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Limit Cycles in Planar Systems of OrdinaryDifferential Equations
Torsten Lindström
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Planar Linear and Linearized Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4First Integral Systems and Gradient Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Monotone Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10The Existence of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12The Liénard (1928) Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Theorems for Absence of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Uniqueness of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Abstract
The idea of a dynamical system is predicting the future of a given systemwith respect to some initial conditions. If the dynamical system is formulatedas a differential equation, then there is usually a direct relation between thedynamical system and the processes involved. Today, we can easily say thatdynamical systems can predict a huge number of phenomena, including chaos.The real question is, therefore, not whether complicated phenomena may occur,but whether restrictions on the possible dynamics exist.
In this chapter, we commence with major theorems that are frequently usedfor justifying phase space analysis. We continue with simple examples that eitherpossess limit cycles and classes of differential equations that never possess limit
T. Lindström (�)Department of Mathematics, Linnæus University, Växjö, Swedene-mail: [email protected]
© Springer Nature Switzerland AG 2020B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_34-1
1
2 T. Lindström
cycles. We end up with the ideas behind two major theorems that put bounds forthe number of limit cycles from above: Sansone’s (Annali di Matematica Pura edApplicata, Serie IV 28:153–181, 1949) theorem and Zhang’s (Appl Anal 29:63–76, 1986) theorem. Both theorems apply to systems that have a clear mechanisticinterpretation. We outline the major arguments behind the quite precise estimatesused in these theorems and describe their differences. Our objective is not toformulate these theorems in their most general form, but we give references torecent extensions.
Keywords
Limit cycle · Global stability · Hamiltonian system · Holomorphicmap ·Phase portrait
Introduction
Let us consider a system of ordinary differential equations of the form
x = f(x), (1)
and x ∈ Rn, with f ∈ C1(Rn). The next theorem (Perko, 2001) implies that f ∈ C1
is sufficient for uniqueness and existence of the solutions.
Theorem 1 (Global existence theorem). For f ∈ C1(Rn) and for each x0 ∈ Rn,the initial value problem
x = f(x)
1 + |f(x)| . (2)
has a unique solution x(t) defined for all t ∈ R, i.e., (2) defines a dynamical systemon Rn and, furthermore, (2) is topologically equivalent to (1).
Since this presentation is entirely two-dimensional (n = 2), we also write
x = P(x, y)
y = Q(x, y) (3)
with x ∈ R, y ∈ R, P,Q ∈ C1(R2) instead of (1).The Poincaré-Bendixson theorem (Hirsch et al., 2013) states that the most
complicated limiting behavior of (3) is periodic. It is a strong theorem, since it rulesout the possibilities for chaos. We formulate it as follows.
Theorem 2 (Poincaré-Bendixson). All non-empty compact limit sets of (3) con-taining no equilibria are closed orbits.
A more precise formulation of the above theorem catalogues the possibilities forlimiting behavior that includes fixed points, too; see, e.g., Perko (2001) or Wiggins
Limit Cycles in Planar Systems of Ordinary Differential Equations 3
(2003). There are extensions of this theorem to other two-dimensional manifolds,e.g., the sphere S2. It cannot be extended to all two-dimensional manifolds. Thetorus T 2 (see Wiggins (2003)) provides a counterexample. The reason is that objectshomeomorphic to a circle do not necessarily divide the torus into two pieces. Somehigher-dimensional extensions exist, too. There are, for instance, extensions tothree-dimensional monotone dynamical systems (Smith, 1995) and even to infinitedimensional delay equations of monotone cyclic feedback type; see Mallet-Paretand Sell (1996) and Smith (2011).
One may construct planar systems that give rise to limit cycles in order to provideexamples. Consider, e.g., the system
r = r(1 − r)
θ = 1 (4)
in polar coordinates. It is easy to see that the system has an isolated periodic orbit(a limit cycle) at r = 1, which is stable. Since the radial equation is simple enough,separates, and is logistic, the stability of the limit cycle can indeed be verified byexplicit computation of the Poincaré map
r2π (r0) = r0e2π
1 − r0 + r0e2π
and its associated derivative at r0 = 1,
0 < r ′2π (1) = e−2π < 1. (5)
For later purposes, we note that the divergence of the vector field (4) is given by
div(r(1 − r), 1) = 1
r
∂(r · r(1 − r))
∂r+ 1
r
∂
∂θ1 = 2 − 3r
in polar coordinates. It takes the value −1 along the unit circle. Integrated along theunit circle, it gives the exponent
∫ 2π
0(−1)dt = [−t]2π
0 = −2π.
of the multiplier (5). A method for computing the stability of a given limit cycle is,therefore, to estimate the sign of the divergence integrated along the limit cycle;see, e.g., Grimshaw (1993) and Perko (2001). This technique is directly relatedto the derivative of the Poincaré map, and the corresponding stability theorem isformulated below. We shall return to this method at a later stage.
Theorem 3 (Poincaré criterion about stability). Assume n = 2. Let � be aperiodic solution of (1) with period T . Consider the Poincaré map P locally alonga straight line normal to the periodic solution �, and let its variable s be the signeddistance from � along the straight line so that for small s, s < 0 if we are inside �
4 T. Lindström
and s > 0 if we are outside �. Then P has a fixed point at s = 0 corresponding tothe periodic solution �, and the derivative of the Poincaré map at this fixed point isgiven by
P ′(0) = exp
(∫ T
0∇ · f(�(t))dt
).
We end this section by concluding that similar techniques can be used for theanalysis of all systems of type
x = −y + xf (
√x2 + y2)
y = x + yf (
√x2 + y2)
since they take the form
r = rf (r)
θ = 1 (6)
in polar coordinates. Indeed, we have f (r) = 1 − r in (4) above. It is therefore,possible to arrange for an arbitrary number of stable and unstable limit cycles ofany multiplicity around the origin for such systems. An explicit computation of thePoincaré map might not be possible or might not provide the information needed,but sign analysis of the involved function f still contains the qualitative information.The key problem here is that despite that the class of system that is considered hereprovides complete information of their limit cycles, it cannot usually be used formaking statements of the appearance of limit cycles in systems of any practicalimportance. The class considered is far too special. We proceed to classes of systemsthat are well-known, but do not possess limit cycles. The resulting catalogue ofclasses of systems without limit cycles shows that systems possessing limit cycleshave special properties and require special analysis.
