Stability region bifurcations of nonlinear autonomous dynamical systems: Type-zero saddle-node bifurcations

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control 2011; 21:591–612Published online 11 June 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1605

Stability region bifurcations of nonlinear autonomous dynamicalsystems: Type-zero saddle-node bifurcations

F. M. Amaral1,2 and L. F. C. Alberto1,∗,†

1Sao Carlos Engineering School, University of Sao Paulo, Sao Carlos, 13566-590, Brazil2Education Department, Federal Institute of Bahia, Eunapolis, 45822-000, Brazil

SUMMARY

The behavior of stability regions of nonlinear autonomous dynamical systems subjected to parametervariation is studied in this paper. In particular, the behavior of stability regions and stability boundarieswhen the system undergoes a type-zero sadle-node bifurcation on the stability boundary is investigated inthis paper. It is shown that the stability regions suffer drastic changes with parameter variation if type-zerosaddle-node bifurcations occur on the stability boundary. A complete characterization of these changesin the neighborhood of a type-zero saddle-node bifurcation value is presented in this paper. Copyright �2010 John Wiley & Sons, Ltd.

Received 17 June 2009; Revised 29 January 2010; Accepted 14 April 2010

KEY WORDS: stability region; basin of attraction; stability boundary; saddle-node bifurcation; Hopfieldartificial neural network

1. INTRODUCTION

The problem of determining stability regions (basins of attraction) of nonlinear dynamical systemsis of fundamental importance for many applications in engineering and the sciences [1–6] andcontinues to play an important role in many emerging research areas such as electrical powersystem applications [7, 8]. The characterization of the stability boundary (the boundary of thestability region) plays an important role in applications because it provides insight into how toobtain optimal estimates of the stability region [9]. Other approaches to estimate stability regionscan also be found in [10–14].

Comprehensive characterizations of the stability boundary of classes of nonlinear dynamicalsystems can be found, for example in [15–17]. The existing characterizations of stability boundariesare proved under the key assumption that all the equilibrium points on the stability boundaryare hyperbolic. Hyperbolicity of equilibrium points is a generic property of nonlinear dynamicalsystems, that is, almost all dynamical systems satisfy it and, as a consequence, the existingcharacterizations of stability boundary are valid almost always.

In this paper, however, we are interested in the study of stability boundaries when a systemis subject to parameter variation. Under parameter variation, local bifurcations may occur onthe stability boundary and the assumption of hyperbolicity of equilibrium points are violated at

∗Correspondence to: L. F. C. Alberto, Electrical Engineering Department, Sao Carlos Engineering School, Universityof Sao Paulo, Av. Trabalhador Sao Carlense 400, Sao Carlos, SP 13566-590, Brazil.

†E-mail: [email protected]

Contract/grant sponsor: Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico (CNPQ)

Copyright � 2010 John Wiley & Sons, Ltd.

592 F. M. AMARAL AND L. F. C. ALBERTO

bifurcation points. Therefore, studying the characterization of the stability boundary at bifurcationpoints is of fundamental importance to understand how stability region behaves under parametervariation.

The behavior of stability regions and stability boundaries when the system undergoes a type-zero saddle-node bifurcation on the stability boundary is studied in this paper. Necessary andsufficient conditions for a type-zero saddle-node equilibrium point lying on the stability boundaryare presented. A complete characterization of the stability boundary is presented if the systempossesses a type-zero saddle-node equilibrium point on the stability boundary. Exploring thischaracterization, results that describe the behavior of the stability boundary in the neighborhoodof a saddle-node bifurcation value are developed. In particular, it is shown that stability regionssuffer drastic changes with parameter variation if type-zero saddle-node bifurcations occur on thestability boundary. A complete characterization of these changes in the neighborhood of a type-zerosaddle-node bifurcation value is presented in this paper.

This paper is organized as follows. Some preliminary concepts about dynamical systems andsaddle-node bifurcation theory are introduced in Section 2. In Section 3, an overview of the char-acterization of the stability boundary of autonomous dynamical systems is presented, includingthe persistence of the characterization under hyperbolicity and transversality assumptions. Neces-sary and sufficient conditions for a type-zero saddle-node equilibrium point lying on the stabilityboundary as well as a complete characterization of the stability boundary in the presence of atype-zero saddle-node equilibrium point are presented in Section 4. In Section 5, results thatdescribe the behavior of the stability boundary in the neighborhood of a type-zero saddle-nodebifurcation value are developed. Section 6 is dedicated to examples that illustrate the proposedresults.

