Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

Following [8], hysteresis is a nonlinear phenomenon which appears in many natural and constructed systems. A system is characterized as a hysteretic system when the equations have a looping behaviour produced by a relation between two scalar time-dependent quantities that cannot be expressed in terms of a scalar function. These loops can be due to different causes, for example, the existence of thermostats for controlling the temperature, voltage threshold on a circuit, etc. In our approach, hysteretic systems appear as a consequence of dimensional reduction in the analysis of slow-fast systems, see the Appendix.

In [10] two characteristics of hysteresis are emphasized. First, the non-linearity has a dependence on previous values of the input (memory effect). Second, hysteretic systems undergo arbitrary quickly transitions, what is an idealization for real systems.

The mathematical analysis of hysteretic systems has emphasized its ability for generating chaotic solutions when the involved dynamics are of focus type. See [6,7,9]. Here, we consider instead symmetric hysteretic systems having two real equilibria of saddle type. After some preparation work, we get a canonical form which is general enough for our purpose. See the appendix for a derivation of the proposed canonical form.

We deal with an upper system of saddle type

{x˙=2γ(x−xE)−(y−yE),y˙=(γ2−1)(x−xE),x⩽1,$$ \begin{equation} \left\{\begin{array}{r c l} \dot{x}&=& 2\gamma (x-x_E)-(y-y_E),\\ \dot{y}&=&(\gamma^2-1)(x-x_E), \end{array}\qquad x\leqslant 1, \right. \end{equation} $$(S_U)

and a similar lower system, namely

{x˙=2γ(x+xE)−(y+yE),y˙=(γ2−1)(x+xE),x⩾−1,$$ \begin{equation} \left\{\begin{array}{r c l} \dot{x}&=& 2\gamma (x+x_E)-(y+y_E),\\ \dot{y}&=&(\gamma^2-1)(x+x_E), \end{array}\qquad x\geqslant -1, \right. \end{equation} $$(S_L)

plus some hysteretic transition mechanism, to be specified later, that allows to build continuous solutions for the global system. Note that, due to the features of our system, we cannot take advantage of any local method for the analysis of dynamics, as done in [5] looking for the characterization of centers in planar dynamical systems. Here, as usual, the dot represents derivatives with respect to the time τ, (x_E, y_E) and (−x_E, −y_E) are the equilibria for the S_U-system and S_L-system respectively. In order to get a dynamics of saddle type we must choose γ ∈ (−1, 1).

Next we define the solutions of the global system (S_U)-(S_L) by using the following transition mechanism. Take (x (0), y (0)) with x (0) < 1 as the initial condition of a solution (x(τ), y(τ)) of the S_U−system. Then this solution is called a valid solution as long as x(τ)⩽1. If there exists a time τ_f, such that x(τ_f)=1 with ẋ > 0, then the point (x(τ_f), y(τ_f)) is assumed to be the initial point for an orbit of the S_L-system and this orbit continues by integrating system (S_L).

Analogously, any solution (x(τ), y(τ)) of the S_L-system with x (0) > −1, is considered as a valid solution as long as x(τ) > −1, and if there exists a time τ_r such that x(τ_r)=−1 with ẋ(τ_r) < 0, then the point (x(τ_r), y(τ_r)) is assumed to be the initial point for an orbit of the S_U-system.

In passing from the S_U-system to the S_L-system, we speak of a fall, which occurs when an orbit of the system (S_U), called upper orbit, hits the falling line

Σ+={(x,y):x=1}.$$ \begin{equation*}\nonumber \Sigma_+=\{(x,y): x=1\}. \end{equation*} $$

Similarly, we define a rise when we pass from the S_L-system to the S_U-system, which occurs when an orbit of system (S_L), called lower orbit, intersects the rising line

Σ−={(x,y):x=−1}.$$ \begin{equation*}\nonumber \Sigma_-=\{(x,y): x=-1\}. \end{equation*} $$

Fig. 1

Note that our formulation is equivalent to write the system

{x˙=2γx−y−H(x)(2γxE−yE),y˙=(γ2−1)(x−H(x)xE),$$ \begin{equation*} \left\{\begin{array}{r c l} \dot{x}&=& 2\gamma x-y-H(x)(2\gamma x_E-y_E),\\ \dot{y}&=&(\gamma^2-1)(x-H(x)x_E), \end{array} \right. \end{equation*} $$

where H(x) is the standard normalized hysteretic function of Figure 2.