Planar Linear and Linearized Systems
All planar linear systems of ordinary differential equations are linearly equivalentto one of the canonical forms
(x
y
)=
(λ1 00 λ2
) (x
y
), (7)
(x
y
)=
(λ 10 λ
) (x
y
), and (8)
(x
y
)=
(α −β
β α
) (x
y
). (9)
Limit Cycles in Planar Systems of Ordinary Differential Equations 5
The cases correspond to (7) existence of two real eigenvectors, (8) existence of asingle real eigenvector, and (9) existence of two complex eigenvectors. The solutionsof these three cases are given in terms of fundamental matrices by
(x(t)
y(t)
)=
(eλ1t 0
0 eλ2t
)(x0
y0
),
(x(t)
y(t)
)=
(eλt teλt
0 eλt
) (x0
y0
), and
(x(t)
y(t)
)=
(eαt cos βt −eαt sin βt
eαt sin βt eαt cos βt
)(x0
y0
),
respectively. Hence, linear systems cannot produce non-constant periodic solutionsunless the real part of the eigenvalues of the defining matrix is equal to zero. Inthis case all solutions are periodic. There cannot exist any isolated non-constantperiodic solutions in linear systems, so there are no limit cycles. We conclude thatlinearization techniques based on the Hartman-Grobman theorem cannot provideany information about limit cycles either.
First Integral Systems and Gradient Systems
Another interesting category of systems are those possessing a first integral. Thisclass include, for instance, systems that can be transformed onto the Hamiltonianform
x = ∂H
∂y(x, y),
y = −∂H
∂x(x, y), (10)
where H : R2 → R is C2(R2). We now differentiate the continuous functionsinvolved here with respect to time and the solution curves of (10) to get
H = ∂H
∂xx + ∂H
∂yy = ∂H
∂x
∂H
∂y− ∂H
∂y
∂H
∂x= 0.
Thus, the C2 function H involved here is a constant of motion or a first integral. Itis constant along every solution curve. In this context, it is most important to notethat systems possessing first integrals cannot have limit cycles. All exceptions arenon-generic.
Theorem 4. Let H be a first integral of a planar system of ordinary differentialequations. If H is not constant on any open set, then there are no limit cycles.
6 T. Lindström
Proof. If there is a limit cycle �, then H must be constant, say H(�) = c along thisperiodic solution. A limit cycle is an isolated periodic solution, so nearby solutionsmust approach it either forward in time or backward in time. Since H is continuous,H evaluated along those solutions must equal c, too. Thus, H is constant on an openset around �. ��
Example 1. The system
x = −y
y = x
is linear and has the first integral H(x, y) = x2 + y2. It can therefore not have limitcycles by both reasons mentioned so far.
Example 2. The Lotka (1925); Volterra (1926) system
dX
dT= X(Y∗ − Y ),
dY
dT= Y (X − X∗), (11)
X ≥ 0, Y ≥ 0, is not Hamiltonian, but it has the first integral
W(X, Y ) =∫ X
X∗
X′ − X∗X′ dX′ +
∫ Y
Y∗
Y ′ − Y∗Y ′ dY ′.
for X > 0, Y > 0. However, on X > 0, Y > 0, the transformation of theindependent variable
dT
dt= 1
XY, X > 0, Y > 0 (12)
brings it to the Hamiltonian form
X = Y∗ − Y
Y
Y = X − X∗X
with respect to the continuous function W(X, Y ), X > 0, Y > 0. In general, itis difficult to characterize first integral systems that are not Hamiltonian (see, e.g.,Dumortier et al. 2006).
Limit Cycles in Planar Systems of Ordinary Differential Equations 7
We now assume that our system is defined as the negative gradient of a functionH(x) that belongs to C2(Rn). Then
x = −∇H
i.e., the direction of the phase curves are perpendicular to the Hamiltonian flowdefined by H (Perko, 2001). In order make general statements concerning thedynamics of (), we refer to a Lyapunov function argument based on the followingtheorem.
Theorem 5 (LaSalle’s (1960) invariance principle). Consider (1) and let V (x) bea scalar C1 function. Assume that (i) V (x) is positive definite and that (ii) V (x) isnegative semidefinite. Let E be the set of points where V (x) = 0 and let M be thelargest invariant set contained in E. Then every solution of (1) bounded for t ≥ 0approaches M as t → ∞.
We can then use H as the scalar function mentioned above and get
H = −||∇H ||2 ≤ 0.
This means that H decreases along the orbits of a gradient system. Accordingto LaSalle’s invariance principle (LaSalle, 1960), the flow will reach the largestinvariant set where H = 0. But this set is exactly the equilibrium points of (), andwe can revert the time in order to get the unstable equilibria. Therefore, a gradientsystem cannot possess limit cycles.
We end this section by mentioning that recent extensions of Lyapunov’s methodand LaSalle’s invariance principle are available in Michel et al. (2015).
Monotone Dynamics
Another category of systems that have received a lot of attention recently aresystems producing monotone dynamics (Smith, 1995). That is, they preserve someorder relation (for instance, the order relation defined by the positive cone) whenwe follow the orbits either forward in time (cooperative systems) or, when possible,backward in time (competitive systems). In many cases it is difficult to ensure thatsuch an order preserving property actually holds and, therefore, the Kamke (1930,1932), Müller (1927a,b), or quasimonotone (Hadeler and Glas, 1983) conditionsare usually used for ensuring this property. They state that if all the off-diagonalelements of the generic Jacobian are positive, then the given ordinary differentialequation generates a monotone flow.
8 T. Lindström
Example 3. Consider (3) in R2 with, P,Q ∈ C1 and
∂P
∂y> 0 and
∂Q
∂x> 0.
Assume now that x(t0) > 0, y(t0) > 0, and x(t) = 0 for some t ≥ t0. Then
x(t) = ∂P
∂xx + ∂P
∂yy > 0
meaning the x cannot become negative. Similar arguments apply to the other coordi-nates and quadrants. The solutions are either monotone or eventually monotone; seeHofbauer and Sigmund (1998). This means that the solutions tend to either infinityor an equilibrium. The system can therefore not have any attracting limit cycles.
The above argument shows that forward monotone (cooperative) dynamical systemscannot have attracting limit cycles and that backward monotone (competitive)dynamical systems cannot have repelling limit cycles. In our two-dimensional case,it is possible to say more: Systems with monotone dynamics can always be reducedto with one dimension less than the given system. A two-dimensional system withmonotone dynamics is one-dimensional; see, e.g., Smith (1995). Thus, forwardmonotone and backward monotone two-dimensional ordinary differential equationscannot have limit cycles. By a similar argument, the Poincaré-Bendixson theoremholds for three-dimensional monotone systems of ordinary differential equations asstated in Sect. ; see Smith (1995).
