This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from a reduction of symmetric 3D systems with slow-fast dynamics. We concentrate our attention on the saddle dynamics cases, determining the existence of periodic orbits as well as their stability, and possible bifurcations. Dealing with reachable saddles not in the central hysteresis band, we show the existence of subcritical/supercritical heteroclinic bifurcations as well as saddle-node bifurcations of periodic orbits.
Following , 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  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
and a similar lower system, namely
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  looking for the characterization of centers in planar dynamical systems. Here, as usual, the dot represents derivatives with respect to the time τ, (xE, yE) and (−xE, −yE) are the equilibria for the SU-system and SL-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 (SU)-(SL) 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 SU−system. Then this solution is called a valid solution as long as x(τ)⩽1. If there exists a time τf, such that x(τf)=1 with ẋ > 0, then the point (x(τf), y(τf)) is assumed to be the initial point for an orbit of the SL-system and this orbit continues by integrating system (SL).
Analogously, any solution (x(τ), y(τ)) of the SL-system with x (0) > −1, is considered as a valid solution as long as x(τ) > −1, and if there exists a time τr such that x(τr)=−1 with ẋ(τr) < 0, then the point (x(τr), y(τr)) is assumed to be the initial point for an orbit of the SU-system.
In passing from the SU-system to the SL-system, we speak of a fall, which occurs when an orbit of the system (SU), called upper orbit, hits the falling line
Similarly, we define a rise when we pass from the SL-system to the SU-system, which occurs when an orbit of system (SL), called lower orbit, intersects the rising line
Note that our formulation is equivalent to write the system
where H(x) is the standard normalized hysteretic function of Figure 2.
Thus, any periodic orbit of global system (SU)-(SL) will lead to a repeated sequence of falls and rises and has at least two pieces, one corresponding to an orbit of system (SU) and the other to an orbit of system (SL).
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
A similar evolution map can be defined for the SL-system by considering the points
This map also induces a transition map L, which is defined by l−=L(l+).
Clearly, the full transition map for the global system (SU)-(SL) 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 xE < −1. The remaining cases, namely real saddles in the central band (∣xE∣⩽1) and virtual saddles (xE > 1) will appear elsewhere.
2 Preliminary results
The symmetry between the SU-system and the SL-system imposes a symmetry property for the functions L and U as follows.
The following statements hold.
If U(y) is well-defined, then L (−y) is well-defined and L (−y)=−U(y).
The full transition map satisfies L∘ U = (−U)∘ (−U).
We only show the first assertion. Under the hypothesis, taking v=U(y), we know that
Since the SL-system is the symmetric one of the SU-system with respect to the origin, we have
and so −v=L (−y) and we are done.
Assume that we start from a point
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.
and reordering equations, we conclude that the pair (−u+, −u−) is also a solution of (1), so that there exists a companion periodic orbit, which is the symmetric one of the assumed periodic orbit with respect to the origin. In the particular case where u+=−u−, the periodic orbit is itself symmetrical with respect to the origin, and the proof is complete.
The existence of a pair of non-symmetric periodic orbits implies, by a standard application of the intermediate value theorem to the function U(y)+y, the existence of a third symmetric periodic orbit.
For the specific case of symmetric periodic orbits, instead of equations (1), we must only consider the equation
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
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 (SU) taking into account that the equilibria is of saddle type, namely
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 (xE, yE) are
In this work, we assume in the sequel xE < −1, corresponding to the case when the saddle of system (SU) is on the left of Σ−. Now, we define the domain of the transition map TU,
For xE < −1 let us introduce the points
We also introduce the point
See Figure 3 for a geometrical view of these distinguished points.
Accordingly, we get as the admissible domain for the map TU the set
so the map U is defined in the interval
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
Now, we write the first and second derivatives with respect to τ, to be used later,
Standard computations show that
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 (SU) and (SL) pieces of the orbit, respectively. So, a non-symmetric periodic orbit has to verify in terms of u− and u+ the following equations
The condition (3) for the stability of a symmetric periodic orbit, in terms of u− and u+, results in
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
where we have used that L(q)=−U (−q). Now, the stability of a periodic orbit requires
It will be useful to introduce the auxiliary parameter
which belongs to the interval (0, 1) since xE < −1.