Fig. 2

Thus, any periodic orbit of global system (S_U)-(S_L) will lead to a repeated sequence of falls and rises and has at least two pieces, one corresponding to an orbit of system (S_U) and the other to an orbit of system (S_L).

There is no loss of generality in taking the initial point of a periodic orbit to belong to an orbit of the upper system. In particular, in what follows we assume x (0)=−1, so that our initial point will be (−1, u₋) ∈ Σ₋, on the rising line, for a certain u₋ ∈ ℝ. Our strategy is to look for the first point where the orbit of the upper system arrives at Σ₊, the falling line, in a point (1, u₊). Thus, we can define an evolution map

TU:Σ−ad⊂Σ−→Σ+ (−1,u−)↦(1,u+)$$ \begin{equation*} \begin{array}{r c l} T_U: \Sigma_-^{ad}\subset\Sigma_-& \rightarrow&\Sigma_+\\ (-1,u_-)&\longmapsto &(1,u_+) \end{array} \end{equation*} $$

where Σ−ad $\Sigma_-^{ad}$ represents the admissible subset of points in Σ₋ for which the forward orbits of the S_U-system reach Σ₊ and fall. This function induces a scalar map U, such that u₊=U(u₋), which will be called the transition map.

A similar evolution map can be defined for the S_L-system by considering the points (1,l+)∈Σ+ad⊂Σ+ $(1,l_+)\in \Sigma_+^{ad}\subset \Sigma_+$ whose forward orbits reach Σ₋ and rise, namely

TL:Σ+ad⊂Σ+→Σ−(1,l+)↦(−1,l−)$$ \begin{equation*} \begin{array}{r c l} T_L: \Sigma_+^{ad}\subset\Sigma_+& \rightarrow&\Sigma_-\\ (1,l_+)&\longmapsto &(-1,l_-) \end{array} \end{equation*} $$

This map also induces a transition map L, which is defined by l₋=L(l₊).

Clearly, the full transition map for the global system (S_U)-(S_L) will be the composition of both maps U and L, that is L∘U, provided that such composition is possible. Obviously, if for a given u₋ we have that u₊=U(u₋) belongs to the domain of L, we can take l₊=u₊ and compute the value L(l₊)=l₋. Then, the condition l₋=u₋ is equivalent to the existence of a periodic orbit.

Once introduced the systems under study and having defined in a precise way how the different orbits behave, our goal is to analyze the existence of periodic orbits and characterize their bifurcations. To this end, after some preliminary results that appear in Section 2, we present our main results in Section 3, see Theorem 9. Such theorem implies that, in the particular saddle case under study, periodic orbits appear either through heteroclinic bifurcations or through saddle-node bifurcations. Furthermore, we show that all the periodic orbits are symmetric, that its maximum number is two, and that at least one of them is stable.

For sake of brevity, the included study of hysteretic symmetric systems with saddle dynamics only considers the case of real saddles out of the hysteresis band, that is x_E < −1. The remaining cases, namely real saddles in the central band (∣x_E∣⩽1) and virtual saddles (x_E > 1) will appear elsewhere.

The symmetry between the S_U-system and the S_L-system imposes a symmetry property for the functions L and U as follows.

TU(−1,y)=(1,v).$$ \begin{equation*} T_U(-1,y)=(1,v). \end{equation*} $$

Since the S_L-system is the symmetric one of the S_U-system with respect to the origin, we have

TL(1,−y)=(−1,−v),$$ \begin{equation*} T_L(1,-y)=(-1,-v), \end{equation*} $$

and so −v=L (−y) and we are done.

Assume that we start from a point (−1,u−)∈Σ−ad $(-1,u_-)\in \Sigma_-^{ad}$ so that T_U(−1, u₋) = (1, u₊). Clearly, if (1,u+)∈Σ+ad $(1,u_+)\in\Sigma_+^{ad}$ and T_L(1, u₊) = (−1, u₋) we have a periodic orbit. In other words, we must have

{U(u−)=u+,L(u+)=u−.$$ \begin{equation} \left\{\begin{array}{r c l} U(u_-)&=&u_+,\\ L(u_+)&=& u_-. \end{array}\right. \end{equation} $$(1)

Reciprocally, if equations (1) have a solution pair (u₋, u₊), then there exists an associated periodic orbit.

Although periodic orbits of four (or more) transitions could be possible, in principle, we omit its consideration in the sequel, and so, when we speak of periodic orbits, we will assume that they have only two transitions.

If we assume that (u₋, u₊) is a solution pair of (1), the following result is straightforward.