Index Theory
Our examples started from the system (4), and in the next step, we proceeded tomore general systems like (6). These systems had just one fixed point at the origin,and it was clear that possible limit cycles had to circumvent that point. If the systemhas several equilibria or more complicated equilibria, it is useful to have somesimple rules for stating whether limit cycles are possible or not. We define the indexof a piecewise, smooth, simple, closed curve (a Jordan curve) relative to a vectorfield; see Perko (2001).
Definition 1. The index IP,Q(γ ) of a Jordan curve γ with respect to the C1 vectorfield (3) on R2, where (3) has no critical point on γ is defined as the integer
IP,Q(γ ) = �
2π
Limit Cycles in Planar Systems of Ordinary Differential Equations 9
where � is the total change in the angle � that the vector (P,Q) makes withrespect to the x-axis, i.e., � is the change in
�(x, y) = arctanQ(x, y)
P (x, y)
as the point (x, y) traverses γ exactly once in positive direction.
It turns out that (1) if there are no fixed points in the interior of γ , then IP,Q(γ ) = 0.(2) If the interior of γ contains just one hyperbolic fixed point, then IP,Q(γ ) = −1if the fixed point is a saddle and IP,Q(γ ) = 1, otherwise. (3) IP,Q(γ ) depends onthe number and types of fixed points in the interior of γ only; we may talk aboutindexes of fixed points rather than Jordan curves. (4) If the interior of γ contains afinite number of fixed points, then IP,Q(γ ) is the sum of the indexes of the fixedpoints that are contained in γ . (5) The index of a limit cycle is 1. (6) If a finite setof fixed points exist, then the sum of their indexes and the index of the infinity pointequals 2 (the Euler characteristic of a sphere).
The fact that the index of a limit cycle is 1 excludes the possibility that limitcycles could surround any constellation of fixed points. In fact, if the limit cyclecontains just hyperbolic fixed points, it contains at least one non-saddle, and if itcontains saddles, it must contain an additional non-saddle for each contained saddle.An explicit example with a limit cycle surrounding one non-saddle is given by (4);an explicit example with a limit cycle surrounding two non-saddles and one saddleis given in Lindström (1993). The next theorem is quite useful for computing theindexes of fixed points (cf. Jordan and Smith 1990).
Theorem 6. Consider the system (3) and traverse Jordan curve γ once in counter-clockwise direction. Let p be the number of changes in Q(x, y)/P (x, y) from +∞to −∞ along γ , and let q be the number of changes in Q(x, y)/P (x, y) from −∞to +∞ along γ . Then IP,Q = 1
2 (p − q).
Example 4. We compute the index of the origin of the system
x = 2xy,
y = 3x2 − y2. (13)
We stretch the theory a bit and use the unit square as a Jordan curve. We beginwith the segment y = 1, −1 ≤ x ≤ 1, and there we have Q/P = (3x2 − 1)/2x.Counterclockwise, we have one transition from −∞ to +∞. A similar transition isfound on the segment y = −1, −1 ≤ x ≤ 1. For the segment x = −1, −1 ≤ y ≤ 1,we have Q/P = (3−y2)/(−2y). Thus, one transition from −∞ to +∞ occurs, anda similar transition occurs counterclockwise for the segment x = 1, −1 ≤ y ≤ 1.Therefore, the index of the origin is −2 for (13).
10 T. Lindström
The Complex Plane
In this section we consider the complex first-order differential equation
z = f (z), z ∈ C, t ∈ R, (14)
where f is analytic in C, except, possibly, at isolated singularities; see Garijo et al.(2007). The family of complex functions that meet these criteria is rather general.Examples are polynomial, rational, entire, and meromorphic functions. Functionswith isolated essential singularities are included. The change of variables z = x+ iy
describes the relation between (14) in the complex plane C and (3) in the real planeR2. Our objective is to elucidate the role of the Cauchy-Riemann equations (cf. Saffand Snider 2003) in the context of limit cycles. Indeed, (14) cannot possess limitcycles.
Theorem 7. Consider (14) and assume that f is analytic in C except at isolatedsingularities. Let � be a periodic orbit with period T in (14). Then all orbits in anopen set containing � are periodic, and they share the period T .
Proof. We first prove that all periodic orbits in a neighborhood of the assumedperiodic orbit inherit its period T . Assume that there is a periodic orbit �ε in aneighborhood of � with period Tε . We have
Tε =∫ Tε
0dt =
∫ Tε
0
z(t)
f (z(t))dt =
∫�ε
1
f (z)dz = T
The second equality holds by (14), and the last equality holds by the residue theorem(Saff and Snider, 2003). Therefore, the periods of possible periodic orbits of (14) aredetermined of sum of residues of 1/f (z) over the zeroes of f surrounded by �.
We continue proving that there are no isolated periodic orbits in (14). Weintroduce the new independent variable
dT
dt= 1
f (z)f (z)
that brings (14) into the conjugate analytic form
z = 1
f (z)(15)
wherever the transformation in question is defined.The conjugate Cauchy-Riemann equations imply now that the vector field is
a divergence free Hamiltonian vector field and that its Hamiltonian is harmonic(Sverdlove, 1979).
Limit Cycles in Planar Systems of Ordinary Differential Equations 11
If f does not have zeroes, it cannot possess limit cycles by index theory, and wecan, thus, assume that the right hand side of (15) has at least one singularity that isnot removable.
Consider now the connected open set (region) in C where the harmonic Hamilto-nian of (15) is defined outside its singularity. It might be multiply defined, so that abranch need to be chosen. Choosing a branch does not remove connectedness. Sincethe singularity is non-removable, the harmonic Hamiltonian cannot be constant ina neighborhood of the singularity. The mean value property for harmonic functions(Conway, 1973) excludes the possibility that it is constant on any open subset of theregion where it was defined. Theorem 4 implies that there are no limit cycles. ��
The above exclusion of isolated periodic orbits based on properties of harmonicfunctions is different from the one in Garijo et al. (2007) that use properties of Liebrackets instead. We continue with a number of simple examples.
Example 5. We return to Example 1. It can be written as z = iz in C, and itscorresponding conjugate analytic system is given by z = −1/iz or
x = −y
x2 + y2,
y = x
x2 + y2 . (16)
The first integral H(x, y) = x2 + y2 given in Example 1 is not harmonic butH∗(x, y) = log(x2 + y2) is. We get
H∗ = 2x · (−y)
(x2 + y2)2+ 2y · x
(x2 + y2)2= 0
along the orbits of (16).
