bifurcations leading to nonlinear oscillations in a 3d piecewise linear memristor oscillator

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International Journal of Bifurcation and Chaos, Vol. 24, No. 1 (2014) 1430001 ( 18 pages)c World Scientific Publishing Company

DOI: 10.1142/S0218127414300018

Bifurcations Leading to Nonlinear Oscillations

in a 3D Piecewise Linear Memristor Oscillator

Marluce da Cruz Scarabello∗ and Marcelo Messias†

Departamento de Matem´ atica e Computac˜ ao,

Faculdade de Ciencias e Tecnologia,

UNESP – Universidade Estadual Paulista,

19060-900, Presidente Prudente, S˜ ao Paulo, Brazil ∗[email protected] †[email protected]

Received September 27, 2012; Revised March 3, 2013

In this paper, we make a bifurcation analysis of a mathematical model for an electric circuitformed by the four fundamental electronic elements: one memristor, one capacitor, one inductorand one resistor. The considered model is given by a discontinuous piecewise linear system of ordinary differential equations, defined on three zones in R

3, determined by |z| < 1 (called thecentral zone) and |z| > 1 (the external zones). We show that the z-axis is filled by equilib-rium points of the system, and analyze the linear stability of the equilibria in each zone. Dueto the existence of this line of equilibria, the phase space R

3 is foliated by invariant planestransversal to the z-axis and parallel to each other, in each zone. In this way, each solution iscontained in a three-piece invariant set formed by part of a plane contained in the central zone,which is extended by two half planes in the external zones. We also show that the system may

present nonlinear oscillations, given by the existence of infinitely many periodic orbits, each onebelonging to one such invariant set and passing by two of the three zones or passing by the threezones. These orbits arise due to homoclinic and heteroclinic bifurcations, obtained varying oneparameter in the studied model, and may also exist for some fixed sets of parameter values. Thisintricate phase space may bring some light to the understanding of these memristor properties.The analytical and numerical results obtained extend the analysis presented in [Itoh & Chua,2009; Messias et al., 2010].

Keywords : Memristor oscillator; discontinuous piecewise linear systems; Filippov conventions;heteroclinic bifurcation; homoclinic bifurcation; nonlinear oscillation.

1. Introduction and Main Results

For 150 years, the known fundamental elements of an electrical circuit were the capacitor , the resistor

and the inductor , discovered in 1745, 1827 and1831, respectively. However, at the beginning of 1970s, Leon Chua, an engineer at the Universityof California at Berkeley, while attempting forthe six known different mathematical relationships

connecting pairs of the four fundamental circuitvariables, the current i, the voltage v, the chargeq and the flux ϕ, observed that five of these rela-tionships were well-known: two of them are givenby the definition of electric current and Faraday’slaw, that is,

i = dq

dt and v =

dϕ

dt

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http://dx.doi.org/10.1142/S0218127414300018



M. da Cruz Scarabello & M. Messias

and the other three are given by the basic equationswhich axiomatically define the three classical circuitelements, that is,

R = dv

di

, L = dϕ

di

and C = dq

dv

,

where R, L and C represent the resistance, theinductance, and the capacitance respectively. Onthe other hand, Chua realized that one relation-ship was undefined, the relationship between theflux ϕ and the charge q . Based on this analysis, hepostulated in the seminal paper [Chua, 1971] that afourth basic two-terminal circuit element character-ized by a ϕ(q ) curve should exist, and he called itthe memristor as a contraction for “memory resis-tor”, because it behaves somewhat like a nonlinear

resistor with memory. In fact, he mathematicallydemonstrated that this hypothetical device couldprovide a functional relationship between the fluxand the charge, similar to what a nonlinear resis-tor provides between voltage and current. He alsoobserved that the device’s resistance would varyaccording to the amount of charge passing throughit and that it would “remember” the resistancevalue even after the current was turned off, deriv-ing from this property the name memory resistor ormemristor . In [Chua, 1971] the author also provedthat the properties of the memristor could not bereproduced by any combination of the other threeelements, so the memristor could be in fact consid-ered as a new fundamental electronic element, the fourth element .

Although theorized in 1971, only 37 years later,in 2008, a group of scientists from Hewlett–PackardCompany announced in [Strukov et al., 2008] thephysical construction of a device having the prop-erties of a memristor. The time elapsed from thetheoretical postulation to the physical constructionof a memristor was due to the dimension in which

the electronic devices are built. Williams [2008]stated that the memristance — a fundamentalproperty of memristors — can be observed and hasphysical meaning only in nanometric or micrometricscales, being unobservable in milimetric and largerscales.

After the ideas presented in [Chua, 1971], giv-ing the mathematical basis for the existence of memristors, and the announcement of the physicalconstruction of it, several papers appeared in theliterature showing the potential applications as wellas theoretical properties of these devices, as can be

seen for instance by a Google Scholar search for theword “memristor”, which gives more than two thou-sands results. In fact this new element attractedworldwide attention due to its potential applica-tions mainly in the construction of the new gen-

eration computers and memories.

1.1. Memristor constitutive relation

According to [Itoh & Chua, 2009; Chua, 1971], amemristor (see Fig. 1) is a passive two-terminal elec-tronic device, characterized by a nonlinear constitu-tive relation between the voltage v and the currenti. It is said to be charge-controlled (flux-controlled)if this relation can be expressed as a single-valuedfunction of the charge q (of the flux-linkage ϕ).