2. PRELIMINARIES

In this section, some classical concepts of the theory of dynamical systems and saddle-nodebifurcations are reviewed. More details on the content explored in this section can be found in[18–22].

2.1. Dynamical systems

Consider the nonlinear autonomous dynamical system

x = f (x) (1)

where x ∈Rn . One assumes that f :Rn −→Rn is a vector field of class Cr with r�1, a sufficientcondition for ensuring the existence and uniqueness of solutions of (1). The solution of (1) startingat x at time t =0 is denoted by �(t, x). The map t →�(t, x) defines in Rn a curve passing throughx at t =0 that is called the trajectory or orbit of (1) through x . A set S ∈Rn is said to be aninvariant set of (1) if every trajectory of (1) starting in S remains in S for all t .

A point x∗ ∈Rn is an equilibrium point of (1) if f (x∗)=0. For an equilibrium point x∗, considerthe linear system

z = J (x∗)z (2)

where z := x −x∗ and J (x∗) is the Jacobian matrix of f (x) calculated at the equilibrium point x∗.An equilibrium point x∗ of (1) is said to be hyperbolic if none of the eigenvalues of the Jacobian

matrix J (x∗) has real part equal to zero. Moreover, a hyperbolic equilibrium point x∗ is of type k ifthe Jacobian matrix J (x∗) possesses k eigenvalues with positive real part and n−k eigenvalues withnegative real part. A hyperbolic asymptotically stable equilibrium point is a type-zero hyperbolicequilibrium point.

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 2011; 21:591–612DOI: 10.1002/rnc

STABILITY REGION BIFURCATIONS 593

Given an equilibrium point x∗ of (1), the space Rn can be decomposed as a direct sum of threesubspaces denoted by E s, Eu and Ec which are invariant with respect to (2)

E s = span {e1, . . . ,es}Eu = span {es+1, . . . ,es+u}Ec = span {es+u+1, . . . ,es+u+c}, s+u+c=n

where {e1, . . . ,es}, {es+1, . . . ,es+u} and {es+u+1, . . . ,es+u+c} are the generalized eigenvectors ofJ (x∗), respectively, associated with the eigenvalues of J (x∗) that have negative, positive and zeroreal part. Subspaces E s, Eu and Ec are, respectively, called stable, unstable and center subspaces.If x∗ is an equilibrium point of (1), then there are local manifolds W s

loc(x∗), W csloc(x∗), W c

loc(x∗),W u

loc(x∗) and W culoc(x∗) of class Cr , invariant with respect to (1) [23, Appendix C], [24]. These

manifolds are tangent to E s, Ec ⊕ E s, Ec, Eu and Ec ⊕ Eu at x∗, respectively, and are, respectively,called local stable, stable center, center, unstable and unstable center manifolds. The local stableand unstable manifolds are unique, but the local stable center, center and unstable center manifoldsmay not be.

The idea of transversality is basic in the study of dynamical systems. The transversal intersectionis notorious because it persists under perturbations of the vector field [25]. The manifolds M andN of class Cr , with r�1, in Rn , satisfy the transversality condition if either (i) the tangent spacesof M and N span the tangent space of Rn at every point x of the intersection M ∩ N ,

i.e. Tx (M)+Tx (N )=Tx (Rn) for all x ∈ M ∩ N

or (ii) they do not intersect at all.

2.2. Saddle-node bifurcation

Consider the nonlinear dynamical system

x = f (x,�) (3)

with x ∈Rn , depending on the parameter �∈R. Let f :Rn ×R−→Rn be a vector field of classCr , with r�2. For each fixed �, one defines the vector field f� = f (·,�) and ��(t, x) denotes thesolution of x = f�(x) passing through x at time t =0.

Definition 2.1 (Saddle-node equilibrium point)A non-hyperbolic equilibrium point x�0 ∈Rn of (3), for a fixed parameter �=�0, is called a saddle-node equilibrium point and (x�0,�0) a saddle-node bifurcation point if the following conditionsare satisfied:

(SN1) Dx f�0 (x�0 ) has a single simple eigenvalue equal to 0 with v as an eigenvector to theright and w to the left.

(SN2) w((� f�/��)(x�0,�0)) �=0.(SN3) w(D2

x f�0 (x�0 )(v,v)) �=0.

A saddle-node equilibrium point or a saddle-node bifurcation point can be classified into typesaccording to the number of eigenvalues of Dx f�0 (x�0 ) with positive real part.

