Angela Slavova

Bulgarian Academy of Sciences

Overview

Introduction

Hysteresis memristor CNN model (HM-CNN)

Dynamical behavior of HM-CNN

Simulations and Applications

Conclusion

Introduction Hysteresis in CNN was considered in [7] in different cases, such as CNN with hysteresis in the

feedback circuit, CNN in which the hysteresis is in the output dynamics, etc.

Such models are strongly connected to the neurobiological studies when the neurons can exhibit very

complicated dynamical behavior.

On the other hand, hysteresis is considered as a model of bi-stability [10]. In this model hysteresis is

described by a hysteresis operators. Usually the system of differential equations incorporates one or

more hysteresis operators together with initial and boundary conditions.

One such example is the model in which the memory operator is in the source term:

𝜕𝑢

𝜕𝑡+

𝜕𝑢

𝜕𝑥+ ℎ 𝑢 = 0, (1)

where ℎ(𝑢) ∈ 𝑄 = 𝑎, 𝑏 × 0, 𝑇 is the hysteresis operator given on Fig. 1.

Fig.1. Hysteresis operator.

Introduction

We shall propose a new hysteresis memristor CNN model which exhibits a variety of dynamic routes in

response to different initial conditions.

In this paper we shall consider instead of standard sigmoid function as the output of our HM-CNN

resonant tunnel diode (RTD) output function which leads to better speed of the computations and is

more flexible in implementations.

We shall study the dynamics of the proposed model by dividing the Euclidean space into different

types of regions – stability region (SR), hysteresis region (HR) and partially stability region (PSR).

Since the range of dynamics and the connection complexity are independent of the number of

processing elements (cells), the implementation is reliable and robust. The easy programming, as well,

is based on the geometric aspects of the cloning templates.

We shall propose application of the HM-CNN model in nano-structures. More precisely, in bounded

piezoelectric domain with heterogeneities under anti-plane load

Hysteresis Memristor CNN Model

In this section we shall consider the following parabolic equation with hysteresis operator:

𝜕𝑢

𝜕𝑡− ∆𝑢 + ℱ 𝑢 = 0 in 𝛺 × 0, 𝑇 (2)

where 𝛺 is an open subset of 𝑹𝑁(𝑁 ≥ 1). In [10] existence of a solution is proved by means of

approximation passage to a limit.

We shall applied spatial discretization and we shall transform (2) into a system of ordinary differential equations. This system will be considered as state equations of CNN with appropriate templates. So we obtain:

𝑑𝑢𝑗

𝑑𝑡= 𝐴1 ∗ 𝑢𝑗 − ℱ 𝑢𝑗 , 1 ≤ 𝑗 ≤ 𝑁 , (3)

where 𝐴1 is one-dimensional discretized Laplace template [1,7], 𝐴1 = [1 − 2 1] , * is the convolution

operator, hysteresis operator ℱ 𝑢𝑗 is a real functional defined in the following way: “upper” function ℱ𝑈 and

“lower” function ℱ𝐿 as it is shown on Fig.2 below. Functions ℱ𝑈 and ℱ𝐿 are real valued, piecewise continuousand differentiable. Moreover, they are odd in the sense that

ℱ𝑈 𝑢𝑗 = − ℱ𝐿(𝑢𝑗) .

Hysteresis Memristor CNN Model

Fig.2. Hysteresis operator

For the output of CNN model (3) we shall consider the following piecewise linear function:

𝑦 𝑢𝑗 = 𝑎 𝑢𝑗 + 𝑏 𝑢𝑗 − 𝑉𝑝 − 𝑢𝑗 − 𝑉𝑣 − 𝑏 𝑢𝑗 + 𝑉𝑝 − 𝑢𝑗 + 𝑉𝑣

Hysteresis Memristor CNN Model where 𝑎 > 0, 𝑏 < 0 are constants, 𝑉𝑝, 𝑉𝑣 0 < 𝑉𝑝 < 𝑉𝑣 are the peak and valley voltages of the resonant

tunnel diode (RTD)-based CNN. The above function is symmetric with respect to the origin and its

graph is given on Fig.3 below.