The voltage across a charge-controlled memris-tor is given by v(t) = M (q )i(t), where

M (q ) = dϕ(q )

dq , (1)

while the current on a flux-controlled memristor isgiven by i(t) = W (ϕ)v(t), where

W (ϕ) = dq (ϕ)

dϕ . (2)

A memristor is characterized by a differen-tiable function q (ϕ) (respectively ϕ(q )) and it ispassive if and only if the derivative of this func-tion M (q ) (respectively W (ϕ)) is non-negative, i.e.M (q ) ≥ 0 (respectively W (ϕ) ≥ 0). The quantityM (q ) in (1) has the unit measure of the resistanceand is called the memristance and W (ϕ) in (2) hasthe unit measure of the conductance and is calledthe memductance .

As an example, in [Itoh & Chua, 2009] theauthors considered a memristor characterized bya monotone-increase and piecewise linear function

+

_

V

i

Fig. 1. Schematic representation of a memristor.

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Bifurcations Leading to Nonlinear Oscillations in a 3D Piecewise Linear Memristor Oscillator

given by,

q (ϕ) = bϕ + 0.5(a − b)(|ϕ + 1| − |ϕ − 1|),

where a, b > 0. Consequently, the memductanceW (ϕ) of the memristor determined by this nonlin-

ear function is given by

W (ϕ) = dq (ϕ)

dϕ =

a, if |ϕ| < 1,

b, if |ϕ| > 1.

This function will be considered ahead, in thememristor oscillator treated here. Other types of nonlinear positive functions representing the mem-ductance were considered for instance in [Ginouxet al., 2010; Messias et al., 2010; Muthuswamy &Chua, 2010]. See also the recent works onother memristor-based oscillators [Buscarino et al.,

2012b; Buscarino et al., 2012a; Corinto & Ascoli,2011, 2012].

1.2. Memristor oscillator obtained

from the canonical Chua ’s

oscillator

Consider the canonical Chua’s oscillator shown inFig. 2(a). Replacing in this circuit the Chua’s diodewith a flux-controlled memristor, Itoh and Chua[2009] obtained the circuit shown in Fig. 2(b).Removing a capacitor in the circuit of Fig. 2(b),

the authors found the third-order oscillator shownin Fig. 3.

Applying Kirchhoff’s circuit laws to node A andloop C of the circuit shown in Fig. 3 we obtain the

i

L

-G+v-

2

C1

+v-

1

Chua’sDiode

C2

(a)

Flux-controlledMemristor

i

L

-G

+v-

2

C1

+v-

1

C2

(b)

Fig. 2. (a) Canonical Chua’s oscillator and (b) canonicaloscillator with a flux-controlled memristor.

C

A

i1

i

i3

L

-R+v-

4

+ v -3

+v-

C1

+v-

1 Flux-controlled

Memristor

Fig. 3. A third-order oscillator with a flux-controlledmemristor.

system i1 = i3 − i,

v3 = v4 − v1.(3)

Integrating (3) with respect to time t, we obtain

the following system of equations, which define therelation between the charge and the flux

q 1 = q 3 − q (ϕ),

ϕ3 = ϕ4 − ϕ1,(4)

where the symbols q 1, q 3 and q denote the chargeon the capacitor C 1, on the inductor L and on thememristor, respectively, and the symbols ϕ1, ϕ3, ϕ4

and ϕ denote the flux on the capacitor C 1, on theinductor L, on the resistor −R, and on the memris-tor, respectively. Considering the following charac-

teristic function for the memristor

q (ϕ) = bϕ + 0.5(a − b)(|ϕ + 1| − |ϕ − 1|)and solving (4) for q 3 and ϕ4, we find

q 3 = q 1 + q (ϕ),

ϕ4 = ϕ3 + ϕ.

Then, q 1, ϕ and ϕ3 may be chosen to be inde-pendent variables. Differentiating (4) with respectto time t, we obtain the system of three first-order

differential equations, which define the relationsamong the variables v1, i3 and ϕ, given by (see[Itoh & Chua, 2009] for details)

C 1dv1dt

= i3 − W (ϕ)v1,

Ldi3dt

= Ri3 − v1,

dϕ

dt = v1,

(5)

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where

dq 1dt

= i1 = C 1dv1dt

, dq 3

dt = i3,

dϕ1

dt

= v1, dϕ3

dt

= v3 = Ldi3

dt

,

dϕ4

dt = v3 = Ri3, W (ϕ) =

dq (ϕ)

dϕ .

Through the change of variables and parame-ters x = v1, y = i3, z = ϕ, α = 1/C 1, ξ = 1/Land β = R/L, system (5) is transformed into thefollowing dimensionless system

x = α(y − W (z)x),

y = −ξx + βy,

z = x,

(6)

where the dots represent derivative with respect totime t, the parameters α, β , ξ , a and b are consid-ered positive and W (z) is given by the piecewise-linear function

W (z) =

a, if |z| < 1,

b, if |z| > 1.(7)

It is important to note that system (6) is notdefined in the planes z = ±1. In fact, the systemis defined in three zones in R

3, determined by the

inequalities |z| < 1 (called here the central zone)and |z| > 1 (called the external zones), having thediscontinuity planes z = ±1. We shall use the Fil-ippov conventions to define the system on theseplanes. We also observe that system (6) may haveno physical meaning for all the parameter values,since it can present for instance orbits escaping toinfinity, which would demand infinite energy supply.However in this note we will not be concerned aboutthis, since the analysis of system (6) presented hereintends to be useful both from the physical and

mathematical points of view.