U(u)=−u,$$ \begin{equation} U(u)=-u, \end{equation} $$(2)

so that any solution of (2) represents a symmetric periodic orbit.

Regarding the stability, a periodic orbit is stable if the absolute value of the derivative of the full transition map is less than one at the fixed point. Taking into account Proposition 1 (b), we get (L∘ U)′(u)=U′(u)² whenever (2) is fulfilled, and so we deduce stability for a symmetric periodic orbit if

|U'(u−)|<1,$$ \begin{equation} |U'(u_-)|\lt1, \end{equation} $$(3)

being u₋ the fixed point.

In any case, according to Proposition 1, we only need to study the transition map U. To this end, let us write the explicit solutions of system (S_U) taking into account that the equilibria is of saddle type, namely

(x(τ)−xEy(τ)−yE)=eγτ(chτ+γshτ−shτ(γ2−1)shτ chτ−γshτ)(x(0)−xEy(0)−yE).$$ \begin{equation} \left(\begin{array}{c} x(\tau)-x_E\\ y(\tau)-y_E \end{array}\right) =e ^{\gamma \tau}\left(\begin{array}{c c} \text{ch}{\tau}+\gamma \text{sh}{\tau} & \quad-\text{sh}{\tau}\\ (\gamma^2-1)\text {sh}{\tau}& \quad \text{ch}{\tau}-\gamma\text{sh}{\tau} \end{array}\right) \left(\begin{array}{c} x(0)-x_E\\ y(0)-y_E \end{array}\right). \end{equation} $$(4)

Note that we use the abridged notation chτ, shτ for ch(τ) and sh(τ) respectively, for convenience.

For the system (S_{_U}), the stable and unstable manifold of the saddle equilibrium (x_E, y_E) are

y=yE+(γ+1)(x−xE),y=yE+(γ−1)(x−xE)$$ \begin{equation*} y=y_E+(\gamma+1)(x-x_E),\quad y=y_E+(\gamma-1)(x-x_E) \end{equation*} $$

respectively.

In this work, we assume in the sequel x_E < −1, corresponding to the case when the saddle of system (S_U) is on the left of Σ₋. Now, we define the domain of the transition map T_U, Σ−ad $\Sigma_-^{ad}$ , by introducing some distinguished points.

Fig. 3

Accordingly, we get as the admissible domain for the map T_U the set

Σ−ad={(−1,u−):u−<u−*},$$ \begin{equation*} \Sigma_-^{ad}=\{(-1,u_-): u_-\lt u_-^*\}, \end{equation*} $$

so the map U is defined in the interval u−<u−* $u_-\lt u_-^*$ . In fact, as we will see later, the map U is not injective in such interval, unless we restrict the domain to the subset with u₋ ⩽ û₋.

Since it is not possible to write an explicit expression of u₊ in terms of u₋, we obtain the parametric expression of the transition map U in terms of the flight time τ, by considering a starting point (−1,u−)∈Σ−ad $(-1,u_-)\in \Sigma_-^{ad}$ and imposing that the orbit corresponding to this point reaches Σ₊ in a point (1, u₊). We follow so a similar approach to the one introduced in [2] for a different case. So, using the expression (4), we get

u−(τ)=yE+e−γτ(xE−1)−(xE+1)(chτ+γshτ)shτ,u+(τ)=yE−eγτ(xE+1)−(xE−1)(chτ−γshτ)shτ.$$ \begin{equation} \begin{array}{l} u_{-}(\tau)=y_E+\dfrac{e^{-\gamma\tau}(x_E-1)-(x_E+1)(\text{ch}\tau+\gamma \text{sh}\tau)}{\text{sh}\tau},\\ \\ u_+(\tau)=y_E-\dfrac{e^{\gamma\tau}(x_E+1)-(x_E-1)(\text{ch}\tau-\gamma \text{sh}\tau)}{\text{sh}\tau}. \end{array} \end{equation} $$(5)

