Here the parameters c,ɛ satisfy the inequalities
- 0 < c ≤ 2, which guarantees the invariance of the interval I;
- 0 < ɛ < 1, which corresponds to positive saddle value in the geometric Lorenz model (see below) and implies infinite one-sided derivatives at the discontinuity point.
The above assumptions imply the following result on the birth of Lorenz attractors.
(Shilnikov, ) If 0 < |A±| < 2 for the map(3), then in the parameter plane (μ,ɛ) there is a region VLA such that (0,0) belongs to the closure
In , the system (4) is shown to appear as the normal form in studying the local bifurcations of the equilibrium state which has zero eigenvalue with multiplicity 3. Also the parameter values corresponding to the existence of the Lorenz attractor in this system have been found numerically. Later, A. Shilnikov studied the Shimizu-Morioka system in more details ([5, 6]). Note that in  the kneading technique was applied for studying the system (4) (see the section 3 below for definitions and some results of the kneading theory). In particular, one of the results of those studies is the following fact: in the parameter plane (α,λ), the boundary of the region corresponding to the existence of Lorenz attractor, contains the system having homoclinic figure-8 loop at a saddle with zero saddle value (see fig. 2). Presence of this bifurcation in the system (4) has been proven analytically in ; however, the problem on finding the estimate on the separatrix value A, which would allow to apply theorem 1, remained open, until recently it was obtained in  that 0.625 < A < 0.627 (using the computer assisted proof).
Note that this change of coordinates is degenerate at μ = 0 and so, the line μ = 0 in the parameter plane (μ,ɛ), which corresponds to systems with homoclinic loops, is transformed to c = ∞ in the parameter plane (c,ɛ). Since the value μ−ɛ can attain any number from 1 to ∞ as c,ɛ →0, it follows that the point (μ = 0, ɛ = 0), which corresponds to the homoclinic butterfly bifurcation of the saddle with zero saddle value in the initial system, is transformed into the whole straight line ɛ = 0. Therefore the map Tc,ɛ with ɛ small is related to the bifurcation of birth of attractor from the separatrix loop with zero saddle value.
The paper is organised as follows. Section 2 contains the construction of bifurcation diagram in the parameter space (c,ɛ). We indicate the bifurcation curves which divides the parameter plane into regions with different number of connected components of attractor and indicate some other dynamical features. We also add the lines of constant topological entropy (the kneading charts) to this diagram.
In Section 3 we consider the entropy and kneading aspects of dynamics of the family Tc,ɛ. We study the problems of continuity and monotonicity of topological entropy as the function of the parameter c. Note that the problems of monotonicity of topological entropy was considered for specific families of one dimensional maps by several authors. In ,  it has been proven that for quadratic maps x2 + c, the topological entropy is a monotone (non strictly) increasing function of c (see fig. 3 for the logistic family, which is actually the same after change of coordinates). In recent paper , the monotonicity result was proven for the family xℓ+ c with ℓ large (not necessarily integer). Our family of maps is different from those families in the sense that we allow infinite derivatives at the discontinuity point, which makes the problem even more complicated because the complex analysis technique doesn’t work here.
We show that for ɛ fixed, in the one-parameter family Tc = Tc,ɛ, the topological entropy is not monotone. We show numerically that the topological entropy as the function of c has a single minimum in certain region. Also we show that in the case when Tc,ɛ is expanding ((DT ≥ q > 1), the topological entropy is monotone increasing in c.
2 Bifurcation diagram
In this section we study principal bifurcations of the map Tc,ɛ in the region 1 ≤ c ≤ 2, 0 ≤ ɛ ≤ 1 of the parameter plane. Note that we consider the map Tc,ɛ not only for ɛ small, and hence, some of our results lie beyond the applications in the geometric Lorenz model (in particular, the condition b) in (2) is satisfied not for all regions we consider here)
The figure 4 presents the bifurcation diagram. It makes sense to consider this diagram together with the chart of the kneading invariants presented in figure 5. The notion of the kneading invariant is discussed in details below in section 3. Now we only stress that under some natural assumptions (like expanding ones) the kneading invariant is the complete invariant of topological conjugation, and in more general case, the points with the same kneading invariant have similar orbits (up to so-called combinatorial equivalence). Thus, the kneading chart along with the bifurcation diagram allow to realize fairly complete picture of dynamical and topological structure.