1.3. Description of the main results

and the organization of the

paper

A first bifurcation analysis of system (6) was pre-sented in [Messias et al., 2010]. In the present paperwe extend such an analysis, showing that anothertype of bifurcations may occur, leading to nonlinearoscillations. In order to do it we shall use in Sec. 2the Filippov conventions to define the discontinuous

system (6) on the planes z = ±1. We prove thatthe points on these planes are almost all sewing(or crossing) points of the new system, except thepoints on two straight lines, which are of fold type(see Definition 2.2 and Proposition 2.1 of Sec. 2).

Then we show that the phase space R3

is foliated byplanes invariant under the flow of system (6), whichare transversal to the z-axis and parallel to eachother, in each zone. This implies that any solutionof system (6) is contained in a three-piece invari-ant set formed by part of a plane contained in thecentral zone, which is extended by two half planesin the external zones. In Sec. 2.1 we study the rel-ative position of these invariant planes. It is easyto see that the z-axis is filled by equilibrium pointsof system (6). The local normal stability of theseequilibria in each zone is analyzed in Sec. 3. Basedon the analytical results obtained, we have devel-oped a detailed numerical bifurcation analysis of system (6), which is presented in Sec. 4. The numer-ical simulations performed bring clear evidence thatthe memristor model presents nonlinear oscillations,given by the existence of infinitely many periodicorbits, each one belonging to one such invariant setand passing by two of the three zones or passing bythe three zones. These orbits arise due to homoclinicand heteroclinic bifurcations, obtained by varyingone parameter in the studied model as shown in

Sec. 4. System (6) may also present periodic orbitsfor some fixed sets of parameters, these cases aretreated in Sec. 5. Some concluding remarks are pre-sented in Sec. 6, ending the paper.

2. Filippov Conventions andGeometric Properties of System (6)

Consider the planes Π1 = F −11 (0) and Π2 = F −12 (0)where F 1, F 2: R3 → R are given by F 1(x,y,z) =z−

1 and F 2(x,y,z) = z + 1, respectively. Then theset M = Π1 ∪ Π2 corresponds to the boundary of the regions

M + = (x,y,z) ∈ R3 | z ≥ 1,

M − = (x,y,z) ∈ R3 | z ≤ −1

and

M C = (x,y,z) ∈ R3 | −1 ≤ z ≤ 1.

Definition 2.1. Let Ωr(R3, M ) be the set of vectorfields X in R

3 such that

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X ( p) =

X +( p), if p ∈ M +,

X C ( p), if p ∈ M C ,

X −( p), if p ∈ M −,

where X +

, X C and X − are Cr

vector fields definedon M +, M C and M −, respectively, with r ≥ 1.We denote by X = (X +, X C , X −) the elements of Ωr(R3, M ). Observe that these vector fields are ingeneral discontinuous in M .

Definition 2.2. Consider the discontinuous vectorfield X = (X +, X C , X −) ∈ Ωr(R3, M ).

(a) The sewing region of X in M is defined as theset

M sw = p ∈ Π1 : X +F 1( p).X C F 1( p) > 0∪ p ∈ Π2 : X C F 2( p).X −F 2( p) > 0.

(b) The escaping region of X in M is defined as theset

M es = p ∈ Π1 : X +F 1( p) > 0, X C F 1( p) < 0∪ p ∈ Π2 : X C F 2( p) > 0, X −F 2( p) < 0.

(c) The sliding region of X in M is defined as theset

M sl = p ∈ Π1 : X +F 1( p) < 0, X C F 1( p) > 0∪ p ∈ Π2 : X C F 2( p) < 0, X −F 2( p) > 0.

Proposition 2.1. Let X = (X +, X C , X −) ∈ Ωr(R3,M ) be the piecewise linear vector field associated to

system (6 ), given by

X ( p) =

X +( p) = (αy − αbx, −ξx + βy, x),

if p ∈ M +,

X C ( p) = (αy − αax, −ξx + βy, x),

if p

∈M C ,

X −( p) = (αy − αbx, −ξx + βy, x),

if p ∈ M −,

(8)

with p = (x,y,z) ∈ R3, M +, M C , M − and M =

Π1∪Π2 as above. The following properties hold.

(a) All the points of M are sewing points of X,except the points over the straight lines

L1 = (0, y, 1); y ∈ R and

L2 =

(0, y,

−1); y

∈R

.

(b) If α = 0 then all the points in L1 ∪ L2 are fold

points of X, except the points (0, 0, ±1), which

are degenerate.

Proof. From the definition of F 1 and F 2, we have

∇F 1 = ∇F 2 = (0, 0, 1), from which we obtain

X +F 1 = X C F 1 = x, X C F 2 = X −F 2 = x.

Then the points in Π1 ∪ Π2 are sewing points of X if and only if x = 0. Moreover X 2+F 1 = X 2C F 1 =X 2C F 2 = X 2−F 2 = αy in L1 ∪ L2. The result thenfollows from the definition of fold and sewing pointsof X .

Theorem 2.1. The following statements hold.

(a) For α(ξ − aβ ) = 0 and α(ξ − bβ ) = 0,system (8 ) has the invariant planes

π1 =

(x,y,z) ∈ R

3 | z

= β

α(ξ − aβ )x − 1

(ξ − aβ )y + k1,

k1 ∈ R, |z| ≤ 1

and

π2 =

(x,y,z) ∈ R

3 | z

= β

α(ξ − bβ )x − 1

(ξ − bβ )y + k2,

k2 ∈ R, |z| ≥ 1

,

with π1 ⊂ M C , π2 ⊂ M − if z ≤ −1 and π2 ⊂M + if z ≥ 1.