Example 6. We compare the result to the system z = z that has the conjugateanalytic form
x = x
x2 + y2
y = y
x2 + y2 (17)
If choose to place the branch-cut on the ray y < 0, x = 0, a harmonic Hamiltoniantakes in this case the form
H∗(x, y) =⎧⎨⎩
arctan(y/x), x > 0.
π/2, x = 0, y > 0arctan(y/x) + π, x < 0
12 T. Lindström
We see that H∗ = 0 along the orbits of (17) and that the origin is the only singularitythat is not connected to the branch-cut choice. We also note that the system preservesvolume outside a source.
Example 7. As a last example, we take z = 1/z which takes the conjugate analyticform z = z. This system has the harmonic Hamiltonian H∗(xy) = xy and weconclude H∗ = 0 here, too.
We close this section by mentioning that there are possibilities to make statementsabout the number of limit cycles in the real plane by adding suitable perturbationsto systems without limit cycles in the complex plane; see, e.g., Álvarez et al. (2010).
The Existence of Limit Cycles
We have so far listed a number of two-dimensional systems where limit cycles donot occur in general: Linear, first integral, gradient, monotone, and systems definedin the complex plane. We have also mentioned that limit cycles cannot surround allconstellations of fixed points due to index theory. The major theorem for provingexistence of limit cycles is the Poincaré-Bendixson theorem (Theorem 2). Notethat the conclusion of the theorem is that closed orbits exists, not necessarily limitscycles.
Example 8. Consider (6) with
f (r) =⎧⎨⎩
1 − r, 0 ≤ r ≤ 1,
0, 1 ≤ r ≤ 2,
2 − r, r ≥ 2.
We see that the annular region 1 ≤ r ≤ 2 consists of closed orbits that are notisolated. Hence, the system contains no limit cycles. Simultaneously the region 1
2 ≤r ≤ 3 is positively invariant and compact and contains no fixed points.
One standard argument to exclude the possibilities alluded to in the above example isto require analytic right-hand sides. This excludes fixed points from accumulating inrelatively prime analytic systems (Perko, 1987) and limit cycles from accumulatingnear limit cycles (use local Taylor series expansions for the Poincaré map; see, e.g.,Perko 2001). Note that such arguments do not prevent fixed points or limit cyclesfrom accumulating at infinity in systems with analytic right-hand sides (see, e.g.,Zhang (1980)) or near polycycles (cycles consisting of fixed points and homo- orheteroclinic loops connecting them) (see Ilyashenko (1991)). In fact, polynomialsystems are required in order to ensure that the number of limit cycles remain finitein unbounded regions of the real plane; see Ilyashenko (1991). Now, we demonstrate
Limit Cycles in Planar Systems of Ordinary Differential Equations 13
the use of the Poincaré-Bendixson theorem in order to prove the existence of limitcycles with the aid of a standard example.
Example 9. We prove that the system (Perko, 2001)
x = x − y − x3
y = x + y − y3
has at least one limit cycle and start by noting that the isoclines of the system arey = x(1 − x)(1 + x) and x = y(y − 1)(y + 1). They intersect at the originonly, and, therefore, the origin is the unique fixed point of the system. The trace-determinant criterion gives that the fixed point is a hyperbolic unstable non-saddle,and limit cycles are, therefore, not excluded by index theory. The vector field ispolynomial and, hence, analytic. Closed orbits are isolated and hence limit cycles.Next introduce the Lyapunov function V (x, y) = 1
2 (x2 + y2). We get
V = x(x − y − x3) + y(x + y − y3) = x2 − xy − x4 + xy + y2 − y4
= (x2 + y2) − (x4 + y4) > (x2 + y2)2 − (x4 + y4)
= x4 + 2x2y2 + y4 − x4 − y4 = 2x2y2 > 0,
if x2 + y2 < 1. Further, (x2 − y2)2 ≥ 0 implies x4 + y4 ≥ 2x2y2. We get
2(x4 + y4) ≥ (x2 + y2)2 > 2(x2 + y2)
for x2 + y2 > 2. Therefore, V < 0 for x2 + y2 > 2 and the region 1 ≤ x2 + y2 ≤ 2contains at least one limit cycle.
The Liénard (1928) Equation
We have so far dealt with more or less artificially constructed equations. We nowturn our attention to an equation that is strongly connected to applications. The mostwidely used system of equations in this context is the Liénard (1928) equation
x + f (x)x + g(x) = 0. (18)
It models a particle subject to friction and a potential and has, thus, an obviousphysical meaning. There are two ways to describe this equation in the phase plane.One may use either the standard phase plane
x = y,
y = −f (x)y − g(x), (19)
14 T. Lindström
or the Liénard plane given by
x = y − F(x)
y = −g(x) (20)
with F(x) = ∫ x
0 f (x′)dx′. The Liénard plane has a large number of advantages.One of them is that the C1-condition for existence and uniqueness in Theorem 1is posed on F and not on its derivative f . We are going to study (20) under thefollowing general conditions:
(A-I) F ∈ C1(R) and F(0) = 0.(A-II) g ∈ C1(R) and xg(x) > 0, x �= 0 and with G(x) = ∫ x
0 g(u)du, we haveG(−∞) = G(∞) = ∞.
Remark 1. If F is defined as F(x) = ∫ x
0 f (x′)dx′ through the Liénard transforma-tion from (18), then assumption F(0) = 0 is redundant. It is needed only when (20)is considered separately from (18).
We first note that we can arrange some additional symmetry in the system by avariable translation, i.e., we replace the potential with a standard one.
Lemma 1. Assume (A-I)-(A-II). System (20) is topologically equivalent to
ξ ′ = η − F(ξ),
η′ = −ξ.
Proof. The relevant variable transformation is
ξ = sgn(x)√
2G(x), η = y anddt
dτ=
√2G(x)
sgn(x)g(x).
It is a homeomorphism because of (A-II). We get
ξ ′ = dξ
dx
dx
dt
dt
dτ= sgn(x)g(x)√
2G(x)(y − F(x))
√2G(x)
sgn(x)g(x)= η − F(x(ξ))
η′ = dη
dy
dy
dt
dt
dτ= −g(x)
√2G(x)
sgn(x)g(x)= −ξ.
The function x(ξ) above is the inverse transformation of the newly introducedvariable ξ . ��
Limit Cycles in Planar Systems of Ordinary Differential Equations 15
Lemma 1 implies that all results that are proved for the system
x = y − F(x),
y = −x (21)
can be translated to (20), too, provided (A-II) holds.It follows from (A-I) that the origin is the only fixed point of (21). The trace-
determinant criterion gives immediately that it is a non-saddle which is stable iff (0) > 0 and unstable if f (0) < 0. Now we use the Poincaré-Bendixson theoremin order to formulate conditions for closed orbits for the Liénard equation.