Further we will describe the regions in the bifurcation diagram as well as the curves which partition these regions. First we explain the reason for considering the values c just from the interval [1,2].
The case when c>2. If c > 2, the dynamical behavior is as follows. The interval I is no longer invariant under Tc,ɛ. Indeed, for such c two unstable fixed point xu > 0 and −xu belong to the interval I (see fig. 6) and any point from subintervals (xu,1) and (−1,−xu) tends to the stable fixed points xs or −xs, respectively, outside the interval I (here we consider the map Tc,ɛ on the whole real line; note that Tc,ɛ is well defined on ℝ due to formula (1)). The same happens to every preimage of these two subintervals. The restriction of Tc,ɛ to the interval [−xu,xu] is an expanding map because, due to monotonicity of the derivative, one has the inequality DT(x) ≤ DT (xu) > 1. So almost all points (with respect to Lebesgue measure) tend to the two stable fixed points. By standard arguments using the expanding condition, the nonwandwering set NW (Tc,ɛ) consists of a Cantor set (the set of points not escaping from the interval I), and moreover, the map Tc,ɛ restricted to this set is conjugate to the one-sided Bernoulli shift with two symbols.
The maximal wandering intervals whose boundary contains this unstable orbit is usually called the trivial lacunae and similar definition is used for initial flow. fig. 10 shows schematically the appearance of the trivial lacuna and the corresponding reconstructing of the attractor.
By passing from LA2 to LA3, a similar bifurcation for the restriction
Note that the set of points corresponding to the moments when the discontinuity point is eventually periodic to the period-2 orbit, is the union of two curves (see two thick red lines in fig. 4). It is not only the boundary between the regions LA1 and LA2, but also the boundary corresponding to the presence of the attractor. Through the point of intersection of these curves, the curve PF passes as well; this corresponds to the moment when a single period-2 orbit exists, and its multiplicator equals 1. In this bifurcation moment the map T is transitive, and the attractor coincides with the whole I.
3 Non-monotonicity of the topological entropy
First recall the main concepts and some results of the kneading theory. The kneading theory was introduced in  for continuous piecewise monotone maps of the interval. In , the kneading theory was developed for (discontinuous) Lorenz maps and in  for unimodal maps along with their Lorenz models.
The following relation holds.
In fact, the problem on monotonicity of topological entropy for symmetric Lorenz maps is equivalent to the problem on monotonicity of the kneading invariant
Along with monotonicity of the topological entropy for the family of maps Tc,ɛ we discuss here the problem of continuity of entropy as the function of the parameters. The next result is a generalization of the theorem from , it provides a criterion for continuity/discontinuity of the topological entropy of Lorenz maps with respect to C0-topology.
(Malkin, Safonov, ) The topological entropy may have a jump at the Lorenz map T in the space of Lorenz maps (not necessarily, symmetric) equipped with the C0-topology if and only if the following two conditions hold:
- 1there is a natural number p > 1 such that Tp−1(1) = Tp−1(−1) = 0 and Ti(1) ≠ 0, Ti(−1) ≠ 0 for all 0 < i< p − 1,
- 2htop(T) = 0.
Note that for our family Tc,ɛ with c > 1, the topological entropy is positive (it can be shown analytically; the numerical calculation see in the graph of entropy in fig. 11). Thus, it follows from the above theorem that in the whole rectangle with c > 1 (in fig. 4), the topological entropy changes continuously.
In the green region (in fig. 4) near the axis c = 1, the entropy equals
For the two red thick lines in fig. 4 (given by formula 7) the kneading invariant is the same as in the yellow region below, i.e.,
Now we want to remark that even without the expanding condition above the map still can possess chaotic attractor. In particular, the next result shows that there are no stable periodic points not only in the region (10) but also in wider region, namely, above the curve PF and satisfying DT2(1) > 1 (see figures 4 and 13).
This work was partially supported by Laboratory of Dynamical Systems and Applications NRU HSE of the Ministry of science and higher education of the RF grant ag. No. 075-15-2019-1931; and by RFBR grant No. 18-29-10081. The work of K.S. was also partially supported by the RSF grant 19-71-10048. The authors thank Dr. Dmitry Turaev for posing the problem.
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The latter conditions could be assumed without loss of generality because outside the interval [T (0+),T (0−)] each point is wandering and dynamics is trivial.