## Abstract

We introduce a mathematical framework for the analysis of the input–output dynamics of externally driven memristors. We show that, under general assumptions, their dynamics comply with a Bernoulli differential equation and hence can be nonlinearly transformed into a formally solvable linear equation. The Bernoulli formalism, which applies to both charge- and flux-controlled memristors when either current or voltage driven, can, in some cases, lead to expressions of the output of the device as an explicit function of the input. We apply our framework to obtain analytical solutions of the *i*–*v* characteristics of the recently proposed model of the Hewlett–Packard memristor under three different drives without the need for numerical simulations. Our explicit solutions allow us to identify a dimensionless lumped parameter that combines device-specific parameters with properties of the input drive. This parameter governs the memristive behaviour of the device and, consequently, the amount of hysteresis in the *i*–*v*. We proceed further by defining formally a quantitative measure for the hysteresis of the device, for which we obtain explicit formulas in terms of the aforementioned parameter, and we discuss the applicability of the analysis for the design and analysis of memristor devices.

## 1. Introduction

According to classical electrical circuit theory, there are three fundamental passive circuit elements: the resistor, the inductor and the capacitor. Back in 1971, Chua (1971) challenged this established perception. He realized that only five out of the six possible pairwise relations between the four circuit variables (current, voltage, charge and voltage flux) had been identified. Based on a symmetry argument, he postulated mathematically the existence of a fourth basic passive circuit element that would establish the missing link between charge and voltage flux. The postulated element was named the *memristor* because, unlike a conventional ohmic element, its instantaneous resistance depends on the entire history of the input (voltage or current) applied to the component (Oster & Auslander 1972; Oster 1974). Hence, the memristor can be understood in simple terms as a nonlinear resistor with memory. Chua & Kang (1976) and Chua (1980) generalized this concept to a broader family of nonlinear dynamical systems, which they termed *memristive*, and demonstrated their usefulness in the modelling and understanding of physical and biological systems such as discharge tubes, the thermistor or the Hodgkin–Huxley circuit model of the neuron. Although in the intervening years, experimental devices with characteristics similar to the memristor (i.e. hysteresis, zero crossing and memory) were investigated, researchers generally failed to associate or explain them in the context of memristive systems (Baatar *et al.* 2009).

It is worth noting that the memristor is not the only resistive element exhibiting non-volatile memory. For example, a decade earlier than Chua, Windrow introduced the *memistor*, a three-terminal passive transistor-like element in which one of its terminals (the control electrode) is used to adjust the resistance between the remaining two electrodes. Although the memistor is passive and exhibits non-volatile, history-dependent resistance as well, it is a three-terminal element and should not be confused with the memristor, which is a two-terminal element (Widrow 1960; Widrow *et al.* 1961). In fact, it has been recently suggested that a memistor can be built from two memristors (Xia *et al.* 2011).

The memristor remained an elusive theoretical component until recently, when scientists at Hewlett–Packard (HP) fabricated a solid-state device that was recognized as a memristor (Strukov *et al.* 2008; Williams 2008). In that work, Strukov and co-workers also provided a simple heuristic model to understand how memristive behaviour could emerge from the underlying physical process. The report of the fabrication of the HP memristor reignited the interest of the community in such circuits, owing to their potential widespread applications. This has resulted in various attempts to build memristor devices based on different underlying physical principles, as well as efforts to understand pre-existing devices in the context of memristive systems. Examples include thin films (Williams 2008; Yang *et al.* 2008; Borghetti *et al.* 2009; Driscoll *et al.* 2009; Pickett *et al.* 2009; Strukov & Williams 2009), nanoparticle assemblies (Kim *et al.* 2009), spintronics (Pershin & Di'Ventra 2008) or neurobiological systems (Pershin *et al.* 2009; Perez-Carrasco *et al.* 2010). On the theoretical front, however, the analysis of these systems has been mostly restricted to numerics (i.e. temporal integration of model differential equations and numerical sweeping of parameters) due to the absence of a general mathematical framework that can provide analytical solutions to their dynamics.