Theorem 8 (Dragilev (1952)). Assume
(i) F ∈ C1(R),(ii) xF(x) < 0, x �= 0 in some neighborhood of the origin, and that
(iii) there exists constants X∗ > 0, K−, K+, with K− < K+ and F(x) < K−, forx < −X∗ and F(x) > K+, for x > X∗.
Then the system (20) has at least one closed orbit.
Remark 2. The most recent extension (up to our knowledge) of this theorem isgiven in Cioni and Villari (2015).
Proof. Consider the Lyapunov functional
V (x, y) = 1
2(x2 + y2). (22)
We have V = x(y − F(x)) + y(−x) = −xF(x) > 0, x �= 0 from condition (ii)in some neighborhood of the origin. This implies that the vector field is directedoutward from or along a sufficiently small neighborhood of the origin.
We continue now constructing an outer boundary, such that the vector field (21)is directed inward with respect to that boundary.
We begin with the region x < −X∗ and use the Lyapunov function
V−(x, y) = 1
2x2 + 1
2(y − K−)2
here. We get
V− = x(y − F(x)) + (y − K−)(−x) = −x(F (x) − K−) < 0.
Similarly, we have in the region x > X∗ with
V+(x, y) = 1
2x2 + 1
2(y − K+)2,
V+ = x(y − F(x)) + (y − K+)(−x) = −x(F (x) − K+) < 0.
16 T. Lindström
We proceed to the region |x| < X∗ and consider the level curves
V0(x, y) = |−(K+ − K−)x + 2X∗y| .
We have for
y > y+ = max
(max|x|≤X∗
F(x) + 2X2∗K+ − K−
,K+ − K−
2
)
that
V0 = −(K+ − K−)(y − F(x)) + 2X∗(−x)
< −(K+ − K−)2X2∗
K+ − K−+ 2X2∗ = −2X2∗ + 2X2∗ = 0.
Similarly, for
y < y− = min
(min|x|≤X∗
F(x) − 2X2∗K+ − K−
,−K+ − K−2
)
we get that
V0 = (K+ − K−)(y − F(x)) − 2X∗(−x)
< −(K+ − K−)2X2∗
K+ − K−+ 2X2∗ = −2X2∗ + 2X2∗ = 0.
For the regions (A) |x| ≤ X∗ with |y| > max(y+, |y−|) and (B) |x| > X∗,it is therefore, possible to construct a trapping region for the solutions provingthat at least one closed orbit exists according to the Poincaré-Bendixson theorem(Theorem 2). ��
We conclude that it is of course, possible to state additional conditions ensuringthat possible closed orbits are isolated. For instance, F analytic is sufficient. Later,we shall deal with uniqueness of limit cycles for (21). If the periodic orbit is unique,then it is isolated and hence, a limit cycle (Fig. 1).
Theorems for Absence of Limit Cycles
The Poincaré-Bendixson Theorem can be used to prove that closed orbits exist. Wehave seen that it can be combined with the fact that the vector field may be analyticor other facts excluding degenerate cases in some relevant part of the phase spacein order to conclude that limit cycles exist, too. We have already proved that a largenumber of quite general systems do not exhibit limit cycles at all.
Limit Cycles in Planar Systems of Ordinary Differential Equations 17
-20 -15 -10 -5 0 5 10 15 20
-5
0
5
10
15
20
25
30
x
y
V0(x, y) = constantV−(x, y) = constant
y = F (x)
V+0(x, y) = constant
x = X∗x = −X∗
y = F (x)
y = K+y = K−
V0(x, y) = constant
V−(x, y) = constant
Fig. 1 The trapping region for the system (21) together with the function F and one closed orbit
However, use of the Poincaré-Bendixson theorem to conclude that limit cyclesexist is the easy part in this field. It can, thus, be used for providing lower boundsfor the number of limit cycles, but it does not provide any answers regarding upperbounds for the number of limit cycles. It does not limit the complexity of planardynamical in any other way than stating that chaotic dynamics do not exist. Thegeneral problem regarding upper bounds for the number of limit cycles is relatedto Hilbert’s 16th problem and is still open. The finiteness theorem for polynomialvector fields in the real plane (Ilyashenko, 1991) and the connection to trophicalgeometry (Viro, 2008) are probably the most interesting achievements so far relatedto Hilbert’s 16th problem (Ilyashenko, 2002).
The simplest theorems giving upper bounds for the number of limit cycles aretheorems for absence of limit cycles. If limit cycles are not excluded by the typeof system considered (it is not linear, first integral, monotone, etc.), then there aremainly two arguments that can be used. Both of them are non-trivial to use andrequire construction of certain user-defined functions. I divide these arguments in
18 T. Lindström
either divergence-based arguments (cf. Theorem 3) or Lyapunov function basedarguments (cf. Theorem 5). The divergence-based argument is in this case calledDulac’s theorem.
Theorem 9 (Dulac’s theorem). Consider (3) and assume that ρ(x, y) ∈ C1(R2).There are no closed orbits in a simply connected domain on which
∂(ρ(x, y)P (x, y))
∂x+ ∂(ρ(x, y)Q(x, y))
∂y(23)
is of one sign.
Proof. This theorem is a consequence of Green’s theorem. We might start byassuming that (23) is positive in a simply connected region R. Let � be a closedtrajectory for (3) in R and let D be the interior of �. We use Green’s theorem in thefirst equality below and get
0 <
∫ ∫D
(∂(ρ(x, y)P (x, y))
∂x+ ∂(ρ(x, y)Q(x, y))
∂y
)dxdy
=∮
�
(−ρ(x, y)Q(x, y)dx + ρ(x, y)P (x, y)dy)
=∮
�
ρ(x, y) (−ydx + xdy) =∫
�
ρ(x, y) (−yx + xy) dt = 0.
The derived conclusion 0 < 0 is a contradiction. We can now repeat argumentsassuming (23) negative in a simply connected region as well. ��
We return to (21). The divergence of the vector field is −f (x), so if f is negative orpositive, then (21) has no closed orbits. In this case it suffices to use ρ(x, y) = 1 inorder arrive in the conclusion. We can, therefore, state the following theorem.
Theorem 10. Assume (A-I). If f (x) > (<)0, ∀x ∈ R, then (21) has no limit cycles.
The standard Lyapunov function argument is LaSalle’s (1960) invariance principle,Theorem 5. Then, we get the following theorem.