Fig.3. Graph of the output function.

Remark. Resonant tunnel diode (RTD) belongs to a class of quantum effect devices. Moreover, it has

both analog and digital applications.

Hysteresis Memristor CNN Model

We shall replace linear resistors in our CNN model (3) by memristors with locally negative differential

resistance. In this way the state equation of our model will be the following:

𝑑𝑢𝑗

𝑑𝑡= 𝐴1 ∗ 𝑢𝑗 − ℱ 𝑢𝑗 − 𝑚(𝑢𝑗), 1 ≤ 𝑗 ≤ 𝑁 , (4)

m(uj) =𝑢𝑗

𝑀(𝑡), where 𝑀(𝑡) is the memristance of the memristor state resistor.

The new hysteresis memristor CNN (HM-CNN) (4) has some advantages. Its structure is simple,

and it has inherent speed and design flexibility which make it easy for fabrication. The proposed model

can have different applications, one of which will be presented in the paper.

Dynamic behavior of HM-CNN model We shall apply topological method for studying the dynamics of HM-CNN model (4). The aim of this

method is to recognize topologically distinct types of dynamics occurring in the model. We shall

consider those members of the family at which topological changes are possible, such as bifurcations,

periodical solutions, chaos. Therefore, we shall solve for all equilibrium points by first determination

of the stable cell equilibrium points. Then using the neighborhood interactive rules in CNN, we will be

able to find the corresponding system equilibrium points. For this reason we divide the Euclidean space

into different types of regions shown of Fig. 4 below.

Fig.4. Different types of regions.

Dynamic behavior of HM-CNN model

Stability region (SR) is defined for all 𝑢, such that 𝑢 > 1 and ℱ ≥ 1. In the hysteresis region (HR),

ℱ < 1 for all 𝑢 ≤ 1. The definition of this region comes from hysteresis nonlinearity. Partial

stability region (PSR) is considered for 𝑢 ≤ 1, ℱ > 1 and 𝑢 > 1, ℱ = 1.

In this paper we shall study the dynamics of HM-CNN model (4) in the hysteresis region. First we shall

define the equilibrium points 𝑢𝑗 as solutions of

𝐹 𝑢𝑗 = 𝐴1 ∗ 𝑢𝑗 − ℱ 𝑢𝑗 − 𝑚 𝑢𝑗 = 0 (5)

The associated linear system in the small neighborhood of an equilibrium is:

𝑑𝑥

𝑑𝑡= 𝐷 𝐹 𝑢𝑗 𝑥,

where 𝑥 = 𝑢𝑗 − 𝑢𝑗, 𝐷 𝐹 𝑢𝑗 = 𝐽 is the Jacobian matrix of the equilibrium point. Now, let us

consider the hysteresis region HR.

Dynamic behavior of HM-CNN model

It is proved in [7] that in HR there is at least one eigenvalue of the Jacobian matrix which has positive

real part and therefore the state equilibrium points are not stable. This instability may lead to complex

behavior of our HM-CNN model (4). The dynamical behavior for different initial conditions is

obtained numerically on Fig. 5 below.

Fig.5.Dynamical behavior of HM-CNN model in HR

Dynamic behavior of HM-CNN model

To prove the emergence of complexity in HR we shall apply local activity theorem [3]. If the cell acts

like a source of small signal for at least one equilibrium point then we can say that it is locally active.

In this case the cell can inject a net small-signal average power into the passive resistive grids.

We calculate the cell coefficients 𝑎11, 𝑎12, 𝑎21, 𝑎22 of the Jacobian matrix at each equilibrium point.

Then we shall define stable and locally active region such as

𝑎22 > 0 or 4𝑎11𝑎22 < 𝑎12 + 𝑎212 and

𝑇𝑟 𝐸𝑘 < 0 and ∆ 𝐸𝑘 > 0.

It is known [3, 8] that the cells of hysteresis CNN can exhibit complexity in the domain of cell

parameters in which the cells are locally active. This parameter domain is called edge of chaos in the

literature [3].

