1 Introduction: topology in computational electronics

Based on the two fundamental attributes (electric charge q and magnetic flux φ) [1,2,3,4], we found that all the (two-terminal) electric elements are topologically homeomorphic [5] in terms of physically interacting charge q (or its αth-order variant q(α)) and flux φ (or its βth-order variant φ(β)), in which $$\alpha$$ or $$\beta$$ can even be a fraction to reflect the fractional coupling between electricity and magnetism. Figure 1 describes such a topological homeomorphism, in which “genus” represents a class, kind, or group marked by common characteristics or by one common characteristic.

As a mathematical study of the structural properties of objects, topology is motivated by the fact that some scientific problems depend not on the exact geometric shape of the objects involved, but rather on the way they are organised and interconnected together [5]. As shown in Fig. 1, a sphere and a cube have a property in common: both separate the space into two parts, the part inside and the part outside. Homeomorphism is the isomorphism in the category of topological spaces [5]—that is, two objects are homeomorphic if one can be continuously deformed into the other. Referring to the change in size or shape of an object, (continuous) deformation includes stretching and bending but excludes cutting or gluing.

In order to describe the physical charge-flux interaction in an electric element in Fig. 2b and then predict the behaviour of a real device, we used the LLG equation:

$$\left( {1 + g^{2} } \right)\frac{{d\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}}{{{\varvec{M}}_{{\varvec{S}}} }} \left( t \right)}}{dt} = - \left| \gamma \right|\left[ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}}{{{\varvec{M}}_{{\varvec{S}}} }} \left( t \right) \times \mathop{H}\limits^{\rightharpoonup} } \right] - \frac{g\left| \gamma \right|}{{M_{S} }}\left[ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}}{{{\varvec{M}}_{{\varvec{S}}} }} \left( t \right) \times \left( {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}}{{{\varvec{M}}_{{\varvec{S}}} }} \left( t \right) \times \mathop{H}\limits^{\rightharpoonup} } \right)} \right],$$

where MS is the saturation magnetization, $$H\propto i$$ is a magnetic field along z, g is the Gilbert damping parameter and ϒ is the gyromagnetic ratio [6, 7]. The following expression was deduced:

$$m\left( t \right) = \tanh \left[ {\frac{q\left( t \right)}{{S_{W} }} + C} \right],$$
(1)

in which $$m\left(t\right)={M}_{z}(t)/{M}_{S}$$ (Mz is the Z component of MS), SW is a switching coefficient and C is a constant of integration such that $$C={tanh}^{-1}{m}_{0}$$ (m0 is the initial value of m) if q(t = 0) = 0 (no accumulation of charge at any point).

By Faraday’s law, the induced voltage v(t) across the two terminals of the conductor is:

$$\mu_{0} S\frac{{dM_{z} }}{dt} = S\frac{{dB_{z} }}{dt} = \frac{{d\varphi_{z} }}{dt} = - v\left( t \right),$$
(2)

where μ0 is the permeability of free space and S is the cross-sectional area.

From Eq. 2, we obtain

$$\varphi = \mu_{0} SM + C^{\prime} = \mu_{0} SM_{S} m + C^{\prime},$$
(3)

where $${C}^{^{\prime}}$$ is another constant of integration.

Assuming $$\varphi \left(t=0\right)=0$$, we have $${C}^{^{\prime}}=-{{\mu }_{0}S{M}_{S}m}_{0}$$, so

$$\varphi \left( q \right) = \mu_{0} SM_{s} \left[ {\tanh \left( {\frac{q}{{S_{W} }} + \tanh^{ - 1} m_{0} } \right) - m_{0} } \right].$$
(4)

A typical q-φ curve with m0 = − 0.964 is depicted in Fig. 6b, which agrees with those experimentally observed q-φ curves [9,10,11]. By nature, it is nonlinear, continuously differentiable, and monotonically increasing (the three ideality criteria [2]). Hence, Eq. 4 is reasonably used as one of the two exemplified constitutive curves for an ideal element [q(α), φ] that is defined based on the q-φ interaction.