(b) The invariant planes π1 and π2 intersect the

discontinuity planes z = ±1 in the parallel

straight lines y = β α

x + (k1 ∓ 1)(ξ − aβ ) and

y = β α

x+(k2∓1)(ξ −bβ ), respectively. Moreover ,these lines coincide on the plane Π1 and on the

plane Π2, that is , for any given k1 ∈ R, there

exists k2 such that the invariant planes inter-

sect on the same line in the planes z = ±1.

Consequently , every plane π1 in the central zone

is extended by half planes π2 in the external

zones , forming a three-piece invariant set to

which the solutions are confined.

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Proof. The plane a1x + a2y + a3z + a4 = 0 i sinvariant for system (8) with |z| ≤ 1 if and only if

(α(y − ax), −ξx + βy, x), (a1, a2, a3) = 0,

from which we obtain(a3 − αaa1 − ξa2)x = 0,

(αa1 + βa2)y = 0.

Solving the system above, we have

a1 = − βa3α(ξ − aβ )

,

a2 = a3ξ − aβ

and a3 ∈ R, a3 = 0.

Thus the invariant planes are given by

z = β

α(ξ − aβ )x − 1

ξ − aβ y + k1,

with k1 = a4a3

∈ R.

On the other hand, the plane b1x + b2y + b3z +b4 = 0 is invariant under the flow of system (8)with |z| ≥ 1 if and only if,

b1 =

−

βb3

α(ξ − bβ )

,

b2 = b3ξ − bβ

and b3 ∈ R, b3 = 0,

from which we have

z = β

α(ξ − bβ )x − 1

ξ − bβ y + k2,

with k2 = b4b3

∈ R.

Thus item (a) of Theorem 2.1 is proved.

In order to prove item (b), note that π1 ∩ z =1 is given by the straight line

y = β

αx + (k1 − 1)(ξ − aβ ) (9)

and π1 ∩ z = −1 is given by

y = β

αx + (k1 + 1)(ξ − aβ ), (10)

from which follows that the plane π1 intersects theplanes z =

±1 in parallel straight lines.

Similarly, we have that π2 ⊂ M + intersects theplane z = 1 on the straight line

y = β

αx + (k2 − 1)(ξ − bβ ) (11)

and π2 ⊂ M − intersects the plane z = −1 on thestraight line

y = β

αx + (k2 + 1)(ξ − bβ ). (12)

Now using the fact that all the points in theplanes z = ±1 are sewing or fold points (see Propo-sition 2.1), we have that any solution of system (8)with initial condition on the straight line (9) mustenter the region M C (in forward or backward time)and stay in one of the planes π1 given above; sim-ilarly the same solution must enter the region M +(in forward or backward time) and stay in one of theplanes π2. As the planes π1 ⊂ M C and π2 ⊂ M + areinvariant under the flow of system (8), the straightline (9) has to coincide with the straight line (11),that is the planes π1 and π2 intersect on the samestraight line in the plane z = 1.

Similarly, as the planes π1 ⊂ M C and π2 ⊂ M −are invariant, any solution with initial condition onthe straight line (10) must belong to both planes.Consequently the straight line (10) must coincidewith the straight line (12).

More precisely, given k1 ∈ R, there alwaysexists k2 given by

k2 = 1 + (k1 − 1)ξ − aβ

ξ − bβ

such that the planes π1 ⊂ M C and π2 ⊂ M + inter-sect the plane z = 1 on the same straight line.Moreover, the planes π1 ⊂ M C and π2 ⊂ M − alsointersect each other over the same straight line inthe plane z = −1, since for any k1 ∈ R there exists

k2 = −1 + (k1 + 1)

ξ

−aβ

ξ − bβ

in such a way that the straight line (10) is equal tothe straight line (12).

Remark 2.1. The planes π1 and π2 described in The-orem 2.1 are generated by the eigenvectors asso-ciated to the nonzero eigenvalues of the equilibria(0, 0, z), z ∈ R, of system (8), see Sec. 3 for details.In each of the external zones M − and M + and inthe central zone M C , the solutions of system (8) arecontained in parallel invariant planes, transversal

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Fig. 4. Geometric representation of planes π1 ⊂ M C andπ2 ⊂ M + ∪ M

−. The planes in the central zone intersect

the planes in the external zones in parallel straight lines con-tained in the planes z = ±1.

to the z-axis. The relative position of these planesdepends on the parameters of system (8) and theplanes located in the external zones M − and M +are also mutually parallel.

For a geometric representation of this result,see Fig. 4. An important fact which follows fromthe results stated above is that for any given ini-tial condition (x0, y0, z0) ∈ R

3 there exists a com-bination of three invariant planes, one containedin M C and the other two contained in M − ∪ M +,which contains the solution of system (8) based on(x0, y0, z0). Indeed all the solutions of system (8)

are contained in a combination of three invariantplanes, obtained from those given in Theorem 2.1(see Fig. 4). This implies that the phase space R

3

is foliated by invariant planes of system (8). Thusthe dynamics of system (8) is essentially planar and,consequently, no chaotic behavior of the solutions ispossible. This fact agrees with the statement madein [Itoh & Chua, 2009], where it is shown that thesystem is equivalent to a second-order differentialequation, hence it is essentially second order.

In the next subsection we will study the rela-tive position of these planes, aiming to determine

how the phase space R3 is filled by the solutions of system (8) contained on them.

2.1. Relative position of the

invariant planes

From the proof of Theorem 2.1 we have that thevectors normal to the planes π1 and π2 are given by

n =

−β

α, 1, ξ − βW (z)

,

where

W (z) =

a, if |z| < 1,

b, if |z| > 1.

