-control input of PVCO (CONT, Fig. 10a), and the loop closes.
External RC circuits in PVCO and MMV (Fig. 10) determine the timing characteristics of rectangular pulses.The pulse periods and width of PVCO are also changed by the voltage (Ufb) connected to CONT (Fig. 10a).Figure 11a shows the experimental dependence of pulse periods on PVCO voltage, as well as its approximation in the range from about 1 to 4.5 V, where it is close to the line:    2.
Figure 11b shows PVCO and MMV rectangular waves without the inclusion of MC and DAC, that is, without FB loop in the circuit.As seen, one Pmv pulse corresponds to four Posc pulses, that is, the multiplicity n=4.
Turning on the FB loop with MC and DAC, one can observe (Fig. 12) for certain circuit parameters (Table 2) an irregular PPM dynamic of pulses with alternating multiples n equal to 4, 3 rarely 2 and even less often 1.This irregularity could be explained by the effect of noise, initiating random switching in PVCO or MMV.
But, as seen in Fig. 13, small changes in parameters or R3) transfer the oscillator into a regular regime, in which PVCO and MMV rectangular waves are synchronized with constant period.This undoubtedly indicates the deterministic nature of the chaotic dynamics of the experimental circuit, observed in Fig. 12.
Note also that the change in resistance R3 (Fig. 13b) corresponds to the variation of the threshold Tth of the model circuit (Fig. 1) and DM (4,5), since it determines (together with the capacitance C3) the width of MMV pulses in the Fig. 10b circuit.

Discussion
In the spike LIF oscillator with chaotic rate coding, which we studied in [7], the point of instability is the inflection of the frequency function from resistance F(R), where its first derivative has a maximum.PPM chaos in this circuit arises because of chaotic frequency oscillations in the vicinity of this point Fо(Rо) between values F > Fo and F < Fo.A similar point of instability of the pulse oscillator in this study is the threshold period Tth, which is both the break point of DM (4,5) and the inflection point of its analytical approximation (6).The instability of rate coding in the proposed scheme (Fig. 1) is created by joint action of MMV and NM, whereas in LIF oscillator [7] it is initiated by second order filter in FB loop.
In nonlinear analysis, there is a well-known DM called the map (or STM), which is defined by the recurrence relation [24]: STM has wide application in engineering, first, due to the simple and highspeed digital algorithm: the operation of doubling and taking the remainder in the binary code is shifting the binary point one bit to the right, and if the bit to the left of the new point is a one, replacing it with a zero.In applications, for example, as pseudorandom number generators, various modifications of STM are also used [28][29][30].
The DM (4,5) developed in this study is very similar to the STM (9).Indeed, if the right (rate ( 4)) and left (rate ( 5)) branches of the DM (Fig. 6a) are replaced by lines Tj = 2Tj-1 and Tj = 2Tj-1−1, then at Tth = 0.5 the map transforms into the relation (9).Generating random sequences with STM ( 9) is analogous to flipping a coin (head, nut) with equal probability of two trials.Modeling shows that DM (4.5) by selecting (five) parameters can also be tuned to generate equiprobable values Tj > Tth or Tj < Tth, i.e.rate (4) or rate (5), and there are infinitely many such combinations of DM parameters.
In fact, there is an analogy between the dynamics of the presented chaotic oscillator (Fig. 1a) and STM of a more general kind: where M is a positive integer.For M > 2 the STM can already have three, four or more conditions of type (9).For example, for M = 3 the mapping (10) has the form:  After the transition to chaos, rare irregular bursts appear in the intervals between quasiregular iterations (Fig. 8a), the average frequency of which gradually increases with increasing parameter A2, that is, with increasing distance from the tangent point Ttg.Interestingly, the bursts of iterations in the background of slightly changing values are very similar to neuron spikes.Indeed, if we compare DM (4,5) with known discrete models of neuronal activity, we can see its surprising similarity to Kurbage-Nekorkin 1D model [21], which is a discrete modificatioin of FitzHugh-Nagumo neuron [33].
In the other presented case (Fig. 7d), the transition into chaos occurs from the limit cycle.As the parameter Tth increases, the limit cycle changes in jumps, increasing and decreasing the number of iteration periods.At a certain point, the limit cyrcle becomes unstable and the DM jumps into chaotic mode.After the transition into chaos, fluctuations appear in multi-period iterations (see, Fig. 8b), which increase with increasing parameter Tth.The mechanism of this bifurcation has not been elucidated, but we can state with certainty that it is not related to either the tangential bifurcation or the scenario of period-doubling cascades.

Conclusion
The dynamics of pulse oscillator with chaotic PPM, which is described by onedimensional DM, is presented.This DM is controlled by five parameters and has a nontrivial bifurcation structure in which the transition to chaos is abrupt from a fixed point or a limit circle preceded by the script of multiple branching regular iterations.DM can be considered a modification of the well-known Bernoulli map (or STM).The developed chaotic model can be of interest for researchers in the field of nonlinear dynamics, as well as in the development of applications that use the phenomenon of dynamic chaos.