The article is structured as follows. In Sect. 1, we introduce topology and identify its potential applications in computational electronics. In Sect. 2, we prove that the constitutive space of an electric element is a differential manifold. In Sect. 3, we highlight that topology captures the essence of what remains unchanged and report super conformality is such a topological invariant. In Sect. 4, we point out that the periodic table of the electric elements can be dramatically reduced to have only six passive electronic elements based on our newly found super conformality. In Sect. 5, we prove a theorem that an electric element is locally active if its mid-point is nonzero. In Sect. 6, we mention an experimental verification in the famous Hodgkin–Huxley circuit and summarize our study.

2 The constitutive space of an electric element is a differential manifold

Topologically, the constitutive q-φ space of an electric element is a differential manifold (a type of manifold that is locally similar enough to a vector space to allow one to do calculus [12]) because it is globally defined with a differentiable but possibly complex structure (especially when characterising a higher-integral-order element).

As a topological space that locally resembles Euclidean space near each point, a manifold (with a possibly complex structure) is something similar to a globe (a 3D spherical model of the Earth), which cannot be unfolded onto a 2D plane without distorting its surface due to the curvature. However, as shown in Fig. 3, the surface of a globe can be approximated by a collection of 2D maps (also called charts), which together form an atlas of the globe. Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts [12].

In topology, each map (chart) locally resembles a linear space near each point, as shown in Fig. 4. These Euclidean pieces can be patched together to form the original manifold. A manifold can be described by two charts U1 and U2. If the transition ($${f}_{1}\bullet {f}_{2}^{-1}$$ or $${f}_{2}\bullet {f}_{1}^{-1}$$) from one chart to another is differentiable, then computations done in one chart are valid in any other differentiable chart.

Topologically, the constitutive space of an electric element is a differential manifold as shown in Fig. 5. Some practical (nonideal) devices exhibit a double-valued q-φ curve, and a pinched i-v curve is asymmetric against the origin [13]. This is a generic case for an electric element in terms of physically interacting electric charge q (or its αth-order variant q(α)) and magnetic flux φ (or its βth-order variant φ(β)).

The mapping function (f1 or f2) can be the so-called conformal differential transformation when characterising a 0th-order element [1, 2] or a more complex transformation [4] when characterising a higher-integral-order element.

Actually, the projection from the q-φ plane to the i-v plane in Fig. 5 is based on the following theorem.

Theorem I: (Conformality theorem)

The fractional differential transformation that covers both integer-order and fraction-order is conformal.

Proof

$$\Psi = arctg\left( {\frac{{d^{\Omega } \hat{\varphi }\left( q \right)}}{{dq^{\Omega } }}} \right) = arctg\left( {\frac{{\frac{{d^{\Omega } }}{{dt^{\Omega } }}\hat{\varphi }\left( q \right)}}{{\frac{{d^{\Omega } }}{{dt^{\Omega } }}q\left( t \right)}}} \right) = arctg\left( {\frac{v\left( t \right)}{{i\left( t \right)}}} \right) = \Psi^{\prime}$$

Where $$\frac{{d}^{-\Omega }}{{dt}^{-\Omega }}$$ is the fraction-order calculus operator ($$0\le\Omega \le 1$$)

The introduction of fraction-order calculus dealing with any order of derivatives or integrals [14, 15] here describes some nonideal memristors whose charge-flux coupling is fractional. This theorem indicates that the differential transformation in the fraction-order still preserves angles in the same way as the (traditional) integer-order transformation does [1, 2, 4].

Obviously, the (differential) memristance of a point on the q-ϕ curve equals the chord memristance of the corresponding point on the i-v curve in the fractional transformation. This chord resistance actually defines a generic “state-dependent Ohm's law” for all the electric elements as follows:

$$v\left( t \right) = \frac{{d^{{\Omega }} \hat{\varphi }^{\left( \beta \right)} \left( q \right)}}{{dq^{{\left( \alpha \right)}{\Omega }} }}i\left( t \right)$$
(5)

Theorem II: symmetry theorem

An ideal memristor has an odd-symmetric voltage-current hysteresis loop and a symmetric memristance hysteresis loop if it is driven by an odd-symmetric periodic excitation current.