The scalar product of these normal vectors with

the vector (0, 0, 1) are given byn, (0, 0, 1) = ξ − βW (z),

from which we obtain

cos θ = n, (0, 0, 1)

|n| = ξ − βW (z)

|n| .

Thus the angle between n and (0, 0, 1) is acuteif ξ − βW (z) > 0 and it is obtuse if ξ − βW (z) < 0.In Fig. 5 we give a schematic representation of therelative positions of the planes π1 in the central zone

Fig. 5. Representation of the possible relative positions of the invariant planes π1 and π2.

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z

z=1

y

z=-1

z

z=1

z=-1

y

(a) (b)

Fig. 6. Possible relative positions of the invariant planes π1 and π2.

z

z=1

y

z=-1

z

z=1

y

z=-1

(a) (b)

Fig. 7. (Continuation of Fig. 6 ) Possible relative positions of the invariant planes π1 and π2.

and π2 in the external zones. Figures 6 and 7 showthe projections on the plane-yz of these invariant

planes, considering all the possible relative posi-tions among them. Looking at these figures onecan see that, depending on the relative position of the planes containing the solutions, there are thefollowing possibilities for the equilibrium points of system (8) restricted to each combination of threeinvariant planes: there are three equilibrium points,one in each zone; there is only one equilibriumpoint in the central zone; there is only one equi-librium point in the external zone M +; there isonly one equilibrium point in the external zone M −.Clearly these equilibrium points are given by the

intersection of the invariant planes containing thesolutions with the z-axis. This analysis will be used

to study and represent the solutions of system (8)in the sections ahead.

3. Normal Linear Stability of theEquilibrium Points

The equilibrium points of system (8) are given bythe set

A = (x,y,z) ∈ R3 | x = y = 0 and z ∈ R,

therefore the system has a line of nonhyperbolicequilibrium points contained in the z-axis. A first

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Table 1. Normal linear stability of the equilibrium (0, 0, z ) of system (8) with 0 < b < a.

Local Stability of (0, 0, z )

Conditions on τ Conditions on ∆ |z | < 1 |z | > 1 Cases

τ < 0 ∆ > 0 and D < 0 if |z | < 1 Saddle Stable foci (a)„β

α < W (z )

« ∆ < 0 if |z | > 1

∆ > 0 and D > 0 if |z | < 1 Stable node Stable foci (b)

∆ < 0 if |z | > 1

τ > 0 ∆ > 0 and D < 0 if |z | < 1 Saddle Unstable foci (c)„β

α > W (z )

« ∆ < 0 if |z | > 1

∆ > 0 and D > 0 if |z | < 1 Unstable node Unstable foci (d)

∆ < 0 if |z | > 1

τ change its ∆ > 0 and D < 0 if |z | < 1 Saddle Unstable foci (e)

sign with ∆ < 0 if |z | > 1

z ∆ > 0 and D > 0 if |z | < 1 Stable node Unstable foci (f)„b <

β

α < a

« ∆ < 0 if |z | > 1

Notation used: τ = β − αW (z ), ∆ = τ 2 − 4D and D = α(ξ − βW (z )).

study of the linear stability of these equilibria wasdone in [Messias et al., 2010], where the authorsconsidered the case 0 < a < b (see Theorem 2.1,p. 439, and Table 1, p. 441, of [Messias et al., 2010]).

Aiming to complete the study presented in [Mes-sias et al., 2010] we analyzed the linear stability of the equilibria (0, 0, z) in the case 0 < b < a andobtained the following result.

Proposition 3.1. The normal linear stability of the

equilibrium point (0, 0, z) of system (8 ) with 0 <b < a is described in Table 1, for positive values of

the parameters α, β,ξ .

Proof. The Jacobian matrix J of system (8) evalu-ated at the equilibrium point (0, 0, z) is given by

J =

−αW (z) α 0

−ξ β 0

1 0 0

which has the eigenvalues

λ1 = 0 and λ2,3 = τ ± √

∆

2 , (13)

where τ = β − αW (z), ∆ = τ 2 − 4D and D =α(ξ

−βW (z)). The corresponding eigenvectors are

v1 = (0, 0, 1),

v2 =

τ − √

∆

2 ,

−(β + αW (z) − √ ∆)(−τ +

√ ∆)

4α , 1

and

v3 =

τ +

√ ∆

2 ,

−(β + αW (z) +√

∆)(−τ − √ ∆)

4α , 1

.

Analyzing the possibilities for the eigenvalues (13),according to the relations among τ, ∆ and D, whichdepend on the parameters α, β , ξ , a, b, we have thecases described in Table 1 for the local stability of the equilibria (0, 0, z) with |z| = 1.

The cases in which ∆ > 0 if |z| < 1 and ∆ < 0if |z| > 1, are possible only if 0 < b < a. In fact, for|z| < 1 we have

∆ = (β + αa)2 − 4ξα > 0 ⇒ (β + αa)2 > 4ξα,

(14)

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and for |z| > 1 we have

∆ = (β + αb)2 − 4ξα < 0 ⇒ (β + αb)2 < 4ξα.

(15)

From (14) and (15) it follows that (β + αb)2 <

4ξα < (β + αa)2

, whence we get, necessarily, thatb < a.

The proof of the normal hyperbolicity of theequilibria (0, 0, z) is analogous to the one made inthe proof of Theorem 2.1 in [Messias et al., 2010].

Based on the analytical results concerning theexistence of invariant planes and on the local normalstability of the equilibrium points, we performed adetailed numerical study of the possible bifurcationswhich can occur with the solutions of system (8).

The main cases which lead to the existence of non-linear oscillations are presented in the next section.