Now, we write the first and second derivatives with respect to τ, to be used later,

u−′(τ)=(xE+1)−(xE−1)e−γτ(chτ+γshτ)sh2τ,u+′(τ)=−(xE−1)+(xE+1)eγτ(chτ−γshτ)sh2τ,$$ \begin{equation} \begin{array}{r c l} u_{-}'(\tau)&=&\dfrac{(x_E+1)-(x_E-1)e^{-\gamma\tau}(\text{ch}\tau+\gamma \text{sh}\tau)}{\text{sh}^2\tau},\\ \\ u_+'(\tau)&=&\dfrac{-(x_E-1)+(x_E+1)e^{\gamma\tau}(\text{ch}\tau-\gamma \text{sh}\tau)}{\text{sh}^2\tau}, \end{array} \end{equation} $$(6)

and

u″−(τ)=−2(xE+1)chτ+(xE−1)e−γτ(1+(chτ+γshτ)2)sh3τ,u″+(τ)=2(xE−1)chτ−(xE+1)e−γτ(1+(chτ−γshτ)2)sh3τ.$$ \begin{equation} \begin{array}{r c l} u_{-}''(\tau)&=&\dfrac{-2(x_E+1)\text{ch}\tau+(x_E-1)e^{-\gamma\tau}(1+(\text{ch}\tau+\gamma \text{sh}\tau)^2)}{\text{sh}^3\tau},\\ \\ u_+''(\tau)&=&\dfrac{2(x_E-1)\text{ch}\tau-(x_E+1)e^{-\gamma\tau}(1+(\text{ch}\tau-\gamma \text{sh}\tau)^2)}{\text{sh}^3\tau}. \end{array} \end{equation} $$(7)

Standard computations show that

limτ→0u−(τ)=limτ→0u+(τ)=−∞,$$ \begin{equation*} \lim_{\tau\rightarrow 0}u_-(\tau)=\lim_{\tau\rightarrow 0}u_+(\tau)=-\infty, \end{equation*} $$

and

limτ→∞u−(τ)=u−*,limτ→∞u+(τ)=u+*.$$ \begin{equation*} \lim_{\tau\rightarrow \infty}u_-(\tau)=u^*_-, \quad \lim_{\tau\rightarrow \infty}u_+(\tau)=u^*_+. \end{equation*} $$

Using the parametric expressions (5), the condition for the existence of symmetric periodic orbits (2), in terms of u₋ and u₊ is

u−(τ)+u+(τ)=0.$$ \begin{equation*} u_-(\tau)+u_+(\tau)=0. \end{equation*} $$

On the other hand, for a non-symmetric periodic orbit of the global system, we have two different flight times τ₁ and τ₂, corresponding to the (S_U) and (S_L) pieces of the orbit, respectively. So, a non-symmetric periodic orbit has to verify in terms of u₋ and u₊ the following equations

{u−(τ2)+u+(τ1)=0,u−(τ1)+u+(τ2)=0.$$ \begin{equation} \left\{\begin{array}{r c l} u_-(\tau_2)+u_+(\tau_1)&=&0,\\ u_-(\tau_1)+u_+(\tau_2)&=&0. \end{array} \right. \end{equation} $$(8)

The condition (3) for the stability of a symmetric periodic orbit, in terms of u₋ and u₊, results in

|U′(u−)|=|u′+(τ)u′−(τ)|<1.$$ \begin{equation*} |U'(u_-)|=\left|\frac{u'_+(\tau)}{u'_-(\tau)}\right|\lt1. \end{equation*} $$

In the case of non-symmetric periodic orbit, to compute the derivative of the full transition map L∘ U, we start by using the notation p=u₋(τ₁), q=u₊(τ₁)=−u₋(τ₂) to write

ddp(L○U)(p)=L′(U(p))U′(p)=L′(q)U′(p)==U′(−q)U′(p)=U′(u−(τ2))U′(u−(τ1)),$$ \begin{equation*} \begin{array}{r c l} \frac{d}{dp}(L\circ U)(p)&=&L'(U(p)) U'(p)= L'(q) U'(p)=\\ &=&U'(-q) U'(p)=U'(u_-(\tau_2))U'(u_-(\tau_1)), \end{array} \end{equation*} $$

where we have used that L(q)=−U (−q). Now, the stability of a periodic orbit requires

|U′(u−(τ2))U′(u−(τ1))|=|u′+(τ1)u′+(τ2)u′−(τ1)u′−(τ2)|<1.$$ \begin{equation*} |U'(u_-(\tau_2))U'(u_-(\tau_1))|=\left|\frac{u'_+(\tau_1)u'_+(\tau_2)}{u'_-(\tau_1)u'_-(\tau_2)}\right|\lt1. \end{equation*} $$

It will be useful to introduce the auxiliary parameter

ρ=xE+1xE−1,$$ \begin{equation*}\nonumber \rho=\frac{x_E+1}{x_E-1}, \end{equation*} $$

which belongs to the interval (0, 1) since x_E < −1.

The following lemmas give preliminary properties to be used later.