We shall determine the so called edge of chaos domain for our HM-CNN model (4). According to [3]

edge of chaos (EC) is a region in which there is at least one equilibrium point which is both locally

active and stable. In our case it is defined in the hysteresis region HR.

Dynamic behavior of HM-CNN model

Simulations show the following result for the edge of chaos domain in our HM-CNN model:

Figure 6. Edge of chaos domain in HR.

Simulations and applications We shall present an application of our HM-CNN model in nano-structures. As far as we know there a

few numerical results for the dynamical behavior of bounded piezoelectric domain with heterogeneities

under anti-plane load [4,5].

Let us consider the square domain 𝐺1𝐺2𝐺3𝐺4 with a side 𝑎, containing a single circular

heterogeneity with a radius 𝑐 = 𝛽𝑎 and center at the square center, see Fig. 7. If 𝛽 < 0.05 the

influence of the exterior boundary 𝜕𝐺 on the solution is expected to be small, while if 𝛽 > 0.2 it is

expected significant influence. For heterogeneities at nano-scale we shall take: the side of the square to

be 𝑎 = 10−7𝑚; material parameters inside 𝐼 to be 0; material parameters on 𝑆 = 𝜕𝐼 to be: shear

stiffness 𝑐44𝑆 = 0.1 𝑐44

𝑀 , piezoelectric constant 𝑒15𝑆 = 0.1 𝑒15

𝑀 , dielectric permittivity 𝜀11𝑆 = 0.1 𝜀11

𝑀 ,

mass density 𝜌𝑆 = 𝜌𝑀 [4,5].

Fig.7. Rectangular nano-matrix with

circle heterogeneity.

Simulations and applications

We programme the hysteresis functional ℱ in the following way:

ℱ(𝑢(𝑡𝑛))=

1, 𝑓𝑜𝑟 𝑢 𝑡𝑛 > −1, ℱ 𝑢 𝑡𝑛−1 = 1

−1, 𝑓𝑜𝑟 𝑢 𝑡𝑛 = −1,

−1, 𝑓𝑜𝑟 𝑢 𝑡𝑛 < 1, ℱ 𝑢 𝑡𝑛−1 = −1

1, 𝑓𝑜𝑟 𝑢 𝑡𝑛 = 1,

where 𝑡𝑛 = 𝑛∆𝑡, 𝑛 = 1,2, … We apply a forward Euler algorithm with a time step size ∆𝑡 = 0.01 to all

computer simulations.

Fig.8. Simulation of the hysteresis functional

Simulations and applications The characteristic that is of interest in nano-structures is normalized Stress Concentration Field (SCF) along the perimeter of the

heterogeneity.

Fig.9. Dynamic SCF at observed point.

On Fig.9 we present the simulation of SCF applying our HM-CNN model to a square elastic isotropic plate under uniform uni-axial time-harmonic traction. The following steps of the numerical procedure are realized:

(a) discretization by the usage of parabolic approximation of the unknowns;

(b) collection method along all boundary nodes;

(c) analytical solution based on the asymptotic behavior of the fundamental solution and its derivatives for small arguments;

(d) computation of the displacement and traction at any point in the PEM plane by the usage of HM-CNN model (4).

Numerical simulations show that the stress concentration field near defects is strongly influenced by the type and the size of the defect (hole) and the defect location and geometry. That is why our HM-CNN model is most suitable for such computations. The advantages are easy programming, flexibility and better speed.

Conclusions

In this paper new hysteresis memristor CNN model is proposed which contains special type of

hysteresis functional consisting of “upper” and “lower” function. This allows us to divide the

Euclidean space into three different regions – stability region, hysteresis region and partial stability

region.

We study the dynamics of our HM-CNN model inside the hysteresis region HR. Applying local

activity we determine edge of chaos domain in this region.

We present an application in nano-structures which is very important in the field of computational

nano-mechanics for finding stress concentration field. Simulations and validation are provided in order

to show the advantages of HM-CNN model.

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