4. Numerical Study of BifurcationsLeading to Oscillations

The fact that the solutions of systems (8) are con-tained in a combination of three invariant planes inR3 resembles the study developed by Freire et al.

[2002], concerning piecewise linear systems definedon three zones in the plane R2. Thus, based on theanalytical study of system (8) developed up to now

and on the results of [Freire et al., 2002], in this sec-tion we present a numerical study of this system inthe cases in which we found nonlinear oscillations,which are relevant in the study of electric circuits.

4.1. Saddle–Focus–Saddle:

Heteroclinic bifurcation

Consider the case in which the eigenvalues of theJacobian matrix of system (8) calculated at theequilibrium point (0, 0, z) are real and with oppo-site signs for |z| > 1 and complex conjugate withpositive real part for |z| < 1 (case (w) of Table 1 in[Messias et al., 2010]). The numerical simulationsdeveloped indicate that in this case a bifurcationinvolving a heteroclinic cycle occurs, leading to thecreation of a periodic orbit, as illustrated in Fig. 8.This type of bifurcation was also studied in [Freireet al., 2002] for piecewise linear systems in the planewith three zones and one equilibrium point in eachzone.

Solving system (8) numerically with the param-

eter values α = 0.32, a = 0.02, b = 2, ξ = 0.2 andβ = 0.2, we get the solution shown in Fig. 9(a), indi-cating the nonexistence of periodic orbits nor het-eroclinic cycles around the z -axis. Taking α = 0.35and keeping the other parameters unchanged wefind a heteroclinic loop formed by the two saddles inthe external zones as shown in Fig. 9(b). Moving theparameter α to 0.38, a periodic orbit arises, passingby the three zones [see Fig. 9(c)]. This is a typi-cal type of bifurcation which occur in the study of piecewise linear systems with three zones and oneequilibrium in each zone, forming a saddle–focus–

saddle configuration [Freire et al., 2002].In Fig. 10(a) is shown the periodic orbit which

can be seen in Fig. 9(c). Figure 10(b) shows thecoordinate z(t) of a solution which tends in the

(a) (b) (c)

Fig. 8. Heteroclinic cycle bifurcation leading to the creation of periodic oscillations.

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(a) (b) (c)

Fig. 9. Numerical simulation of the solutions of system (8) with the parameter values: (a) α = 0.32, a = 0.02, b = 2, ξ = 0.2and β = 0.2 and time [−20, 180]: there is no periodic nor heteroclinic orbits, (b) α = 0.35, a = 0.02, b = 2, ξ = 0.2 andβ = 0.2 and time [−20, 180]: there is a heteroclinic cycle formed connecting the saddles in the external zones and (c) α = 0.38,a = 0.02, b = 2, ξ = 0.2 and β = 0.2 and time [−20, 180], there is a stable periodic orbit encircling the z -axis.

(a) (b)

Fig. 10. (a) Periodic solution of system (8) with the parameters α = 0.38, a = 0.02, b = 2, ξ = 0.2 and β = 0.2 and time[40, 80] and (b) coordinate z (t) of the solution shown in (a), which tends to the periodic orbit.

past to the equilibrium (0, 0, z) and in the futureto this periodic orbit. The fact that the phase spaceis foliated by invariant planes implies that there isone periodic orbit related to each unstable equilib-

rium point (0, 0, z) for |z| < 1, as shown in Fig. 11.Indeed, in each three-piece invariant set contain-ing the solutions of system (8), there is a periodicorbit. The amplitude of these periodic orbits changeaccording to the relative position of the parallelplanes containing the solutions. See Fig. 11.

4.2. Focus–Saddle–Focus: Figure

eight like bifurcation

Consider now the case in which the equilibriumpoints of system (8) are unstable foci in the external

(a) (b)

Fig. 11. Numerical solution of system (8) with α = 0.8,a = 0.02, b = 2, ξ = 0.2 and β = 0.2 and time [40, 80],showing the existence of periodic orbits associated to severalequilibrium p oints (0, 0, z ) with |z | < 1.

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(a) (b) (c)

Fig. 12. Sketch of the figure eight like bifurcation for system (8). (a) There are two heteroclinic orbits and no periodic orbits,(b) there are two homoclinic orbits, forming a “figure eight” and (c) there are two small periodic orbits around the z -axis,each one passing by two zones.

zones and saddles in the central zone (case (e) of Table 1). In this case, when a solution enters thecentral zone there are two possibilities: the solutioncoincides with the stable (unstable) manifolds of the saddle and tend to it in the future (past) orit leaves this zone in the future (past). After a care-ful numerical study of the solutions in this case, we

observe that a “figure eight” like bifurcation occurs,as sketched in Fig. 12, leading to the creation of small and large periodic oscillations. This type of bifurcation can be compared to the one described

in [Freire et al., 2002], Propositions 4.2 and 4.3, inthe study of planar piecewise-linear systems withthree zones.

Taking the parameter values a = 5, b = 2,ξ = 25 and β = 6 and numerically integratingsystem (8), varying the parameter α from α = 1.8to α = 3.1, we obtained the solutions shown in

Figs. 13(a)–13(c), showing the creation of smallperiodic orbits, after the appearance of a doublehomoclinic orbit. Each one of these periodic orbitspass by two zones, see Fig. 13(c).

(a) (b) (c)

Fig. 13. Numerical solution of system (8) with the following parameters: (a) α = 1.8, a = 5, b = 2, ξ = 25 and β = 6 andtime [−20, 20], (b) α = 2.08, a = 5, b = 2, ξ = 25 and β = 6 and time [0, 30] and (c) α = 3.1, a = 5, b = 2, ξ = 25 and β = 6and time [−10, 20]. Observe the creation of two stable periodic orbits, after the appearance of a double homoclinic loop.