Figure
Figure 1a shows the general circuit of a chaotic pulse oscillator.The circuit consists of a PVCO and FB loop in series with a pulse width shaper, which is a non-retriggerable MMV, period-voltage converter (PVC) and non-linear voltage converter (nonlinear module, NM).The output of the circuit can be both a PVCO output and an MMV output, in the form of square wave signals (pulse trains Posc and Pmv).

Fig. 1 .
Fig. 1.(a) General circuit of chaotic pulse oscillator.FB is a feedback unit: MMVmonostable multivibrator, PVCperiod-voltage converter, NMnonlinear module.P osc and P mv are output pulse trains of PVCO and MMV.(b) The pulse oscillator based on logic elements with variable resistance R. (1), (2) and (3) are output, intermediate and input nodes, respectively.

Fig. 2 3 .
Fig.2 shows the time dependences of the MMV pulse periods for two values of the parameter Tth.Oscillograms of PVCO (Posc) and MMV (Pmv) pulse trains, corresponding Fig.2, are shown in Fig.3.Figures 2a and 3a are examples of chaotic oscillator dynamics with characteristic (irregular) 1-hold states of MMV, when Tosc become less than threshold value Tth.Regular regime of rate coding is demonstrated at the second value of Tth (Fig.2b and 3b), where there are also 1-hold states, but which already repeat with some period Trs.There is, of course, trivial regular dynamics for

)Fig. 2
Fig.2 Oscillograms of MMV periods (T mv ) in chaotic (a) and regular (b) regimes.T rs is the period of regular oscillations.

Fig. 3 .Fig. 4 .
Fig. 3. Oscillograms of PVCO (P osc ) and MMV (P mv ) output pulse trains in chaotic (a) and regular (b) regimes.The periods of the 1-hold state (T osc =2T mv ) and its absence (T osc =T mv ) are shown.The calculation parameters are in Table1.(a) linearly controlled by voltage as resistance R or capacitance C, we get a linear PVCO.In another (experimental) PVC version based on the integrated circuit (IC), presented in Sec. 4, the pulse periods are directly controlled by the voltage.The PVCO periods, transformed into voltage in the FB circuit, change their value always discrete, which is the basis for developing the DM of the pulse oscillator dynamics.Indeed, values of MMV pulse periods are converted with PVC into voltage (Upvc = Uj) only after the arrival of the next pulse, that is at certain moments of time Uj = a1Tj, where j = 1, 2, 3... fixes the appearance of the next MMV pulse (Tmv = Tj).Combining the dependence of resistance on the pulse period (2) with the linear function (3), we obtain the following DM function, which we called rate:

Fig.5a andFig. 5 .
Fig.5a and bshow the calculation by the DM (4,5) of pulse periods and their correlation function, in which the mapping parameters correspond to the simulation parameters of Fig.2a.Further, we use in the calculations (Fig.6-9) the dimensionless form of DM (4.5) by dividing A1, A2, To, Tth by 1 ms and multiplying β by 1 ms, which is presented in Table1.

Fig. 7
Fig.7shows the bifurcation diagrams of DM(4,5) when changing the parameters A2 and Tth.The windows of continuous filling, where chaos should be observed, are clearly visible.Adjacent to these windows are areas where iterations are filled pointwise (discretely), which corresponds to the regular regime.

Fig. 7 . 6 (
Fig. 7. Bifurcation diagrams of iterations based on DM (4,5) when changing parameters A 2 (a and b) and T th (c and d).The vertical dotted line is the "regular regime -chaos" transition boundary: A 2 0.8463(b) and T th 0.3723 (d).The dotted arrows in the figures (a and c) indicate a sharp transition from chaos to a stable fixed point.The calculation parameters are the same as in Fig. 6 (except for A 2 and T th ).
Fig.12 PVCO, Fig.11a, Line (8) A 1 = 0.22 ms A 2 = 3 ms β = 5000 s -1 T o = 2.5 ms c o = 0.47 ms c 1 = 0.29 ms/V The MC control program (Arduino IDE) first measures the pulse periods that come from the MMV (Fig.1b).Then, the received values are converted with rate (4) and transferred into the input DAC binary levels Udac = 0 ÷ 4095, i.e., into its output voltage Ufb from 0 to 5 V.In fact, MC and DAC replace the second (PVC) and the third module (NM) in the FB loop of the model circuit Fig.1a.The circuit and MC program parameters were approximated by DM simulation (4,5) togetherwith the linear approximation(8), which are represented in Table2.

Fig. 11 .Fig. 12 .Fig. 13 .
Fig.11.(a) Experimental dependence (blue dots) of PVCO pulse periods on U fb voltage and its linear approximation (solid red curve) for the range 1÷4.5 V. (b) Photo oscillograms of PVCO (green) and MMV (yellow) rectangular waves in the absence of FB.CONT voltage of PVCO U fb is 2.5 V. Parameters of the circuit (without rate (4)) in Table2.

1 Fig. 14 .
Fig.14.(a) The view of DM function (4,5) with five fixed points: T 1 and T 5 (blue circles) -stable points, T 2 −T 4 (red circles) -unstable points.(b) Tangential bifurcation of chaos birth with the growing of parameter A 2 : black curve → blue curve → red curve.The green solid line is the bisector T j =T j-1 .