Motivated by this lack of tools for the analysis of memristors, we introduce here a mathematical framework for their study that allows us to provide analytic solutions for their dynamics and input–output characteristics. The framework is based on the compliance of a general class of ideal memristor dynamics with Bernoulli's differential equation, a classic nonlinear equation that can be solved analytically. This formulation provides a powerful and systematic methodology for the analysis, characterization and design of devices governed by Bernoulli dynamics that does not rely on computationally expensive sweeping of parameters (Drakakis 2000; Drakakis & Payne 2000; Drakakis *et al.* 2010). Moreover, our analytical results can be used to reveal the relevant combinations of parameters that govern the response characteristics of the memristor.

The paper is structured as follows. Firstly, we introduce the general theoretical framework for memristor dynamics and the analytical solutions that follow from the Bernoulli differential equation. We then illustrate its application through the HP memristor (Strukov *et al.* 2008), which is shown to comply with the Bernoulli framework, and we obtain analytical expressions of the output of the model when excited by three fundamental inputs: sinusoidal, bipolar square and triangular waves. To exploit further the insight provided by the framework, we define a quantitative measure of the hysteresis of the memristor in terms of the work done by the driving signal, and we use the derived analytical solutions to show that the hysteresis of the device depends on a specific dimensionless lumped quantity that combines all the parameters of the model. This lumped parameter relates fabrication and device properties together with characteristics of the input signal and shows how such experimental parameters can be used to design and control the response of the memristor.

## 2. Memristor dynamics under time-varying inputs

### (a) Definitions and background

Consider the four fundamental circuit variables: voltage *v*, current *i*, charge *q* and voltage flux *φ*. There are six distinct relations linking these variables pairwise. Two of these relations correspond to the definitions of charge and voltage flux as time-integrated variables,
Three other links are given by the implicit equations that define the constitutive laws of the generalized fundamental circuit elements,
2.1
2.2
2.3
In order to complete the symmetry of the system-theoretic structure, Chua's insight was to postulate that the remaining link between *q* and *φ* should be completed by another constitutive relation
2.4
which would correspond to a missing element: the *memristor*. In this sense, the memristor complements the other three fundamental circuit elements as the fourth ideal passive two-terminal component (Chua 1971, 1980; Oster & Auslander 1972).

Assume a time-invariant memristor with no explicit dependence on time, i.e. . Then, the time derivative of the constitutive relation (2.4) leads to
2.5
which serves us to define the memristance (Chua 1971; Oster & Auslander 1972),
2.6
Since we are considering *strictly passive* elements, with , the constitutive law (2.4) defines a strictly monotonically increasing function in the (*q*,*φ*) plane. It then follows that we can express the memristor constitutive law as an explicit function either in terms of the flux, , or in terms of the charge, , indistinctly and to our convenience (Apostol 1957).

Take, for instance, a current-driven memristor, in which *i*(*t*) is the time-dependent input under our control, then we can formally express the input–output relationship in terms of *q*, an integrated variable of the input *i*(*t*) over its past,
2.7
a form that makes explicit the memory of the device.

Note, in contrast, that a time-invariant resistor governed by (2.1) has a differential resistance , 2.8 The input–output relationship of a current-driven passive resistor is then given by 2.9 which is in general nonlinear but, unlike equation (2.7), it exhibits no memory.

Hence, the memory property follows from the integration necessary to go from the *i*–*v* plane, which is readily accessible to experiments, to the *q*–*φ* plane in which the memristor constitutive law is defined (Chua 1971; Oster 1974). The memristance (2.6) and the resistance (2.8) are thus dimensionally congruent, but they only coincide if they are both constant, i.e. for the ohmic (linear and constant) resistor (Chua 1971).

### (b) Solution of memristor dynamics under time-dependent inputs: the Bernoulli equation

Although the constitutive relation of the memristor is defined in the *q*–*φ* plane, such devices are characterized experimentally in the *i*–*v* plane. Specifically, it is typical to characterize their output dynamics in response to time-dependent inputs. The particular relationship imposed by the definition of the memristor translates into a restricted form of the differential equations that govern the dynamics in the *i*–*v* time domain. In particular, the dynamics obey a Bernoulli differential equation (BDE) that can be reduced to an associated linear time-dependent differential equation (LDE). In some cases, this structure can be exploited to provide analytical solutions for the input–output dynamics without the need for explicit numerical integration of the model.