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(a) (b)

Fig. 14. Numerical solution of system (8) with the parameters α = 2.2, a = 5, b = 2, ξ = 25 and β = 6 and time [−20, 20]: thesystem has two small stable periodic orbits passing by two of the three zones and a big unstable periodic orbit encompassingthe two small ones.

Beyond these small stable periodic orbits pass-ing by two zones, for α = 2.2 we numerically detectthe existence of a large unstable periodic orbit coex-isting with and surrounding the small ones. Suchlarge periodic orbit pass by the three zones, asshown in Fig. 14(a). The nice Fig. 14(b) shows the

coordinate z(t) of solutions which in the past tend

to the large unstable periodic orbit and in the futuretend to the small periodic orbits.

In Figs. 15(a) and 15(b) are shown thesmall and large periodic orbits separately. To plotFig. 15(a) we took the time interval [−25, −20] andto plot Fig. 15(b) we took the time [20, 30] (observe

that the scales in these figures differ from each other

(a) (b)

Fig. 15. Numerical solution of system (8): (a) with time [−25, −20], showing only the large periodic orbit passing by thethree zones and (b) with time [20, 30], showing two small periodic orbits passing by two zones.

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in order to get a better visualization of the periodicorbits).

In our numerical study of system (8) we did notdetect other large amplitude periodic orbits in thecase above, as it happens for example in the study

of piecewise linear systems in the plane with threezones [Freire et al., 2002] and also in the smoothfigure eight bifurcation [Kuznetsov, 2004; Perko,2001].

5. The Existence of Oscillations forFixed Sets of Parameters

In this section we study two cases in which sys-tem (8) presents nonlinear oscillations for fixed setsof parameters. The oscillations exist due to thechange in the normal linear stability of the equi-

librium point (0, 0, z), as z belongs to the regions|z| < 1 and |z| > 1.

5.1. Focus–Node–Focus

Consider the case where the equilibria of system (8)are stable nodes for |z| < 1 and unstable foci for|z| > 1. The numerical simulations performed indi-cate that a solution having one equilibrium point inthe central zone as its α-limit set can either tendin the future to a periodic orbit or tend to infinity.

These two situations are illustrated in Fig. 16.Taking specific values for the parameters inorder that system (8) satisfies these conditions, wecompute the solutions numerically, they are shown

in Fig. 17. The initial conditions used were pointsbelonging to straight lines generated by the eigen-vectors v2 and v3 associated with the nonzero eigen-values associated to the equilibria (0, 0, z) for |z| <1. In Fig. 17(a) we used the parameters α = 2.9,

a = 5, b = 2, ξ = 31 and β = 6; in this case thenumerical solution tends to infinity in the future.When we change the parameter value for α = 2 aperiodic orbit arises, which is the ω-limit set of asolution starting near the equilibrium point (0, 0, z)with |z| < 1, as seen in Fig. 17(b).

In Fig. 18(a) we have the same values of param-eters of Fig. 17(b) with time [−18, −16], showingonly the periodic orbit. We can observe that thereis a change in the local stability of the equilib-rium point when z crosses the plane |z| = 1. Thischange in the stability of the points generates peri-odic oscillations (unstable). In Fig. 18(b) we takeinitial conditions in |z| > 1, i.e. when the equilibriaare unstable foci. Note that with the same parame-ter values and time interval [20, 30] we also observethe existence of periodic orbits.

5.2. Stable focus–Unstable

focus–Stable focus

Consider the case in which the Jacobian matrixassociated to the equilibrium (0, 0, z) of system (8)

has complex conjugate eigenvalues with positivereal part in the central zone, i.e. for |z| < 1, andnegative real part in the external zones, i.e. |z| > 1,leading to changes in the local normal stability of

(a) (b)

Fig. 16. Sketch of system (8) solutions: (a) when there is a periodic orbit and (b) when there is no periodic orbit.

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(a) (b)

Fig. 17. Numerical solution of system (8). (a) Parameters α = 2.9, a = 5, b = 2, ξ = 31 and β = 6 and time [−8, 5] and(b) parameters α = 2, a = 5, b = 2, ξ = 31 and β = 6 and time [−8, 4], showing the existence of a periodic orbit.

the equilibria when z crosses the planes z = ±1.Considering an orbit that has an equilibrium point(0, 0, z) in the central zone as its α-limit set, wenumerically found two possibilities: its ω-limit setcan be either a periodic orbit or the empty set(i.e. the solution tends to infinity) as described in

Fig. 19. In fact, from the numerical simulationsdeveloped for system (8) with the Runge–Kuttamethod of fourth-order and taking the parametervalues satisfying the conditions for this case, wehave evidence that a periodic orbit can exist, asshown in Fig. 20.

(a) (b)

Fig. 18. Numerical solution of system (8). (a) Parameters α = 2, a = 5, b = 2, ξ = 31 and β = 6 and time [−18, −16],showing only the periodic orbit and (b) parameters α = 2, a = 5, b = 2, ξ = 31 and β = 6 and time interval [20, 30], showingperiodic orbits that arise when the initial conditions are taken on the external zones.

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(a) (b)

Fig. 19. Sketch of the solution of system (8): (a) when the ω -limit set is empty, i.e. the solution tends to infinity as t → +∞and (b) when the ω -limit set is a periodic orbit.