The following lemmas give preliminary properties to be used later.
For xE < −1 and γ ∈ (−1, 1), the function u−(τ) is increasing for all τ ∈ (0, +∞).
From (6), we see that
and then, to get the conclusion, as ρ ∈ (0, 1), it suffices to see that
Effectively, since η (0)=1, and η′(τ) = (1−γ2)e−γτshτ > 0, we are done.
For xE < −1 and γ ∈ (−1, 1), the transition map U defined in (5) is concave down.
The transition map U is concave down if and only if
where we omit the argument in the above functions for brevity.
From Lemma 4, u′−>0 and so, we need to study the sign of
The conclusion comes from the fact that e2γτρ2−2γρ eγτshτ−1 is a quadratic polynomial in ρ negative for ρ ∈ (0, 1), and we are done.
3 Main results for real saddles out of the hysteresis band
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.
(Necessary conditions for non-symmetric periodic orbits)
If there exists a non-symmetric periodic orbit, the two following statements are true.
The function u+(τ)−u−(τ) is not injective.
The function u+(τ)+u−(τ) takes opposite values.
Adding and subtracting equations (8), we can write
and the conclusion follows.
When xE < −1 and γ ∈ (−1, 1), there cannot be non-symmetric periodic orbits.
Let us show that the function u+(τ)−u−(τ) is injective and so the conclusion will follow from Lemma 6. We will show that u′+(τ)−u′−(τ) < 0 for all τ > 0.
From (6), we see that
where ξ(τ;γ)=eγτ(chτ−γshτ)−1. Since ξ (0;γ)=0 and ξ′(τ;γ)=eγτ(1−γ2)shτ > 0, we conclude that u′+(τ)−u′−(τ) < 0 for all τ > 0, and the conclusion follows.
From Proposition 7, there can be only symmetric periodic orbits. By equation (2), the number of symmetric periodic orbits of the global system correspond to the intersections of the transition map U with the secondary diagonal. Since the map U is concave down from Lemma 5, we conclude that the maximum number of symmetric periodic orbits is two.
In what follows, we assume a fixed value xE for the abscissa of the saddle point, and we study equation (2) looking for the solution values of yE, γ and τ. It is worth noting that the effect of the parameter yE in (5) is only a translation.
Next, by considering γ and yE as principal bifurcation parameters, we give the complete bifurcation set for the case xE < −1. We classify the parameter regions according to the number of symmetric periodic orbits, see Figure 4. The case xE ⩾ −1 will be the subject of future works.
If we define the function yH(γ)=1+γ xE, then at the points of the straight line yE=yH(γ) in the parameter plane (γ, yE), the system has an heteroclinic bifurcation.
For γ ∈ (−1, 0] this bifurcation is supercritical, namely, for yE > yH(γ) the heteroclinic connection gives rise to a stable symmetric periodic orbit.
For γ ∈ (0, 1) the bifurcation is subcritical, so that when yE=yH(γ) the heteroclinic connection coexists with a stable symmetric periodic orbit, while for yE < yH(γ) the heteroclinic connection gives rise to an unstable symmetric periodic orbit that coexists with the stable one.
Furthermore, for yE > yH(γ) there exists only one stable symmetric periodic orbit.
For γ ∈ (−1, 0] and yE < yH(γ) there are no periodic orbits. For γ ∈ (0, 1), there exists a function ySN(γ) such that ySN(γ) < yH(γ) and
so that at the points (γ, ySN(γ)), the system undergoes a saddle-node bifurcation of symmetric periodic orbits. More precisely, when γ ∈ (0, 1) in the interval
the system has two symmetric periodic orbits with opposite stability, while for yE < ySN(γ) there are no periodic orbits.