Our objective is to study the existence, uniqueness and stability of the possible periodic orbits. Our first result gives two necessary conditions for the existence of non-symmetric periodic orbits.

and the conclusion follows.

sign[u′+(τ)−u′−(τ)]=sign[(xE+1)ξ(τ;γ)+(xE−1)ξ(τ;−γ)],$$ \begin{equation*} \text{sign}[u'_+(\tau)-u'_-(\tau)]=\text{sign}[(x_E+1)\xi(\tau;\gamma)+(x_E-1)\xi(\tau;-\gamma)], \end{equation*} $$

where ξ(τ;γ)=e^γτ(chτ−γshτ)−1. Since ξ (0;γ)=0 and ξ′(τ;γ)=e^γτ(1−γ²)shτ > 0, we conclude that u′₊(τ)−u′₋(τ) < 0 for all τ > 0, and the conclusion follows.

In what follows, we assume a fixed value x_E for the abscissa of the saddle point, and we study equation (2) looking for the solution values of y_E, γ and τ. It is worth noting that the effect of the parameter y_E in (5) is only a translation.

Next, by considering γ and y_E as principal bifurcation parameters, we give the complete bifurcation set for the case x_E < −1. We classify the parameter regions according to the number of symmetric periodic orbits, see Figure 4. The case x_E ⩾ −1 will be the subject of future works.

Fig. 4

limu−→u−*U(u−)=u+*,U″(u−)<0,for allu−<u−*.$$ \begin{equation*} \lim_{u_-\to u^*_-} U(u_-)=u^*_+,\quad U''(u_-)\lt0, \;\; \mbox{for all}\,\, u_-\lt u^*_-. \end{equation*} $$

In the open interval γ ∈ (−1, 0), we have also

0<U′(u−)<1,limu−→u−*U′(u−)=0;$$ \begin{equation*} 0\lt U'(u_-)\lt1, \quad \lim_{u_-\to u^*_-} U'(u_-)=0; \end{equation*} $$

while, in the particular case γ=0,

−ρ<U′(u−)<1,limu−→u−*U′(u−)=−ρ.$$ \begin{equation*} -\rho\lt U'(u_-)\lt1, \quad \lim_{u_-\to u^*_-} U'(u_-)=-\rho. \end{equation*} $$

In Figure 5, the above properties are illustrated.

The heteroclinic connection arises when u+*=−u−* $u^*_+=-u^*_-$ , and taking into account Definition 1, we get the expression y_H(γ)=1+γ x_E. Since U′(u₋) > −1 for γ ∈ (−1, 0], it is not possible to get a tangent point of the map U with the line u₊=−u₋. Then, the maximum number of symmetric periodic orbits is one in this case. So, for γ ∈ (−1, 0] we have the following situations.

If y_E < y_H(γ), then there are no periodic orbits because the map U does not intersect with u₊=−u₋.

Fig. 5

When γ ∈ (0, 1) the situation is more involved, due to the lack of injectivity for the map U and the fact that its derivative U′(u₋) can be −1, see Figure 6. We still have U″(u₋) < 0 for all u−<u−* $u_-\lt u^*_-$ but we now have U′(û₋), so that

0<U′(u−)<1,for allu−<u^−,andU′(u−)<0,for allu−>u^−,$$ \begin{equation*} 0\lt U'(u_-)\lt1, \;\;\mbox{for all} u_-\lt\hat{u}_-, \mbox{and}U'(u_-)\lt0,\;\; \mbox{for all} u_-\gt\hat{u}_-, \end{equation*} $$

along with

limu−→u−*U(u−)=u+*,limu−→u−*U′(u−)=−∞.$$ \begin{equation*} \lim_{u_-\to u^*_-} U(u_-)=u^*_+, \quad \lim_{u_-\to u^*_-} U'(u_-)=-\infty. \end{equation*} $$

From now on, we emphasize the dependence of γ for the functions. The heteroclinic connection is obtained as before, while the saddle-node bifurcation comes from applying the saddle-node theorem in [4]. This theorem assures the existence of a saddle-node bifurcation of periodic orbits under several conditions for the map G:= U (−U(u₋;γ);γ)+u₋, which can be translated to the map U as follows.

Periodic orbit condition: U(u₋;γ)=−u₋,

Since we have the expression of the transition map U in a parametric form, the above conditions are equivalent to

u₊(τ;γ)+u₋(τ;γ)=0,

Next, we study the equations (a) and (b) in order to check if their solutions satisfy the other two conditions (c) and (d). In such a case, the global system (S_U)-(S_L) undergoes a saddle-node bifurcation of periodic orbits for the obtained values of the parameters.

