Novel Impedance Sensor based on the Chaotic Van der Pol and Damped Dufﬁng Circuits Coupled

: The signature of chaotic systems can be characterized either by the sensitivity of the 1 initial conditions or by the change of its parameters. This feature can be used for manufacturing 2 high sensitivity sensors. Sensors based on chaotic circuits have already been used for measuring 3 water salinity, inductive effects, and both noise and weak signals. This article investigates an 4 impedance sensor based on the Van der Pol and Dufﬁng damped oscillators. The calibration 5 process is a key point and therefore the folding behavior of signal periods was also explored. A 6 sensitivity of 0.15 k Ω / Period was estimated over a range from 89.5 to 91.6 k Ω . This range can 7 be adjusted according to the application by varying the gain of the operational ampliﬁer used 8 in this implementation. The development of this type of sensor might be used in medical and 9 biological engineering for skin impedance measurements, for example. This type of chaotic sensor 10 has the advantage of sensing small disturbances and then detect small impedance changes within 11 biological materials which, in turn, may not be possible with other detectors. 12 obtain a change of state in a duplication oscillator. The bifurcation theory studies and classiﬁes phenomena characterized by sudden changes


Introduction
Many experiments in the biological, engineering and physical sciences have shown 15 that a fundamental implicit property of chaos is a sensitive dependence on the initial 16 conditions. According to [1] [2], the change in state of a dynamic non-linear system that 17 can be obtained through small changes in the system parameters is also the result of 18 a sensitive dependence on the initial conditions. Therefore, the parameter's sensitive 19 depend on the initial conditions. These properties could significantly improve the 20 characteristics of various types of sensory systems. The hypothesis of chaotic sensors is 21 that weak changes in system parameters can cause significant changes in its behavior, 22 in contrast to linear oscillators in which small changes in parameters only cause small 23 changes in oscillation dynamics [3]. For this reason, sensors based on chaotic circuits 24 have been increasingly studied and used in the sensing area, among some examples, 25 this approach has been used to determine the salinity of water [4], also on inductive 26 sensors [3], noise-activated non-linear sensors [5][6] [7], weak signal detection sensors 27 [8] [9] and impedance sensors [10]. There are several ways to implement a chaotic circuit 28 for sensing, in [5] a description of the system dynamics is discussed, which allows for 29 the use of a measurement technique based on the monitoring of the permanence time of 30 the system in its stationary states. And in [1] three different case studies are presented 31 that explore the sensitivity dependence on the initial conditions to build a sensory device 32 that can be implemented in hardware. One of the experiments uses the control manifold 33 theory to build a sensor that uses the change in the character of a fixed point from 34 elliptical to hyperbolic as a detection mechanism. A second experiment uses the escape 35 from the basin of attraction from a fixed point as a detection mechanism. And a third 36 one uses a bifurcation process to obtain a change of state in a duplication oscillator. The 37 bifurcation theory studies and classifies phenomena characterized by sudden changes system is characterized by a single point in the phase space and chaos when the system 48 shows chaotic behavior in that same phase space [12]. In particular, equilibrium points 49 can be created or destroyed, or have their stability altered. These qualitative changes in 50 dynamics are called bifurcations and the parameter values in which they occur are called 51 bifurcation points. In other words, bifurcation is a change in the topological type of the 52 system when its parameters pass through a critical value. Bifurcations are scientifically 53 important, as they provide models of transitions and instabilities when some parameters 54 are varied [11]. There are some types of bifurcation in continuous dynamic systems, in 55 which equilibrium points can be created or destroyed, or have their stability altered by 56 varying one or more parameters [13]. In continuous dynamic systems, there are two main 57 bifurcations by which an equilibrium has its stability altered by varying one or more 58 parameters: the saddle-node bifurcation and the Hopf bifurcation. The saddle-node 59 bifurcation is the basic mechanism for creating and destroying equilibrium points, and is 60 also called the fold, tangent, limit point or return point bifurcation. The Hopf bifurcation 61 of a system implies in the generation of a limit cycle. If a Hopf bifurcation grows by 62 adequately varying the system parameters, it can generate a periodic limit or attractor 63 cycle. In addition to the equilibrium, the most common form of behavior is a limit cycle.

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A limit cycle is an isolated closed path, isolated in the sense that neighboring paths are 65 not closed, so they spiral towards or away from the limit cycle. Stable limit cycles are very 66 scientifically important, as they model systems that exhibit self-sustaining oscillations.

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In other words, these systems oscillate even without periodic external forces. One of 68 the possible bifurcations of a limit cycle generated by Hopf bifurcation is the period 69 doubling bifurcation. The original limit cycle becomes unstable as a family of duplicate 70 period solutions emerges. In general, a series of n duplications may arise after each 2 n T 71 period limit cycle is obtained. Several systems have an infinite series of bifurcations 72 with a finite accumulation point; the associated movement beyond that point is chaotic, 73 characterizing a route to chaos via a period doubling or Feigenbaum's scenario [11].

