The number of limit cycles of piecewise linear Liénard systems

Abstract: For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open. Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided.

• The paper confirms the lower bounds (n for fold points, 2n for jump points, and n+m for mixed) of the maximum number of limit cycles of piecewise linear Liénard systems, as predicted by Tonnelier's conjecture (2002).
• The authors provide explicit constructions verifying these lower bounds, and a classification of all possible dynamics near infinity on the Poincaré disc, crucial for bounding the number of limit cycles.
• For configurations where $F(x)$ is symmetric but only outside a switching line, or where parameters shift the slopes, boundedness near infinity and accordingly cycle stability remain unresolved.

Overview

The paper studies the planar Liénard system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is piecewise linear on $n+1$ intervals, written in the unified form

$$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$

with switching lines $L_i: x=x_i$. A point on a switching line is a fold point when $F$ is continuous but its one-sided derivatives differ ($b_i=0$, $a_i\neq 0$), and a jump point when $F$ is discontinuous ($\dot{y}=x$0). Tonnelier conjectured in 2002 that the maximum number of limit cycles is $\dot{y}=x$1 when $\dot{y}=x$2 has only fold points and $\dot{y}=x$3 when it has only jump points. Prior to this work the conjecture was verified only for $\dot{y}=x$4 in the fold case (2608.19542) and for $\dot{y}=x$5 in the jump case; all other cases were open. The paper establishes two main contributions: (i) confirmation of the lower bound part of Tonnelier's conjecture for all $\dot{y}=x$6, via explicit constructions, and (ii) a complete classification of the dynamics near infinity of these systems on the Poincaré disc.

Dynamics near infinity

The authors first settle the qualitative behavior near infinity using Poincaré compactification. Under the transformations $\dot{y}=x$7 with $\dot{y}=x$8, equilibria at infinity correspond to real roots of $\dot{y}=x$9 (right half-plane) and $F(x)$0 type equations (left half-plane), where

$F(x)$1

are the outermost slopes of $F(x)$2. The resulting classification is sharp: if $F(x)$3 there are no equilibria at infinity in the right half-plane, if $F(x)$4 exactly one (a saddle-node), and if $F(x)$5 two (a stable node and a saddle); analogously for $F(x)$6 in the left half-plane. This partitions the parameter space of outermost slope pairs into 15 cases, each associated with an explicitly drawn phase portrait near infinity.

A key structural finding is that, away from a degenerate situation, the behavior at infinity is governed entirely by the pair $F(x)$7. In case (3c) ($F(x)$8, $F(x)$9) there are no equilibria at infinity, and the paper proves that infinity is a center if $n+1$0 for $n+1$1, and otherwise a focus whose stability is determined by the sign of $n+1$2 (and, when $n+1$3, by comparing $n+1$4 and $n+1$5): orbits sufficiently close to infinity are positively bounded if $n+1$6 and negatively bounded if $n+1$7. The proof is by direct comparison of integral curves of $n+1$8 using variation-of-constants estimates.

One limitation is stated plainly: when $n+1$9 holds only for $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$0 but not globally for $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$1, the comparison integral degenerates to a pure exponential whose sign cannot be determined, so boundedness near infinity remains unresolved in that scenario — an open center–focus problem at infinity.

This classification is not merely descriptive: it supplies the outer boundary of Poincaré–Bendixson annular regions used throughout the constructive proofs, since stability at infinity depends only on $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$2 regardless of whether $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$3 lies in cases (3c) or elsewhere.

Lower bounds for the maximum number of limit cycles

The central results are three existence theorems, proved by induction:

• Fold points only ($$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$4): a lower bound for the maximum number of limit cycles is $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$5, confirming the lower bound of part (i) of Tonnelier's conjecture.
• Jump points only ($$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$6): a lower bound is $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$7, confirming part (ii).
• Mixed case ($$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$8 jump points, $$F(x)=ax+b+\sum_{i=1}^n a_i|x-x_i|+\sum_{i=1}^n b_i\,\operatorname{sgn}(x-x_i),$$9 fold points): a lower bound is $L_i: x=x_i$0, supporting the authors' proposed Conjecture that this is also the upper bound — a generalization of Tonnelier's conjecture that was previously completely open.

All constructed systems satisfy restrictive structural properties: all switching lines lie in $L_i: x=x_i$1, the outermost slope pair belongs to case (3c), and the outermost limit cycle crosses every region. The induction proceeds by appending a new switching line strictly to the right of the current outermost limit cycle, so that the modification leaves all previously constructed cycles untouched.

For a new fold point, the parameter $L_i: x=x_i$2 shifts the rightmost slope $L_i: x=x_i$3 without changing $L_i: x=x_i$4, allowing the authors to flip the sign of $L_i: x=x_i$5 and hence reverse the stability of infinity. Together with an inner boundary built from an orbit arc through the point where the new segment meets $L_i: x=x_i$6 — across which the vector field has fixed transversality because $L_i: x=x_i$7 there — this produces a Poincaré–Bendixson annulus containing a new crossing or grazing cycle. Each fold point thus contributes exactly one additional cycle.

For a new jump point, two cycles arise per step. First, choosing the jump height $L_i: x=x_i$8 so that the left and right limits of $L_i: x=x_i$9 straddle the orbit arc creates a Filippov sliding segment on $F$0; closing this arc with the sliding segment yields a sliding limit cycle, shown to be unstable by evaluating $F$1 along the segment. Second, the annular region between this sliding cycle and the equator (with infinity an unstable focus under property (V)) contains a further crossing cycle. Hence each jump point contributes two additional limit cycles.

The mixed-case theorem combines both mechanisms: starting from a system with $F$2 jump points and $F$3 cycles, fold points are appended one at a time, each adding one crossing/grazing cycle while preserving prior ones, yielding $F$4 cycles. The classification into crossing, sliding, grazing, and composite limit cycles is essential here, since sliding sets exist precisely at jump points.

It should be emphasized that the constructed systems realize the lower bound only; matching upper bounds remain unproved except for $F$5 (fold) and $F$6 (jump). Moreover, the authors concede that the imposed arrangement — jump points first, then fold points, all at positive abscissas, with $F$7 in case (3c) — is sufficient rather than necessary, and that more general arrangements can be handled by their method.

Limitations and open questions

Three gaps are acknowledged within the paper itself. First, the center–focus dichotomy at infinity in case (3c) is incomplete when $F$8 coincides with its even reflection only outside the outermost switching line; distinguishing center from focus there requires further analysis. Second, the paper establishes lower bounds only; whether the maxima equal $F$9 (fold-only), $b_i=0$0 (jump-only), or $b_i=0$1 (mixed) remains unproved for $b_i=0$2 and mixed configurations, respectively. Third, the constructions rely on placing every new switch point outside all existing limit cycles and on tuning the stability of infinity via the outermost slopes; whether the same counts are attainable without these structural restrictions is not addressed.

Conclusion

The paper confirms the lower-bound half of Tonnelier's conjecture for piecewise linear Liénard systems with arbitrary numbers of fold and jump points, and extends it to a mixed regime via the sharper statement that $b_i=0$3 jump points plus $b_i=0$4 fold points yield at least $b_i=0$5 limit cycles. Its complete Poincaré-disc classification of the dynamics at infinity — showing that, generically, everything near infinity is dictated by the outermost slopes $b_i=0$6 and $b_i=0$7 — provides the technical backbone for the constructions. The remaining challenge, left open here, is proving matching upper bounds, i.e., that no system of this class exceeds $b_i=0$8 limit cycles.