Considering the values α = 0.17, a = 1, b = 12,ξ = 50 and β = 2, we numerically solve system (8)obtaining the solutions shown in Fig. 20(a), wherethe ω-limit set of a solution starting near the z-axis is empty, that is the solution tends to infinity.Changing the value of the parameter α to 0.5 weobtained the solution shown in Fig. 20(b), whichhas a stable periodic orbit as its ω-limit set. In both

cases, the α-limit of the orbit is an equilibrium pointin the central zone, that is, (0, 0, z) with |z| < 1.

The numerical evidences show that there existsone periodic orbit associated to each unstable equi-librium point (0, 0, z) for |z| < 1 in the following

sense: for each −1 < z < 1 there is in correspon-dence a solution which has (0, 0, z) as its α-limit setand a periodic orbit as its ω-limit set, as illustratedin Fig. 21. In this way, the periodic orbits are notisolated, but appear in a continuum set (one orbitin each combination of three-piece invariant set con-taining the solutions).

It is important to observe that, in this case, the

existence of solutions tending to infinity, as shownin Fig. 19(a), is due to the relative position of theplanes invariant by the flow of system (6). Indeed,the half planes in the external zones, when extendedto the central zone, intersect the z-axis at points

(a) (b)

Fig. 20. Phase portrait of the system (8): (a) with α = 0.17, a = 1, b = 12, ξ = 50 and β = 2 and (b) with α = 0.5, a = 1,b = 12, ξ = 50 and β = 2, showing the possible existence of a stable periodic orbit around the z -axis.

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(a) (b)

Fig. 21. Phase portraits of the system (8) with α = 0.5, a = 1, b = 12, ξ = 50 and β = 2 showing the existence of a stableperiodic orbit around the z -axis, corresponding to an equilibria (0, 0, z ) with |z | < 1.

Fig. 22. Illustration of the relative position of the invariantplanes in the case stable focus–unstable focus–stable focus ,showing that the global dynamics of a solution starting nearthe equilibrium (0, 0, z ) with |z | < 1 is given by a combinationof two unstable foci.

(0, 0, z), with |z| < 1, which are unstable foci, seeFig. 22. In this way, although the equilibria in theexternal zones are stable foci, each global solutionof the system considered in this section, is contained

in a three-piece invariant set and is a combinationof two unstable foci. Hence, the existence of solu-tions tending to infinity as well as the existence of periodic orbits depend on the rate of expansion of these two unstable foci. This is one case which can-not occur in the study of planar piecewise linearsystems with three zones.

6. Concluding Remarks

In this paper, we have developed an analytical andnumerical study of the phase space of system (6), for

several different parameter values. As the systemis not defined on the planes z = ±1, we used theFilippov conventions to define it on these planesand study the solutions on the entire space R

3.The obtained system (8), which is given by the vec-tor field X = (X +, X C , X −), is bi-valuated on theplanes z = ±1 and almost all points on these planesare sewing (or crossing) points of the system in thefollowing sense: a solution starting for example inthe central zone M C is governed by the vector fieldX C until it reaches the plane z = 1; then it passes

to be governed by X +, entering the external zoneM +; and so on.

We have proved in this paper that the dynam-ics of system (8) is essentially planar and no chaoticbehavior is possible, which agrees with the calcula-tions in [Itoh & Chua, 2009]. Thus the system isessentially second order. Mathematically it can beexplained in two ways: from the equivalence of thestudied system to a second-order equation, provedin [Itoh & Chua, 2009], and from the foliation of the phase space by invariant planes containing the

solutions, proved in this paper. From the circuit the-ory point of view, the planar behavior of the solu-tions of system (8) is certainly related with the waythe memristor is connected to the other elementsin the circuit. Another element to be considered isthe positive function chosen to represent the mem-ductance of the memristor, which has a relevanteffect on the circuit dynamics. In the particular caseconsidered in this paper, taking the memductancegiven by the expression (7), the dynamics is essen-tially planar. However other arrangements of theelements in the circuit as well as the consideration

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of other memductance function could lead to morecomplicated dynamics, truly three-dimensional andeven chaotic (see, for instance, [Ginoux et al., 2010;Muthuswamy & Chua, 2010]). The derivation of explicit ODE models for systems involving mem-

ristors is considered in [Riaza, 2011].Furthermore, in this paper we also numerically

detected the existence of periodic orbits within eachthree-piece invariant set, for some parameter values.These orbits arise due to homoclinic and hetero-clinic bifurcations, obtained varying the parameterα in system (8), and may also exist for some fixedsets of parameter values. The intricate configurationof the phase space obtained through the analysispresented here may contribute to the understand-ing of properties of a circuit involving the four basicelectronic elements: one memristor; one capacitor;one resistor; and one inductor. It may also indicatesome new properties of the memristor itself. On theother hand, the analysis motivated by the study of this physical model suggests that new and interest-ing dynamical phenomena can appear in the studyof piecewise linear discontinuous systems of ordi-nary differential equations defined on R

3 and havinga line of equilibria. Therefore further mathematicalanalysis of the general systems of this type may beworthy.

Acknowledgments

This work was supported by FAPESP-Brazil underthe Process 2009/11699-0. The second author issupported by CNPq-Brazil, under the project308315/2012-0 and by FAPESP-Brazil under theProcess 2012/18413-7. The authors would like tothank Paulo Ricardo da Silva and Claudio Buzzifrom IBILCE/UNESP for the valuable discussionsand suggestions during the development of thiswork. The authors also thank the anonymous ref-erees for their valuable comments and suggestions

which allowed them to improve the presentation of the paper.

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bifurcations leading to nonlinear oscillations in a 3d piecewise linear memristor oscillator

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