Theorem 11. Consider (21) and assume
(i) F ∈ C1(R)
(ii) xF(x) > 0, x �= 0
then the system (21) has no limit cycles.
Limit Cycles in Planar Systems of Ordinary Differential Equations 19
Proof. Consider the functional (22). It is scalar and C1. It is positive definite, too.Now, we get
V = x(y − F(x)) + y(−x) = −xF(x) (24)
which is negative semidefinite by assumption (ii). It follows that all boundedsolutions approach the set x = 0, and there the only invariant part of this set isthe origin. Hence, all bounded orbits approach the origin and no closed orbits exist.��
Note that there are conditions for which both Dulac’s and LaSalle’s invarianceprinciple apply but that there are exist cases for which just one of these criteriaapply in order to exclude closed orbits. The construction of the Lyapunov functionalV above or a Dulac function ρ can be a delicate mathematical problem if no naturalor obvious choices exist.
Uniqueness of Limit Cycles
When limit cycles cannot be excluded, the next step is to prove that they exist,and we have already concluded that Poincaré-Bendixson’s theorem combined withsome argument excluding degenerate cases suffices for this. However, estimating thenumber of limit cycles from above was an acknowledged mathematical problem.The only theory that describes the evolution of limit cycles and limit cyclebifurcations in a precise manner is the theory of general rotated vector fields; see,e.g., Duff (1953), Ye et al. (1986), and Perko (1993). The difficulty in applying thistheory is finding suitable rotation parameters for systems occurring in applications.In the end of the proof of Theorem 12, we give an example of how this theory canbe applied.
Also in this case the so far most efficient methods are either divergence basedor Lyapunov functional based. In both cases we assume that two limit cycles existand compare either the divergence integrated over these cycles or the changes of thevalue of the Lyapunov functional when integrated over the assumed cycles. Alsohere, the Liénard system serves as a good example. It is simple enough in orderto generate strong conclusions, and yet it is an example that is clearly based onreal-world applications. We first present the divergence-based method.
Theorem 12 (Zhang (1986)). Consider (21) and assume
(i) F ∈ C1(R)
(ii) xF(x) < 0, x �= 0 in some neighborhood of the origin(iii) f/id is non-decreasing on R− and R+
then the system (21) has at most one limit cycle, and if it exists, it is stable.
20 T. Lindström
Remark 3. Cherkas and Zhilevich (1970) proved a theorem that at the first glanceseems more general and easier to prove than the above one. The key problemwith their formulation is one of the conditions of their theorem. Symmetry is notassumed, and it is not obvious how to verify that all cycles encircle the prescribedinterval on the x-axis before using their theorem. In the proof of Theorem 13 later,we give an argument that could be used in order to remedy this problem.
Remark 4. When sufficient smoothness is added, xF(x) < 0, x �= 0 impliesf (x) < 0 in some neighborhood of the origin.
Proof. We consider (22) and conclude that the origin is an unstable non-saddle ifxF(x) < 0, x �= 0 in some neighborhood of the origin. Index theory gives thatall limit cycles must encircle the origin. We assume that two limit cycles existand baptize them as �i , i = 1, 2. Let their parameter description be given by(xi(t), yi(t)), i = 1, 2. We assume that �1 is the one that is closest to the origin.Then limit cycle �1 must be stable from the inside meaning that
∫�1
f (x1(t))dt ≥ 0. (25)
according to Theorem 3. The next step is to prove∫
�2
f (x2(t))dt >
∫�1
f (x1(t))dt (26)
meaning that �2 is stable. This is a contradiction if we can exclude the possibility forsemistable limit cycles. Our plan to prove the uniqueness of limit cycles, by provinginequality (26) first and then by removing the possibility of semistable limit cycles.We first circumvent the problem that we addressed in the theorem of Cherkas andZhilevich (1970). By continuity, the interior limit cycle �1 must intersect the graphof the isocline y = F(x) at exactly two points; we call them Q1 and P1. We assumethat their horizontal coordinates are xQ1 < 0 and xP1 > 0 and construct a newfunction defined as
f∗(x) = f (x) − f (xQ1)
xQ1
x.
We note that f∗ has the following properties: First, f∗(x) = f (0) ≤ 0 follows fromcontinuity and xF(x) negative definite in some neighborhood of the origin. Second,
f∗(x)
x= f (x)
x− f (xQ1)
xQ1
is non-decreasing on R− and R+. Third, f∗(x)(x −xQ1) negative definite for x < 0since it can be reorganized as
f∗(x)(x − xQ1) = xf∗(x)
x(x − xQ1).
Limit Cycles in Planar Systems of Ordinary Differential Equations 21
We already knew that f∗/id was non-decreasing on R− and that it was constructedin order to have a zero at xQ1 . Similarly, we note that
f∗(x)(x − xN) = xf∗(x)
x(x − xN)
is positive definite on x > 0 for exactly one point N with xN > 0. We have xN <
xP , and if this does not happen, then we must have
∫�1
f∗(x1(t))dt =∫
�1
f (x1(t))dt − f (xQ)
xQ
∫�1
x1(t)dt (27)
=∫
�1
f (x1(t))dt + f (xQ)
xQ
∫�1
dy1(t) =∫
�1
f (x1(t))dt < 0
contradicting (25). From the second equality of (27), we conclude that
∫�i
f∗(xi(t))dt =∫
�i
f (xi(t))dt, i = 1, 2,
too. We have now circumvented the problem with the formulation of the Cherkasand Zhilevich (1970) theorem and have derived sufficiently precise statementsregarding the position of the limit cycle with respect to the zeros of the constructedfunction f∗ from the conditions stated in the theorem. These statements can now beused for distinguishing between negative and positive contributions in the Poincarécriterion about stability (Theorem 3). Now consider Fig. 2. We start from the arcsQ1A1 and F2A2. Along these arcs, we have x > 0 meaning that we can representthese arcs as functions yi(x), i = 1, 2, xQ1 < x < xN . We then get
∫F2A2
f∗(x2(t))dt −∫Q1A1
f∗(x1(t))d =∫ xN
xQ1
f∗(x)
y2(x) − F(x)dx −
∫ xN
xQ1
f∗(x)
y1(x) − F(x)dx =
∫ xN
xQ1
f∗(x)(y1(x) − y2(x))
(y2(x) − F(x))(y1(x) − F(x))dx > 0.