The Bernoulli equation has the form (Polyanin & Zaitsev 2003)
2.10
where . If *n*={0,1}, the equation is linear. For *n*>1, this family of nonlinear ODEs can be solved analytically using the nonlinear change of variable *z*=*y*^{1−n} to give the LDE
from which the general solution follows,
2.11
Here, is an integrating factor and *C* is the constant of integration.

In the case of memristor dynamics, the mathematical transformation that relates the BDE/LDE pair is not only a convenient manipulation, but is also associated with the experimental measurement setup and the intrinsic description of the memristor: both the BDE and the LDE can appear depending on the choice of externally driven variable (*i* or *v*) in conjunction with the internal variable (*q* or *φ*) through which the memristive behaviour is controlled.

In particular, consider an experiment in which the memristor is described in terms of the integrated input, e.g. a current-driven memristor with input *i*(*t*) whose memristance is charge controlled, . The dynamical system can then be written in the canonical form of a memristive system,
2.12
This leads directly to the following LDE:
2.13
which has the standard analytical solution shown in table 1.

However, in certain situations, it might be more meaningful (or convenient) to express the memristance as a function of the integrated *dependent* variable. For instance, if the memristor is voltage driven with input *v*(*t*), but the memristance is given in terms of the charge , we have
2.14
and the resulting equation is
2.15
which has a Bernoulli form. Indeed, this is the case for the first HP memristor model (Strukov *et al.* 2008), as shown in the next section.

In general, the output dynamics of the general memristor leads to four cases. Differentiating (2.6) with respect to time yields
2.16
If the device is current driven, the dynamical equations are
2.17a
2.17b
while for the voltage-driven memristor, we have
2.18a
2.18b
where we have emphasized in the equations the explicit dependence on time of the externally controlled drives. Note that the dynamics are formally either a BDE (equations (2.17b) and (2.18b)) or an LDE (equations (2.17a) and (2.18a)), which are related through the nonlinear transformation given above. Assuming *strictly passive* memristors (Chua 1971; Chua & Kang 1976) (for which , or equivalently , is always positive), ensures that the *q*–*φ* curve is a bijective function such that its inverse exists (Apostol 1957). Hence, the memristance can be represented as a function of either the independent or dependent variable, as convenient (i.e. ). Therefore, once we specify the intrinsic integrated variable (i.e. *q* or *φ*) that controls the specific memristor, we can obtain BDEs and LDEs as summarized in table 1 with their corresponding general solutions.

The above discussion shows that the constitutive relationship of the memristor leads generally to Bernoulli dynamics that can be reduced to the corresponding time-varying linear dynamics through a nonlinear transformation. The analytical solutions obtained can be used to understand explicitly the parameter dependence of memristor models as well as to aid in the design process without the need for parameter sweeping via numerical simulations of the dynamics. In particular, we obtain below analytical expressions for the behaviour of a memristor model when driven with different input waveforms. Furthermore, our analytical solutions allow us to show that the behaviour of the system is fully characterized by a particular lumped parameter that incorporates all the physical parameters of the memristor. This renormalized parameter is shown to encapsulate both the hysteretic and nonlinear characteristics of the device and its memristive character.

### (c) Dynamics of the Hewlett–Packard memristor under time-varying inputs

We exemplify our approach with the HP model, which was introduced by Strukov *et al.* (2008) to describe the behaviour of their fabricated memristor. Although this model describes an idealized memristor without detailed consideration for the underlying physical mechanisms, it can still reproduce the fundamental constitutive characteristics of memristive behaviour over a range of conditions. Indeed, both experimental *i*–*v* measurements (Yang *et al.* 2008; Jo *et al.* 2010) and predictions from more detailed models (Strukov *et al.* 2009) show close resemblance to the responses of the HP model (with or without windowing). Therefore, the HP model provides a simple, yet useful approximation, which is valid over given regimes or when a detailed description is unwarranted.