Proof

If and only if the two branches (U1 and U2) in the q-φ plane (Fig. 5) overlap and the two smooth mapping functions (f1 and f2) are the same, i.e. if and only if $${U}_{1}={U}_{2}$$ and $${f}_{1}={f}_{2}$$, the two i-v loops [$${\widehat{v}}_{1}\left(i\right) and {\widehat{v}}_{2}\left(i\right)$$] in the local coordinate systems will become identical:

$$\hat{v}_{1} \left( i \right) = f_{1} \left( {U_{1} } \right) = f_{2} \left( {U_{2} } \right) = \hat{v}_{2} \left( i \right)$$
(6)

or

$$\hat{v}_{1} \left( i \right) = f_{1} f_{2}^{ - 1} \left( {\hat{v}_{2} \left( i \right)} \right) = \hat{v}_{2} \left( i \right)$$
(7)

After rotating the 2nd coordinate system in Fig. 5 by π, an odd-symmetric voltage-current hysteresis loop arises. Based on this theorem, an ideal electric element with memory should be characterised by a single-valued, unique and time-invariant constitutive curve complying with the following three criteria [2, 13]: 1. Nonlinear; 2. Continuously differentiable; 3. Strictly monotonically increasing. Note that a curve is single-valued if it is strictly monotonic increasing. We need to unambiguously exclude the special case, in which a nonideal memristor has a doubled-valued, strictly monotonic-increasing q-φ curve. Furthermore, the “single-valued” feature vividly depicts Noether's theorem [16] in the sense that every differentiable symmetry of the action of a physical system (that is the aforementioned odd-symmetry of the voltage-current hysteresis loop) has a corresponding conservative quantity (that is the number of the constitutive curve in this case).

In mathematics, conformality means the condition (of a map) of being conformal. Conventional conformality ($$\psi ={\psi }^{^{\prime}}, \gamma ={\gamma }^{^{\prime}}, \eta ={\eta }^{^{\prime}}, \theta ={\theta }^{^{\prime}}$$) exists as a topological invariant under the conformal differential transformation from the constitutive (q, φ) plane to its first-order differential ($$\dot{q}$$,$$\dot{\varphi }$$) plane for an element.

From the concept of local passivity (the origin-crossing of the v-i loci) [17], the following local passivity theorem is reasoned:

Theorem III: (Local passivity theorem)

The first-order electric element is locally passive.

Proof

Criteria 4 (strictly monotonically increasing) means that, if A < B, f(A) < f(B) by monotonicity, thus the slope of $$\widehat{\varphi }(q)$$ is nonnegative ($${\mathrm{f}}^{\mathrm{^{\prime}}}\left({\mathrm{x}}_{\mathrm{t}}\right)\ge 0$$); hence, this ideal memrisor is locally passive at each point on the φ–q curve. If the order “ ≤ ” in the definition of monotonicity is replaced by the strict order “ < ”, then one obtains a strictly monotonically increasing function. Note that $${\mathrm{f}}^{\mathrm{^{\prime}}}\left({\mathrm{x}}_{\mathrm{t}}\right)=0$$ only at those isolated points, rather than any continuous range, otherwise it violates the definition of monotonicity.

The origin-crossing signature leads to local passivity, which can be proved by contradiction: a cell is said to be locally active at a cell equilibrium point $${\mathbb{Q}}$$ if, and only if, there exists a continuous input time function $${i}_{\alpha }\left(t\right)\in {\mathbb{R}}^{m}, t\ge 0$$, such that at some finite time $$T, 0<T<\infty$$, there is a net energy flowing out of the cell at $$t=T$$, assuming the cell has zero energy at t = 0, namely, $$w\left(t\right)={\int }_{0}^{T}{v}_{\alpha }\left(t\right)\bullet {i}_{\alpha }\left(t\right)dt<0$$, for all continuous input time functions $${i}_{\alpha }\left(t\right)$$ and for all $$T\ge 0$$, where $${v}_{\alpha }\left(t\right)$$ is a solution of the linearized cell state equation about $${\mathbb{Q}}$$ with zero initial state $${v}_{\alpha }\left(t\right)=0$$ and $${v}_{b}\left(t\right)=0$$ [17]. If $$i=0, v\ne 0$$ or vice versa, we should have an intersectional point crossing the $$i=0$$ or the $$v=0$$ axis, which implies that the v-i curve must inevitably enter either the second or the fourth quadrant of the v-i plane. Therefore, a cell with $$i=0, v\ne 0$$ or vice versa must be locally active since $$i(t)$$ and $$i(t)$$ always have opposite signs in the second or the fourth quadrant of the v-i plane hence $$w\left(t\right)={\int }_{0}^{T}{v}_{\alpha }\left(t\right)\bullet {i}_{\alpha }\left(t\right)dt<0$$.