A similar computation verifies
∫C2E2
f∗(x2(t))dt −∫C1Q1
f∗(x1(t))dt =
−∫ xN
xQ1
f∗(x)(y1(x) − y2(x))
(y2(x) − F(x))(y1(x) − F(x))dx > 0,
22 T. Lindström
-4 -3 -2 -1 0 1 2 3-6
-4
-2
0
2
4
6
8
10
N
Q1
Q2
P1
P2A1
B2
A2
C1
D2
C2
E2
F2
y
x
y = f∗(x)
y = f(x)
y = F (x)
Fig. 2 Two limit cycles in the system (21) together with the functions F , f , and f∗
too. We then consider the arcs B2D2 and A1C1. These arcs can be represented asfunctions xi(y), i = 1, 2. We get
∫B2D2
f∗(x2(t))dt −∫A1C1
f∗(x1(t))dt =∫ yA1
yC1
(f∗(x2(y))
x2(y)− f∗(x1(y))
x1(y)
)dy ≥ 0
since xN > 0 and f∗/id was non-decreasing on R+. We also have
∫A2B2
f∗(x2(t))dt ≥ 0,
∫D2C2
f∗(x2(t))dt ≥ 0, and∫E2F2
f∗(x2(t))dt ≥ 0,
so (26) holds.
Limit Cycles in Planar Systems of Ordinary Differential Equations 23
The last step is to exclude the possibility for semistable limit cycles, and we usethe theory of rotated vector fields (Duff, 1953; Perko, 1993) to do so. First we takea line x = x� > 0 intersecting �1 and construct a new system
x = y − Fε(x),
y = −x, (28)
ε > 0, where
Fε(x) ={
F(x), x < x�,
F (x) + ε(x − x�)2, x ≥ x�.
Since
fε(x) ={
f (x), x < x�,
f (x) + 2ε(x − x�), x ≥ x�.
is continuous and (x − x�)/x is strictly increasing; (28) satisfies all conditions ofTheorem 12. We have now that
∣∣∣∣ y − Fε1(x) x
y − Fε2(x) x
∣∣∣∣ = x(Fε2(x) − Fε1(x)) ={
0, x < x�,
x(ε2 − ε1)2, x ≥ x�
has fixed sign with respect to ε2 > ε1 > 0 meaning that (28) is a family of generalrotated vector fields. If �1 is semistable for ε = 0 and stable from the inside, itwill split into at least two limit cycles as we get 0 < ε << 1. We baptize the mostinterior of these limit cycles as �11 and the most exterior of these limit cycles as �12 .We have that �12 encloses �11 and that �12 is unstable from the outside, whereas�11 is stable from the inside. The Poincaré criterion about stability (Theorem 3)gives
∫�11
f (x11(t))dt ≥ 0 and∫
�12
f (x12(t))dt ≤ 0
which is impossible by (26). The theorem is proved. ��
We proceed to the Lyapunov functional-based method and formulate the follow-ing theorem. We note that the conditions for uniqueness are quite different.
Theorem 13 (Sansone (1949)). Assume
(i) F ∈ C1(R)
(ii) There exists an X∗ such that F is nondecreasing for |x| > X∗ and for x �={−X∗, 0, X∗} we have F(x)x(x − X∗)(x + X∗) > 0.
Then the system (21) has a unique limit cycle, which is stable.
24 T. Lindström
Remark 5. This theorem requires some symmetry regarding the location of ±X∗.Recent work that address this problem are available; see, e.g., Hayashi et al. (2018).
Proof. Consider (21). Theorem 8 implies that at least one limit cycle exists. Theinnermost cycle is stable from the inside, and the outermost cycle is stable from theoutside.
The derivative of (22) with respect to (21) is given by (24). Therefore,
dV
dy= dV
dt
dt
dy= −xF(x)
−x= F(x).
meaning that dV = F(x)dy. We commence by estimating the location of the limitcycle by proving that its leftmost point must take a smaller value than −X∗ and thatits rightmost point must take larger value than X∗, i.e., all limit cycles encircle theinterval [−X∗, X∗] on the x-axis. We first assume that no part of the limit cycle � isoutside the region |x| ≤ X∗. Moving along the cycle when x > 0, we have dy < 0and F(x) < 0, i.e., dV > 0. Similarly for x < 0, we get dV > 0. Therefore,
∮�
dV > 0.
However, V should return to its original value when moving around any closedcurve once. This contradicts the existence of a limit cycle that does not possess anypart outside the strip −X∗ < x < X∗. Next suppose that the leftmost point Q ison the right of x = −X∗ and that the rightmost point is on the right of x = X∗.Then � intersects x = X∗ at a point P below the horizontal axis, and we must haveOP > X∗. If we move along � from P , then dV > 0 but we have OQ < X∗. Asimilar argument excludes the possibility that the leftmost point of the limit cycle ison left of x = −X∗ and the rightmost point is on the left of x = X∗. We concludethat all limit cycles of (21) must intersect x = ±X∗ at four points.
We now assume that two limit cycles �i , i = 1, 2 exist and label their fourintersection points with x = ±X∗ as Ai , Ci , Ei , and Gi , i = 1, 2; see Fig. 3. Theidea is now to compare two integrals that are supposed to be zero, i.e.,
∮�1
dV and∮
�2
dV
and show that both of them cannot equate to zero since they are different. First wehave
∫B2D2
dV =∫B2D2
F(x)dy ≤∫A1C1
F(x)dy =∫A1C1
dV
Limit Cycles in Planar Systems of Ordinary Differential Equations 25
-1 -0.5 0 0.5 1 1.5
-3
-2
-1
0
1
2
3
4y
x
A1
A2
B2
C1
C2
D2
E1
E2
F2
G1
G2
H2
Fig. 3 Two limit cycles in the system (21) together with F
since F is positive and non-decreasing on x > X∗. Similarly, we have
∫F2H2
dV =∫F2H2
F(x)dy ≤∫E1G1
F(x)dy =∫E1G1
dV
since F is negative and non-decreasing on x < −X∗. Next, we have
∫G2A2
dV =∫G2A2
−xF(x)
y − F(x)dx <
∫G1A1
−xF(x)
y − F(x)dx =
∫G1A1
dV
and
∫C2E2
dV =∫C2E2
−xF(x)
y − F(x)dx <
∫C1E1
−xF(x)
y − F(x)dx =
∫C1E1
dV,
26 T. Lindström
that together with
∫H2G2
dV < 0,
∫A2B2
dV < 0,
∫D2C2
dV < 0,
∫E2F2
dV < 0,
implies
∮�2
dV <
∮�1
dV.