HP's device consists of a thin-film semiconductor of TiO_{2} with thickness *D* placed between two metal contacts made of Pt. The film has two regions: a doped region of thickness *w* with low resistance due to the high concentration of dopants (positive oxygen ions) and an undoped region with thickness (*D*−*w*) and high resistance . The total resistance of the film is modelled as two variable resistors in series whose resistance depends on the position of the boundary between the doped and undoped regions. Applying an external voltage across the device will have the effect of moving the boundary between the doped and undoped region, effectively changing the total resistance of the device. Assuming ohmic electronic conductance and linear ionic drift with average vacancy mobility *μ*_{v}, the device is modelled by the following pair of equations (Strukov *et al.* 2008):
2.19
and
2.20
Equation (2.20) means that the position of the boundary between regions is proportional to the total charge that passes through the device. Hence, equation (2.19) is a memristor relationship of the form given in equation (2.14) with memristance controlled by charge,^{1}
2.21
When the device is voltage driven (as is the case in the original model of Strukov *et al.* (2008)), the dynamics is governed by equation (2.15). A simple inspection of table 1 shows that the solution of the corresponding BDE leads to an *i*–*v* relationship with the following explicit form:
2.22
where the constant of integration is the resistance of the device at *t*_{0}, the initial time at which the input signal is applied.

We can use the explicit solution (2.22) to calculate the output response of the HP memristor to different voltage drives. Here, we will study periodic input signals *v*(*t*) with period *T*_{0}=2*π*/*ω*_{0}, amplitude *A*, and zero mean value, . Such inputs are standard test signals for electronic devices and lead to the classic characterization of the memristor through hysteresis and Lissajous-type *i*–*v* curves (Chua 1971). In particular, we consider three test signals widely used in engineering settings, namely, the sinusoidal, bipolar piece-wise linear and triangular waveforms, all defined over a period 0<*t*≤*T*_{0} as follows:

— sinusoidal voltage input 2.23

— bipolar piece-wise linear wave 2.24

— triangular wave 2.25

For the bipolar piece-wise linear wave, the parameter determines the rise and fall time, assumed equal here. In particular, *m*=0 corresponds to the (Heaviside) square wave, while is the triangular wave (2.25). The three input waveforms are shown in figure 1*a*–*c*.

It is convenient to normalize the *i*–*v* characteristics (2.22) in terms of rescaled variables,
2.26
to give
2.27
where we have defined *β*, a positive dimensionless quantity that combines the parameters of the device and of the input drive,
2.28
Note that the renormalized parameter *β* includes material and fabrication properties of the device (*μ*_{v}, , , *D*); parameters dependent on the preparation of the state of the device (*R*_{0}); as well as properties of the driving signal (*A*, *ω*_{0}). We will show below that the memristive behaviour of the device is fully encapsulated by the parameter *β*.

Using the solution (2.27), the response to inputs (2.23)–(2.25) over a period 0≤*x*≤2*π* is

— sinusoidal voltage input 2.29

— bipolar piece-wise linear wave 2.30

— triangular wave 2.31

Figure 1 shows the response of the HP memristor for different values of *β* under the three periodic inputs shown in figure 1*a*–*c*. Figure 1*d*–*f* shows the (nonlinear) output current from the HP memristor for three values of *β*, with the more positive values having a larger deviation from the linear output from an ohmic (linear) resistor with resistance *R*_{0} (dashed black lines). The response dynamics give rise to Lissajous-type *i*–*v* characteristics, which are typical of memristors when driven by a periodic excitation with zero mean value (Chua 1971). In such cases, the output is a double-valued function that forms a simple closed curve with no self intersections, except at the origin. Equivalently, this corresponds to a hysteresis loop crossing the origin in the *i*–*v* plane (Chua & Kang 1976; Chua 1980). As our results show, the amount of hysteresis in the device is dependent on *β*, which changes as a combination of different experimental factors. Figure 1*g*–*i* illustrate the effect of *β* on the *i*–*v* characteristics: the hysteretic behaviour of the device is enhanced for the most positive values of *β*. Again, the dashed black line is the limiting case when the memristor tends to a linear resistor *R*_{0}, which corresponds to the limiting case of *β*=0.