3 Super conformality as a topological invariant

In this section, super conformality in the generic form (x, y) was proposed based on strict mathematical deduction and reasoning. In Fig. 6, (x, y) represent the two constitutive attributes for an electric element. Taking a 2nd-order memristor (α = − 2, β = − 2) as an example, beyond the 1st-order setting, it requires double-time integrals of voltage and current, namely, $$x=\int qdt=\iint idt, \dot{x}=q=\int i dt, \ddot{x}=i$$ and $$y=\int \varphi dt=\iint v dt, \dot{y}=\varphi =\int v dt, \ddot{y}=v$$.

A 3rd-order ideal inductor [3] should be characterized by a time-invariant $$\int q-\iint \varphi$$ curve, thus $$x=\int q dt, \dot{x}=q, \ddot{x}=i$$ and $$y=\iint \varphi dt, \dot{y}=\int \varphi dt, \ddot{y}=\varphi$$.

A 3rd-order ideal capacitor [3] should be characterized by a time-invariant $$\iint q-\int \varphi$$ curve, thus $$x=\iint q dt, \dot{x}=\int qdt, \ddot{x}=q$$ and $$y=\int \varphi dt,\dot{y}=\varphi =\int v dt$$, $$\ddot{y}=v$$.

From $${y}_{t}={f}_{{x}_{t}}({x}_{t})$$ in the x + jy plane in Fig. 5, we have

$$\dot{z}_{t} = \dot{x}_{t} + j\dot{y}_{t} = \dot{x}_{t} + j\frac{{d f\left( {x_{t} } \right)}}{dt} = \dot{x}_{t} + jf_{{x_{t} }}^{^{\prime}} \left( {x_{t} } \right)\dot{x}_{t}$$
(8)

Actually, Eq. 8 verifies Theorem I (Conformality Theorem) between the x + jy plane and the $$\dot{x}+j\dot{y}$$ plane (not shown in Fig. 6): the line tangent $${f}^{^{\prime}}({x}_{t})$$ to the $${y}_{t}=f({x}_{t})$$ curve at any operating point ($${x}_{t}, {y}_{t}$$) in the x + jy plane is equal to the (chord) slope of a straight line connecting the projected point ($${\dot{x}}_{t}, {\dot{y}}_{t}$$) to the origin in the $$\dot{x}+j\dot{y}$$ plane [1, 2, 4].

Observing Eq. 8, we should obviously have $${\dot{z}}_{t}={\dot{x}}_{t}+j{\dot{y}}_{t}{=j\bullet f}_{{x}_{t}}^{^{\prime}}\left({x}_{t}\right)\bullet {\dot{x}}_{t}=j\bullet 0$$ when $${\dot{x}}_{t}=0$$. That is, the curve in the $$\dot{x}+j\dot{y}$$ plane must go through the origin, as stated in Theorem III: (Local Passivity Theorem).

Furthermore, we can use differentiation-by-parts to obtain

$$\mathop y\limits_{t} = \frac{{d \dot{y}_{t} }}{dt} = f_{{x_{t} }}^{^{\prime\prime}} \left( {x_{t} } \right)\dot{x}_{t}^{2} + f_{{x_{t} }}^{^{\prime}} \left( {x_{t} } \right)\mathop x\limits_{t}$$
(9)

That is, when $${\ddot{x}}_{t}\left(t\right)=0$$, we obtain

$$\mathop z\limits_{t} = \mathop x\limits_{t} + j\mathop y\limits_{t} = jf_{{x_{t} }}^{^{\prime\prime}} \left( {x_{t} } \right)\dot{x}_{t}^{2}$$
(10)

Equation 10 is a key finding of this topological electronics, which bridges the constitutive x + jy complex plane and its second-order differential $$\ddot{x}+j \ddot{y}$$ complex plane.