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Impedance sensors are very well established in the industry due to their versatility 75 and their main appeal for being non-invasive and non-contact. The first attribute is 76 given by the adjustable penetration of the electric field in matter, while the last one 77 is due to the charge induction (capacitive coupling). Impedance sensors can either be 78 individually classified as capacitive and inductive for single frequency applications.

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In the case of multifrequency systems, impedance sensors measure a spectrum in a 80 wide frequency range. Therefore, impedance sensors can measure the change in either 81 resistance (the real part of the impedance) or reactance (the imaginary part of the 82 impedance) or both. It is known that the resistance depends mostly on both resistivity of 83 the material and electrode (sensor) geometry [14]. Many biological applications use the 84 phase angle given by arctan(reactance/resistance) in order to characterize the sample 85 under study. Therefore, this type of sensor can have great applicability within the 86 context of biomedical engineering, since skin hydration levels are typically characterized 87 by measurements of electrical skin impedance or thermal conductivity, or by optical 88 spectroscopic techniques, including reflectivity. Indirect methods include assessing the 89 mechanical properties of skin or its surface geometry. Among these methods, electrical 90 impedance provides the most reliable and established evaluation, due to its instrumental 91 simplicity and minimized cost [15].
Sensors can also be classified as being passive or active devices. Active sensors 93 require an external power supply to produce a signal at the output, which may be 94 changed according to its properties. Unlike an active sensor, a passive sensor does not 95 need any additional power supply, but it generates an output signal in response to some 96 external stimulus. For example, a thermocouple that generates its own voltage output 97 when exposed to heat. Therefore, passive sensors are direct sensors that change their 98 physical properties, such as resistance, capacitance or inductance [16].

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Some applications, in which passive impedance sensors based on chaotic circuits 100 are of great interest, are: i) monitoring of skin hydration and temperature changes [15]; ii) 101 investigations of skin nonlinear properties [17]; iii) investigations related to Bioelectrical 102 Impedance Analysis (BIA) [18]; iv) optimization of bio-oscillators [19] [20]. The second 103 application suffers from low sensitivity when measuring its properties. Sensitivity signal 104 may also be related to precision and/or accuracy. Either precision or accuracy of the 105 measured output voltage depends on many things, such as movement artifacts. These 106 artifacts (noise) change significantly the skin impedance which, in turns, cannot not 107 be separated from the total measured impedance. It is known from [17] that skin has 108 a memristive characteristic, where the skin resistance depends on the applied voltage 109 used to characterize it. Therefore, a low output voltage may imply in a low resistance  The chaotic region is also shown for the position of periodic window islands, some of 122 which exhibit period doubling phenomena. The latter case, in turn, can be analyzed 123 using a bifurcation diagrams obtained experimentally that will provide a relationship 124 between the resistance, which represents one of the system parameters, and the attractor 125 period [13]. 126

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The dynamics of the system that linearly couples the van der Pol and damped Duffing oscillators is given by: where µ is a positive parameter that controls one of the model's nonlinear behaviors, α 128 corresponds to the dissipation term and k the coupling constant.

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Lets make x = x 1 , y = y 1 ,ẋ = x 2 andẏ = y 2 , then the system can be rewritten as: This way, an integrating circuit can be built following the schematic shown in figure   135 1, where τ is the time of integration (R b = 100 kΩ and C b = 10 nF).

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An experimental bifurcation diagram was obtained from the coupled van der Pol 167 and Duffing damped oscillators integrated circuit, as shown in Figure 3.  To obtain the most appropriate curve fit, it is necessary to select in the software a suggestion about the mathematical structure. For this reason, the equation named by normal form was sought, this is obtained when expanding the equations that describe the Taylor series system model around the fixed point. There are several types of bifurcation and for each one of them a specific normal form equation [22]. For the period doubling fork, according to [23], the following equation must be used to adjust the curve: where P represents the system period, R the sensor resistance and a, b and c are coeffi-176 cients.

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The curve fit for the points in table 1 using equation 7 is shown in Figure 5. Based on Figure 5, coefficient values of equation (7) are calculated, so that: Based on the sensor's characteristic resistance curve, its sensitivity is calculated.

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This is defined as the variation of the input parameter required to produce a normalized 180 variation at the output. Generically, sensitivity is defined as a variation of the output 181 signal, for a given variation of the physical input parameter. According to [24], the 182 sensitivity S of a transducer can be calculated as: where RFS is the full-scale resistance and PFS is the measured full-scale period.

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For the sensor being proposed, according to data in Table 1, it is shown that:

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Even with various ways for implementing chaotic circuits for sensing mentioned in 187 [5] and [1], none of these are focused on impedance sensing for bioimpedance applica-    We believe that the behavior presented by the bifurcation diagram shown in figure 4 268 is such a case, and visually there is no way to count the period doublings after the 4th