Since symmetric periodic orbits correspond with the intersections of the graph of U with the secondary diagonal of the plane (u−, u+), some properties of map U are studied. To prove the theorem, we distinguish the two cases γ ∈ (−1, 0] and γ ∈ (0, 1).
In the open interval γ ∈ (−1, 0), we have also
while, in the particular case γ=0,
In Figure 5, the above properties are illustrated.
The heteroclinic connection arises when
If yE < yH(γ), then there are no periodic orbits because the map U does not intersect with u+=−u−.
If yE=yH(γ), then the heteroclinic connection is produced.
If yE > yH(γ), then there is a stable symmetric periodic orbit because the map U has exactly an intersection point with the secondary diagonal.
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
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 . 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−,
Non-hyperbolicity condition: U′(u−;γ)=−1,
Non-degeneracy condition: U″(u−;γ)≠ 0.
Since we have the expression of the transition map U in a parametric form, the above conditions are equivalent to
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 (SU)-(SL) 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
and condition (b) can be rewritten as f(τ)=ρ, where
Now, we study if the equation (10) has a solution for each γ ∈ (0, 1).
Some properties of f(τ) are
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
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
Dividing by (xE−1), and using condition (b) we get
which is a contradiction because all the factors are non-vanishing.
To get the saddle-node curve yE=ySN(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 yE=ySN(γ). For this concrete value of yE, 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 yE, we have either no symmetric periodic orbits when yE < ySN(γ) or two of them if yE > ySN(γ).
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 . 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.
Communicated by J. Llibre
Authors are partially supported by the Spanish Ministerio de EconomÍa, Industria y Competitividad, in the frame of projects MTM2014-56272-C2-1-P and MTM2015-65608-P, and by the ConsejerÍa de EconomÍa y Conocimiento de la Junta de AndalucÍa under grant P12-FQM-1658.
A. Visintin, (1994), Differential Models of Hysteresis, Springer-Verlag, New York.
We start by assuming a symmetric 3D piecewise linear system with the following structure
where X, Y, Z ∈ ℝ are the states variables, A = (aij) is a 2 × 2 matrix with coefficients in ℝ, b = (b1, b2) is a real vector, 0 < µ≪ 1 and φ(Z) is a piecewise linear function defined by
where c, m, x0 and z0 are positive real numbers, and x0=cz0 so that φ(Z) is a continuous function. Here, the dot represents the derivative with respect to the time s.
In the limit when µ → 0, the last equation of (3DPWL) represents a surface which is usually called slow manifold. In Figure 8 we can see a graph of φ(Z) and the generated surface for some values of the parameters c, m, x0 and z0.
Regarding Figure 8, we can define the two half-planes ZU and ZL, namely
The set, ZU (resp. ZU will be called upper half-plane (resp. lower half-plane). Now, for δ≠ 0 and small, consider a half-plane which is parallel to ZU
Then, using the last equation of (3DPWL) with the suitable evaluation of
and so, ZU is attractive. Something similar occurs for the lower half-plane ZL. However, the intermediate stripe between ZU and ZL turns out to be repulsive.
Therefore, we can consider that the motion in ℝ3 happens only in the two attractive half-planes ZU and ZL (upper and lower). When an orbit reaches the boundary of one half-plane (ZU or ZL), it jumps instantaneously to the other half-plane by keeping the same values of X and Y and changing only the value of Z. Clearly, the dynamics on each half-plane is essentially two-dimensional; in fact, we can eliminate the third variable by projecting the orbits on the plane Z=0 by using the equation of each half-plane. Then we have on ZU for Z=z0, the dynamical system
and similarly on ZL for Z ⩽ −z0, the system
Starting from these two systems, which form a symmetric pair of dynamical systems, some changes of variables are needed in order to arrive at the system (SU)-(SL) of Section 1. Basically, after a rescaling to put the boundary lines at x=± 1, all what is required is to write the systems in the reduced Liénard form following the approach in .