Next, let us use proof-by-contradiction [19] to prove that $$f_{{x_{t} }}^{{''}} \left( {x_{t} } \right) \ne 0$$. If $${f}_{{x}_{t}}^{{''}}\left({x}_{t}\right)=0$$ at every operating point, we should have $${f}_{{x}_{t}}^{^{\prime}}\left({x}_{t}\right)=k$$ where k is an arbitrary constant and consequently $${f}_{{x}_{t}}\left({x}_{t}\right)=k{x}_{t}+C$$ where C is another arbitrary constant. Obviously, $${f}_{{x}_{t}}\left({x}_{t}\right)=k{x}_{t}+C$$ is a linear function, which is in contradiction to Criterion 1 (nonlinearity) for an ideal electric element defined in Sect. 2. Therefore, $${\ddot{y}}_{t}={f}_{{x}_{t}}^{{''}}\left({x}_{t}\right)\bullet {\dot{x}}_{t}^{2}\ne 0$$ under any excitation $${\dot{x}}_{t}\ne 0$$.

The following local activity [17] theorem for a 2nd-order or higher-integral-order electric element can then be obtained from Eq. 10:

Theorem IV: (Local activity theorem)

A second-integral-order or higher-integral-order electric element is locally active.

If we assume that an excitation $${x}_{t}\left(t\right)=1-\mathrm{cos}t$$, we should have $${\dot{x}}_{t}\left(t\right)=\mathrm{sin}t=1$$ and $${\ddot{x}}_{t}=\mathrm{cos}t=0$$ when $$t=\frac{\pi }{2}, \frac{3\pi }{2}, \dots$$. From Eq. 10, we obtain the coordinate $${\ddot{y}}_{t}={f}_{{x}_{t}}^{{''}}\left({x}_{t}\right)\bullet {\dot{x}}_{t}^{2}={f}_{{x}_{t}}^{{''}}\left({x}_{t}\right)$$ when the second differential curve $${\ddot{y}}_{t}\left(t\right)=h({\ddot{x}}_{t})$$ crosses the y axis, i.e. $${\ddot{x}}_{t}\left(t\right)=\mathrm{cos}t=0$$. Therefore, the following super conformality theorem is reasoned from Eq. 10:

Theorem V: (Super conformality theorem)

The second derivative $${f}_{{x}_{t}}^{{''}}({x}_{t})$$ at the mid-point ($${x}_{t}=1, {y}_{t}$$) of the $${y}_{t}={f}_{{x}_{t}}({x}_{t})$$ curve in the x + jy complex plane is equal to the slope of a straight line connecting the point $${\ddot{z}}_{t}={\ddot{x}}_{t}+j{\ddot{y}}_{t}=-1$$ to the point $${\ddot{z}}_{t}={\ddot{x}}_{t}+j{\ddot{y}}_{t}=j\bullet {f}_{{x}_{t}}^{{''}}\left(1\right)$$ in the $$\ddot{x}+j\ddot{y}$$ complex plane.

This super conformality theorem is vividly depicted in Fig. 6. A critical angle $${f}_{{x}_{t}}^{{''}}({x}_{t}=1)$$ is preserved between the constitutive x + jy complex plane and its second-integral-order differential $$\ddot{x}+j\ddot{y}$$ complex plane.

4 A reduced periodic table of only six passive electric elements

Finding a correct, accurate electric element table in electrical/electronic engineering is similar to the discovery of Mendeleev’s periodic table of chemical elements [20]. An electric element table would help understand the complex world of modern electronics and request rewriting the electrical/electronic engineering textbooks. Unfortunately, the previous unbounded table predicted 40 years ago [21, 22] has an infinitive number of electric elements, as shown in Fig. 7.

Based on our newly found Theorem IV (Local Activity Theorem) and Theorem V (Super Conformality Theorem), a reduced periodic table (in green) of six electric elements is also displayed in Fig. 7. In contrast, the Standard Model of elementary particles counts six flavours of quarks and six flavours of leptons [23]. Such a periodic table may help reveal the deep physical origin of elements, categorise the existing elements and predict new elements.