### (d) The lumped parameter *β* encapsulates the dynamical response of the Hewlett–Packard memristor

Our analysis above shows that the dynamical response of the HP memristor is governed by the lumped parameter *β*. This dimensionless parameter represents in a single quantity the collective effect of device parameters of diverse origin including: material properties, such as carrier mobility and doping ratio (, , *μ*_{v}); device fabrication, such as the depth of the thin-film layer (*D*); preparation of the initial state of the device (*R*_{0}); as well as properties of the driving input (*A*, *ω*_{0}). Therefore, one does not need to consider each individual parameter separately—our analytical expressions show that similar responses can be achieved by changing different properties of the device if they lead to the same value of *β*. Furthermore, for a given fabricated device, one can tune the properties of the input drive in the experiment to produce particular output responses.

We now examine some of the characteristics of the parameter *β*. From the definition (2.28), it follows that *β*>0 since . As *β*→0, the memristor tends to the behaviour of a linear resistor with resistance *R*_{0}. Hence, for a given fabricated device with particular physical characteristics, the linear behaviour will be revealed experimentally in the limit of low-amplitude/high-frequency drive (*A*→0 and/or ). In the process of designing memristors, linear behaviour will be more likely when the dimension of the device is large (); when the mobility of the carriers is low (*μ*_{v}→0); when the doping ratio is small (); or with a combination of all of those. In fact, note that *β*≡0 when . This corresponds to building an ohmic resistor, rather than a memristor, with just one (undoped) region. In this case, the resistor always responds linearly with the same unique resistance *independently* of history or preparation protocol, i.e. .

In the case of the HP model, the value of *β* is also bounded from above, as follows from requiring that the arguments in the square roots in equations (2.29)–(2.31) be positive, so that the output current is real and finite. This leads to the following bounds for *β* for each type of input:
2.32
In order to make the comparison across input drives more direct, it is thus helpful to define the following rescaled parameter:
2.33
such that for all three drives. When , the memristor becomes a linear resistor, while as , the HP memristor becomes maximally nonlinear and hysteretic and the separation of the two branches in the *i*–*v* plane is maximized. In the next section, we make this notion more precise through the introduction of a quantitative measure of hysteresis.

It should be noted that the upper bound of *β*, which stems from the requirement that the output remains real and finite, is a manifestation of a limitation of the first simplified HP memristor model (Strukov *et al.* 2008), i.e. that the device does not saturate or break down as the memristance is driven towards its lower bound . Although such a mechanism is not accounted for by equations (2.19) and (2.20), this limitation has already been addressed by models that contain a windowing function, in which it becomes increasingly difficult to change the memristance as the boundary between the doped and undoped region reaches either of the two ends of the device (Strukov *et al.* 2008; Joglekar & Wolf 2009; Pershin & Di'Ventra 2011).

## 3. Quantitative measure of hysteresis

### (a) Definition

Controlling and designing the hysteretic pinch of the *i*–*v* curve of the memristor is crucial for the use of these devices, both individually or as part of larger circuits. It is important, therefore, to have a method to control the hysteretic effect of a particular model in order to achieve the required specifications for an application. Here, we propose a quantitative measure of the hysteresis of the *i*–*v* curve in terms of the work carried out by the driving input on the device (refer to Nedaaee Oskoee & Sahimi (2011) for another computational measure of hysteresis). We will then use the analytical solutions obtained for the HP model in the previous section to calculate the hysteresis of the model for the three different drives in terms of the parameter . Finally, we will show how these expressions may be used as an aid to fabricate a memristor with a prescribed *i*–*v* curve.

Our measure of hysteresis is based on the calculation of the work done by the input signal through the device. Let *H* denote the positive scalar quantity measuring the difference between the work done while traversing the upper and lower branches of the hysteresis loop in the *i*–*v* plane,
3.1
where *t*_{1} to *t*_{2} and *t*_{2} to *t*_{3} is the time required to move along the upper and lower branch of the hysteresis loop, respectively. Here, *P*(*t*)=*i*(*t*)*v*(*t*) is the instantaneous power and, as it is customary by now, corresponds to the rescaled quantity.

Clearly, *H* is defined in terms of the energy dissipated by the device and becomes zero when the memristor tends to a linear resistor since *W*_{+} and *W*_{−} coincide in that case. Therefore, it is useful to scale the hysteresis by *W*_{0}, the work done on the linear resistor *R*_{0} when driven for one complete cycle by the same signal for which *H* is evaluated,
3.2
We define the scaled hysteresis as
3.3

### (b) Analytical expressions for the hysteresis of the Hewlett–Packard memristor

We now use our analytical results for the HP model to obtain the hysteresis of this device under the three drives specified in the previous section.