This means that �2 and �1 cannot simultaneously exist. The remaining limit cycleis unique and stable from both sides and, thus, stable. ��
Summary
In this chapter we have considered a quite special nonlinear phenomenon: limitcycles or isolated periodic orbits. Such phenomena are not encountered in manyclasses of systems that in general are considered as well-understood. Completequalitative analysis of systems possessing limit cycles with clear connections to realworld problems require thus, in many cases, precise estimates and well-selectedmethods.
Our last remark is that many of the methods that we have treated above apply tothe generalized Liénard equation
x = φ(y) − F(x),
y = −g(x), (29)
too. The usual conditions set on the involved functions are (A-I), (A-II), and
(A-III) φ ∈ C1(R) and yφ(y) > 0, y �= 0 and φ nondecreasing with φ(±∞) =±∞.
The function φ cannot be removed by a transformation similar to the one thatremoved g in Lemma 1. A natural Lyapunov function still exists and is given by
V (x, y) =∫ x
0g(u)du +
∫ y
0φ(v)dv.
This makes it harder to use the various symmetry arguments above. This gener-alization is necessary for translating results for the generalized Liénard equationinto a biological context; see, e.g., Kuang and Freedman (1988) and Xiao andZhang (2003), Lindstrom (2019), Mathematics and recurrent population outbreaks,Springer, 2019.
Limit Cycles in Planar Systems of Ordinary Differential Equations 27
References
Álvarez MJ, Gasull A, Prohens R (2010) Topological classification of polynomial complexdifferential equations with all critical points of centre type. J Differ Equ Appl 16(5–6):411–423
Cherkas LA, Zhilevich LI (1970) Some criteria for the absence of limit cycles and for the existenceof a single cycle. Differ Equ 6:891–897
Cioni M, Villari G (2015) An extension of Dragilev’s theorem for the existence of periodicsolutions of the Liénard equation. Nonlinear Anal 127:55–70
Conway JB (1973) Functions of one complex variable. Springer, New York/Heidelberg/BerlinDragilev AV (1952) Periodic solutions of the differential equation of the differential equation of
nonlinear oscillations. Acad Nauk SSSR Prikl Mat Meh 16:85–88Duff GFD (1953) Limit cycles and rotated vector fields. Ann Math 57(1):15–31Dumortier F, Llibre J, Artés JC (2006) Qualitative theory of planar differential systems. Springer,
Berlin/HeidelbergGarijo A, Gasull A, Jarque Z (2007) Local and global phase portrait of equation z = f (z). Discret
Contin Dyn Syst 17(2):309–329Grimshaw R (1993) Nonlinear ordinary differential equations. CRC Press, Boca RatonHadeler KP, Glas D (1983) Quasimonotone systems and convergence to equilibrium in a population
generic model. J Math Anal Appl 95:297–303Hayashi M, Villari G, Zanolin F (2018) On the uniqueness of limit cycle for certain Liénard
systems without symmetry. Electron J Qual Theory Differ Equ 55:1–10. http://www.math.u-szeged.hu/ejqtde
Hirsch MW, Smale S, Devaney RL (2013) Differential equations, dynamical systems, and anintroduction to chaos. Academic, Oxford
Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge Univer-sity Press, Cambridge
Ilyashenko Y (1991) Finiteness theorems for limit cycles. Translations of mathematical mono-graphs, vol 94. American Mathematical Society, Providence
Ilyashenko Y (2002) Centennial history of Hilbert’s 16th problem. Bull Am Math Soc 39(3):301–354
Jordan DW, Smith P (1990) Nonlinear ordinary differential equations, 2nd edn. Clarendon Press,Oxford
Kamke E (1930) Über die eindeutige Bestimmtheit der Integrale von Differentialgleichungen. MatZ 1:101–107
Kamke E (1932) Zur Theorie der Systeme gewöhnlicher Differentialgleichungen II. Acta Math58:57–85
Kuang Y, Freedman HI (1988) Uniqueness of limit cycles in Gause-type models of predator-preysystems. Math Biosci 88:67–84
LaSalle JP (1960) Some extensions of Lyapunovs second method. IRE Trans Circuit Theory CT–7:520–527
Liénard A (1928) Étude des oscillations entretenues. Revue Gén Électr 23:906–924Lindström T (1993) Qualitative analysis of a predator-prey system with limit cycles. J Math Biol
31:541–561Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, BaltimoreMallet-Paret J, Sell GR (1996) The Poincaré-Bendixson theorem for monotone cyclic feedback
systems with delay. J Differ Equ 125:441–489Michel AM, Hou L, Liu D (2015) Stability of dynamical systems, 2nd edn. Birkhäuser, ChamMüller M (1927a) Über das fundamentaltheorem in der theorie der gewöhnlichen differentialgle-
ichungen. Mat Z 26(1):619–645Müller M (1927b) Über die Eindeutigkeit der Integrale eines systems gewöhnlicher Differential-
gleihungen und die Konvergenz einer Gattung von Verfahrenzur Approximationdieser Integrale.Sitzungsber Heidelb Akad Wiss Math-Natur Kl 9, 2–38
Perko L (1987) On the accumulation of limit cycles. Proc Am Math Soc 99:515–526
28 T. Lindström
Perko LM (1993) Rotated vector fields. J Differ Equ 103:127–145Perko L (2001) Differential equations and dynamical systems. Springer, New YorkSaff EB, Snider AD (2003) Fundamentals of complex analysis with applications to engineering
and science, 3rd edn. Pearson Education, Upper Saddle RiverSansone G (1949) Sopra l’equazione di A. Liénard delle oscillazioni di rilassimento. Ann Mat Pura
Appl Ser IV 28:153–181Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and
cooperative systems. American Mathematical Society, ProvidenceSmith HL (2011) An introduction to delay differential equations with applications to life sciences.
Springer, New YorkSverdlove R (1979) Vector fields defined by complex functions. J Differ Equ 34:427–439Viro O (2008) From the sixteenth Hilbert problem to tropical geometry. Jpn J Math 3:185–214Volterra V (1926) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi.
Memorie della R. Accademia Nationale dei Lincei 6 2:31–113Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos, 2nd edn.
Springer, New YorkXiao D-M, Zhang Z-F (2003) On the uniqueness and nonexistence of limit cycles for predator-prey
systems. Nonlinearity 16:1185–1201Ye Y-Q et al (1986) Theory of limit cycles, 2nd edn. American Mathematical Society, ProvidenceZhang Z-F (1980) Theorem of existence of exact n limit cycles in |x| ≤ n for the differential
equation x + μ sin x + x = 0. Sci Sin 23(12):1502–1510Zhang Z-F (1986) Proof of the uniqueness theorem of limit cycles of generalized Liénard
equations. Appl Anal 29:63–76