5 Mid-point theorem

Our study can be simplified to the following mid-point theorem when $${x}_{t}\left(t\right)=1-\mathrm{cos}t$$:

Theorem VI: (Mid-point theorem)

If and only if the second-derivative at the mid-point of the constitutive curve is nonzero for a two-terminal electric element, it is locally active.

Actually, this theorem vividly describes our finding: all higher-integral-order passive memory electric elements [memristor (α ≤ − 2, β ≤ − 2), higher-integral-order meminductor (α ≤ − 3, β ≤ − 2), and higher-integral-order memcapacitor (α ≤ − 2, β ≤ − 3)] must have an internal power source (Fig. 8).

According to the mid-point theorem, the passive version of a higher-integral-order electric element should not exist in nature. Even if it had existed, it would have degenerated into a zeroth-order negative nonlinear element. Nevertheless, the active version of a higher-integral-order electric element may still be found in nature (with either an electric, optical, chemical, nuclear or biological power source). For example, both the first-state-order potassium memristor and the second-state-order sodium memristor in the Hodgkin–Huxley circuit are higher-integral-order memristors with an internal power source [24, 25]. The active version may also be built as an artefact in the laboratory with the aid of transistors, operational amplifiers, and/or power supplies.

By coincidence, this phenomenon is quite similar to Mendeleev’s periodic table of chemical elements, in which a chemical element with a higher atomic number is unstable and may decay radioactively into other chemical elements with a lower atomic number [20].

6 Conclusion and discussions

Topology is such a subject that examines geometric objects and captures the essence of what remains unchanged when they smoothly, continuously transform into one another. To date, topology has been used in many applications, such as biology, computer science, physics, robotics, games and puzzles, and fibre art. For example, the 2016 Nobel Prize in physics was awarded for theoretical discoveries of topological phase transitions of matter [26].

Motivated by the above achievements, in this work, we used topology as a powerful tool to categorize the electric elements via a pair of integer/fraction numbers ($$\alpha , \beta$$). When an electric element is described by the right pair of numbers ($$\alpha , \beta$$), we found a brand new feature that is super conformality. As a result, a reduced periodic table of only six passive electric elements was then proposed, including resistor [27], inductor [28], capacitor [29], memristor [1], meminductor [3], and memcapacitor [3]. In our opinion, this reduced table reveals the topological homeomorphisms of the electric elements.

Our claim that all the higher-integral-order (≥ 2) electric elements must be active was experimentally verified by the fact that both higher-integral-order memristors in the famous Hodgkin–Huxley circuit are locally active [24]. Here, it is worth clarifying the relationship between an ideal electric element (normally in theory) and those real devices (in practice) belonging to the same family led by that ideal electric element [4]. An analogue in Chemistry is that Mendeleev predicted the existence of the (pure, ideal) chemical elements in his periodic table of chemical elements [20] and other chemists kept claiming they had found those elements one by one in spite of their purities are less than 100% in the real world. In 1971, to link flux φ and charge q, Chua predicted an ideal memristor, whose state variable is q or φ [1]. 37 years later, HP announced “The missing memristor found” [29]. As a real device that is highly unlikely to be ideal, the HP memristor is not ideal as its state variable is the width "w" of the active TiO2 domain (doped with ions), which is still a function of charge q (as well as other factors). Simply speaking, the state variable of the HP memristor is still q (with the “purity” less than 100%). Despite its non-ideality, the HP memristor is still well-recognized as the first physical (1st-order) memristor since it exhibits the zero-crossing signature as predicted [1]. That is, the 1st-order memristor is passive. Similar to the physical HP memristor, both the 1st-state-order potassium memristor and the 2nd-state-order sodium memristor (in practice) are not ideal as their state variable are ionic gate probabilities, which are still a function of q (with the “purity” less than 100%). The important thing is that these two higher-integral-order memristors are active with an internal battery [24], which agrees with our claim.

By coincidence, this dramatically reduced number of electric elements with memory as key building blocks of modern electronics is analogous to a many-awards-winning breakthrough in mathematics: the bounded gap between two primes has been reduced from 70,000,000 to 6 [30]. Actually, two primes that differ by six are called sexy primes since “sex” is the Latin word for “six” [31]. In a similar way, this reduced periodic table of only six passive electric elements may also be called a sexy table.