#### (i) Sinusoidal drive

Consider an HP memristor with sinusoidal input *v*(*t*)=*σ*(*t*) and the corresponding output current , given by equation (2.29). The hysteresis is then
3.4
The work done by the input signal *σ*(*t*) for a complete cycle on the linear resistor *R*_{0} is
Using the symmetry of the functions, the scaled hysteresis is given by
3.5
which can be rewritten as (Byrd & Friedman 1971; Gradshteyn & Ryzhik 2007)
3.6
where and . Here, *F*(*ϕ*,*k*) and *E*(*ϕ*,*k*) are the incomplete elliptic integrals of the first and second kind, respectively, while *K*(*k*) and *E*(*k*) denote the complete elliptic integrals of the first and second kind, respectively (Byrd & Friedman 1971; Gradshteyn & Ryzhik 2007).

#### (ii) Bipolar piece-wise linear input

After some algebra, the scaled hysteresis of the bipolar input waveform *v*(*t*)=⊓(*t*) is obtained as (Gradshteyn & Ryzhik 2007)
3.7
where *m*′=1−2*m* and .

#### (iii) Triangular wave input

The hysteresis for the triangular wave input *v*(*t*)=∧(*t*) is obtained by particularizing equation (3.7) for ,
3.8

### (c) The dependence of the hysteresis on the parameter *β*

The definition of hysteresis (3.1) is based on the integration of the instantaneous power consumed by the device over the course of an input cycle. Figure 2 shows a plot of the instantaneous power for the HP model driven by the bipolar wave ⊓(*t*). Each of the areas *w*_{1} to *w*_{4} indicate the work done by the input signal for the corresponding time interval. The hysteresis is given by *H*=(*w*_{2}+*w*_{3})−(*w*_{1}+*w*_{4}), where *W*_{+}=*w*_{2}+*w*_{3} and *W*_{−}=*w*_{1}+*w*_{4}. Note that *w*_{2}=*w*_{3}≠*w*_{1}=*w*_{4}. This asymmetry is a reflection of the difference in the work carried out on each of the two branches of the *i*–*v* curve due to the nonlinearity of the device and giving rise to the hysteresis.

The expressions of the memristor hysteresis subject to the three drives given by equations (3.6)–(3.8) are all explicit functions of , the lumped parameter that combines the physical parameters of the device together with the properties of the drive. Figure 3*a* shows that for all drives, the hysteresis of the device is zero when and increases as . It is important to remark that the device has a *finite* maximum value of hysteresis it can exhibit, i.e. the value of does *not* diverge as . In fact, we can use our analytical expressions to calculate the upper bound of the hysteresis for each drive. For instance, from equation (3.5), the maximum hysteresis for the HP memristor driven by a sinusoidal input is
while for the triangular input, the upper bound of the hysteresis is slightly lower,
Therefore, the maximum hysteresis exhibited by the HP memristor, understood as the difference between the work along the upper and lower branches of the *i*–*v* over a period of the input, is equivalent to approximately 50 per cent of the energy dissipated by the equivalent ohmic resistor.

The dependence of the hysteresis on the lumped parameter *β* can be used during the design process of a memristor or during the characterization of a fabricated device. If the aim is to design a memristor with a pre-specified *i*–*v* response that needs to operate under a particular input, one may calculate first the hysteresis *H* of the desired *i*–*v* curve and the value of *β* that will produce the desired response. The identified *β* value can then be used to restrict our fabrication parameters. Similarly, for a given fabricated memristor, one can generate different *i*–*v* characteristics under a particular type of drive with varying frequency and/or amplitude, and/or under a different type of drive. For each *i*–*v* curve, the scaled hysteresis can be obtained from the experimental data. If the data is well described by the HP model, our expressions could be used to fit some of the intrinsic parameters of the device that are contained in *β*. The interplay between hysteresis and *β* is exemplified in figure 3*b* for the *i*–*v* curves generated by a memristor with *β*_{0}=0.9 under the three input drives.