After some computations, we get that condition (a) is equivalent to

(yE−γxE)shτ=chτ+ch(γτ)+xEsh(γτ),$$ \begin{equation} (y_E-\gamma x_E)\text{sh}\tau=\text{ch}\tau+\text{ch}(\gamma\tau)+x_E\text{sh}(\gamma\tau), \end{equation} $$(9)

and condition (b) can be rewritten as f(τ)=ρ, where

f(τ):=e−γτ(chτ+γshτ)+1eγτ(chτ−γshτ)+1=xE+1xE−1=ρ.$$ \begin{equation} f(\tau):=\frac{e^{-\gamma\tau}(\text{ch}\tau+\gamma\text{sh}\tau)+1}{e^{\gamma\tau}(\text{ch}\tau-\gamma\text{sh}\tau)+1}=\frac{x_E+1}{x_E-1}=\rho. \end{equation} $$(10)

Now, we study if the equation (10) has a solution for each γ ∈ (0, 1).

Some properties of f(τ) are

f(0)=1,limτ→+∞f(τ)=0.$$ \begin{equation*} f(0)=1, \quad \lim_{\tau\rightarrow +\infty}f(\tau)=0. \end{equation*} $$

Also, standard computations show that for γ ∈ (0, 1), the function f is positive and decreasing. Then, there is only one solution of (10), in other words, there is only one point of the map U which has slope −1. Finally, we also need to check the conditions (c) and (d) for the solution obtained before. Condition (c) is equivalent to

−2xEshτ+γ(eγτ(xE+1)+e−γτ(xE−1))shτ≠0,$$ \begin{equation*} -\frac{2x_E \text{sh}\tau +\gamma(e^{\gamma\tau}(x_E+1)+e^{-\gamma\tau}(x_E-1))}{\text{sh} \tau}\neq 0, \end{equation*} $$

which is automatically satisfied for γ ∈ (0, 1) and τ > 0.

Condition (d) is more involved and we prove it by contradiction. Using equations (7), to deny condition (d) is equivalent to assume the equality

−2(xE+1)chτ+e−γτ(xE−1)(1+(chτ+γshτ)2)+2(xE−1)chτ−eγτ(xE+1)(1+(chτ−γshτ)2)=0.$$ \begin{eqnarray*}-2(x_E+1)\text{ch} \tau+ e^{-\gamma\tau}(x_E-1)(1+(\text{ch}\tau+\gamma\text{sh}\tau)^2)+\\ 2(x_E-1)\text{ch}\tau-e^{\gamma\tau}(x_E+1)(1+(\text{ch}\tau-\gamma\text{sh}\tau)^2)=0. \end{eqnarray*} $$

Dividing by (x_E−1), and using condition (b) we get

e−γτ(1−γ2)sh2τ(−2γeγτshτ−e2γτ+1)=0,$$ \begin{equation*} e^{-\gamma\tau}(1-\gamma^2)\text{sh}^2\tau(-2\gamma e^{\gamma\tau}\text{sh}\tau-e^{2\gamma\tau}+1)=0, \end{equation*} $$

which is a contradiction because all the factors are non-vanishing.

To get the saddle-node curve y_E=y_SN(γ), shown in Figure 4, we fix γ ∈ (0, 1) and solve equation (10) for τ. Assume that τ* is that solution. Then, we put τ* in (9) and we obtain only one solution y_E=y_SN(γ). For this concrete value of y_E, the point of the map U with slope −1 is exactly on the line u₊(τ)+u₋(τ)=0. As a consequence, perturbing slightly the parameter y_E, we have either no symmetric periodic orbits when y_E < y_SN(γ) or two of them if y_E > y_SN(γ).

Fig. 6

We show in Figure 7 two symmetric periodic orbits coexisting for γ ∈ (0, 1) and y_SN(γ) < y_E < y _H(γ). This parameter region is indicated in Figure 4.

Fig. 7

The dynamical richness regarding the existence of periodic orbits in bidimensional hysteretic linear systems with symmetry, for the specific case of equilibria not in the hysteretic band, has been shown through a bifurcation analysis. The corresponding bifurcation set is mainly organized by a locus of heteroclinic bifurcations, what indicates the relevance of heteroclinic orbits, as it has been recently emphasized in this journal, see [1]. The bifurcation set also includes a locus of saddle-node bifurcations of periodic orbits, leading to a parameter region where two different periodic orbits coexist.