A practical scenario that requires the capability to optimize the hysteretic properties of memristors is when designing memristor-based memories. In such applications, the memristor plays the role of the fundamental memory cell by mapping its resistance state to digital values. Hence, increasing the hysteresis of the device is desirable in order to make the two resistance levels well defined and reliably distinguishable (Xia *et al.* 2009). As explained above, increasing the hysteresis of the device translates to selecting device parameters that maximize the parameter *β*, while, at the same time, satisfying application requirements such as input voltage margins and frequency of operation.

## 4. Discussion

In this paper, we have presented a mathematical framework for the analysis of the input–output dynamics of strictly passive memristors. The passivity assumption is generally met in practical settings. Although these are nonlinear elements, the form of the constitutive relation of the memristor leads in general to Bernoulli dynamics, which can be directly linked to an associated linear differential equation through a nonlinear transformation. Table 1 highlights this general form of memristor dynamics and the duality that emerges when devices with distinct internal control variables are interrogated with different externally controlled variables, i.e. charge- and flux-controlled memristances, which can be voltage and current driven.

Our methodology allows us to obtain, in some cases, the output of the device as an explicit function of the input. We exemplified our analysis with the recently introduced HP memristor model (Strukov *et al.* 2008), which is shown to be a Bernoulli memristor, for which we obtain *analytically* the output current explicitly as a function of the voltage for three typical input signals, namely, the sinusoidal, bipolar piece-wise linear and triangular waveforms. It is worth noting that, although the HP memristor is an idealization, which does not account for dopant drift nonlinearities, it can provide a good empirical description when such physical detail is not necessary (Jo *et al.* 2010). The mathematical analysis of this model has led us to the identification of the governing parameter *β*. This dimensionless lumped parameter combines physical parameters of the device as well as properties of the input drive. We have shown above how this parameter completely dictates the output of the device and hence the hysteretic properties of its *i*–*v* characteristics. Consequently, the parameter *β* can be used as a means to design specific input–output characteristics. We have shown elsewhere (Georgiou *et al.* 2011) that the same parameter also quantifies the amount of nonlinearity in the output spectrum of the device. Consequently, *β* controls both the hysteresis of the memristor and the harmonic distortion that the device introduces. The functional form (2.27) makes apparent how *β* controls the memristive character of the device and reveals the fundamental interlinking between nonlinearity, hysteresis and memory in these devices.

The explicit solutions thus obtained can be of use not only for the understanding of memristive dynamics, but also as a means to parametrize experiments and to study the deviation from idealized models. They can also be used to devise experiments that can reveal the memristive properties of particular devices. For instance, it is possible to use *β* to design an input drive (e.g. its functional form, amplitude and frequency) that will enhance the hysteretic response of a particular memristor, dependent on the physical properties of the device and the underlying transport mechanism of the charge carriers. The parameter *β* also indicates which fabrication parameters are likely to enhance (or reduce) memristive behaviour. In the HP model, the small dimensions of the device, a large carrier mobility and large differences between the doped and undoped resistances all contribute to make the memristive behaviour more conspicuous. The results presented could be applicable to experiments given that the simple HP model studied here has already been shown to provide a reasonable approximation of the behaviour of particular experimental realizations (Jo *et al.* 2010).

Although we have exemplified our work here through a particular example of memristor, this analytical approach has been extended to more complex memristor models that include windowing functions and saturation (Drakakis *et al.* 2010; Cai *et al.* 2011). Furthermore, the current framework can be used to obtain analytical input–output relations for *networks* of memristors and other mem-elements connected to circuit parasitics (Georgiou *et al.* 2011). Since nonlinear hysteretic elements are key components in application fields such as multi-scroll chaotic oscillators (Lu *et al.* 2004; Lu & Chen 2006), an interesting path of future work would be the application of our framework to the design of such nonlinear devices using memristors. The study of such directions as well as the investigation of more realistic memristor models (in particular those that account for nonlinear dopant kinetics) will be the object of further research.

## Footnotes

↵1 Note that

*k*_{2}, the corresponding constant in Georgiou*et al.*(2011), is the negative of*κ*here:*κ*=−*k*_{2}.

- Received September 26, 2011.
- Accepted February 9, 2012.

- This journal is © 2012 The Royal Society