Definition 2.2 (Saddle-node bifurcation type)A saddle-node equilibrium point x�0 of (3), for a fixed parameter �=�0, is called a type k saddle-node equilibrium point and (x�0,�0) a type k saddle-node bifurcation point if Dx f�0 (x�0 ) hask eigenvalues with positive real part and n−k−1 with negative real part.

Remark 2.1The parameter value �0 of Definition 2.2 is called a type k saddle-node bifurcation value.



Figure 1. The local manifolds W c+loc,�0

(x�0 ) and W sloc,�0

(x�0 ) are unique, whereas there are infinite choices

for W c−loc,�0

(x�0 ). Three possible choices for W c−loc,�0

(x�0 ) are indicated in this figure.

In this paper, we will be mainly interested in type-zero saddle-node bifurcations. If x�0 is atype-zero saddle-node equilibrium point, then the following properties hold [19]:

(1) The unidimensional local center manifold W cloc,�0

(x�0 ) of x�0 can be split into three invariantsubmanifolds:

W cloc,�0

(x�0 )=W c−loc,�0

(x�0 )∪{x�0}∪W c+loc,�0

(x�0 )

Figure 1, illustrates the division of W cloc,�0

(x�0 ). If p∈W c−loc,�0

(x�0 ) then ��0(t, p)−→ x�0 as

t −→+∞ and p∈W c+loc,�0

(x�0 ) then ��0(t, p)−→ x�0 as t −→−∞. Moreover, W c+

loc,�0(x�0 )

is unique while W c−loc,�0

(x�0 ) is not.(2) The (n−1)-dimensional local stable manifold W s

loc,�0(x�0 ) of x�0 exists and is unique.

Moreover if p∈W sloc,�0

(x�0 ) then ��0(t, p)−→ x�0 as t −→+∞.

The global center and stable manifolds of the type-zero saddle-node equilibrium point x�0 aredefined extending the local manifolds through the flow, i.e.

W s�0

(x�0 ) := ⋃t�0

��0(t,W s

loc,�0(x�0 ))

W c�0

(x�0 ) := W c−�0

(x�0 )∪{x�0}∪W c+�0

(x�0 )

where

W c−�0

(x�0 ) := ⋃t�0

��0(t,W c−

loc,�0(x�0 ))

and

W c+�0

(x�0 ) := ⋃t�0

��0(t,W c+

loc,�0(x�0 ))

Obviously, if p∈W s�0

(x�0 ), then ��0(t, p)−→ x�0 as t −→+∞, while p∈W c−

�0(x�0 ) implies

��0(t, p)−→ x�0 as t −→+∞ and p∈W c+

�0(x�0 ) implies ��0

(t, p)−→ x�0 as t −→−∞.Next theorem is proved in [19], it studies the dynamical behavior of system (3) in the neigh-

borhood of a type-zero saddle-node bifurcation point.

Theorem 2.1 (Type-zero saddle-node bifurcation point)Let x�0 be a type-zero saddle-node equilibrium point of (3), for a fixed parameter �=�0. Thenthere is a neighborhood N of x�0 and �>0 such that depending on the signs of the expressions in(SN2) and (SN3), there is no equilibrium point on N when �∈ (�0 −�,�0)[�∈ (�0,�0 +�)] and twoequilibrium points ys

� and yu� on N for each �∈ (�0,�0 +�)[�∈ (�0 −�,�0)]. The two equilibrium

points on N are hyperbolic, more specifically ys� is of type-zero and yu

� is of type-one. Moreover,



Figure 2. Phase portrait of system (3) in the neighborhood N of Theorem 2.1 for �∈ (�0 −�,�0 +�).For �<�0, system (3) has two hyperbolic equilibrium points, an equilibrium point of type-zero and oneof type-one. At �=�0, system (3) has a unique equilibrium point x�0 which is a type-zero saddle-node

equilibrium point. For �>�0, system (3) has no equilibrium point on N .

the stable manifold of the type-zero equilibrium point and the unstable manifold of the type-oneequilibrium point, intersect along a one-dimensional manifold.

In this paper, we assume, without loss of generality, that in Theorem 2.1 the two equilibriumpoints exist on N for �∈ (�0 −�,�0) and no equilibrium point exists on N for �∈ (�0,�0 +�).Figure 2 illustrates Theorem 2.1.

In order to gain more insight into the dynamical behavior of system (3) in the neighborhood ofa type-zero saddle-node equilibrium point x�o , a neighborhood U ⊆ N of x�o will be decomposedinto subsets U+ and U−. For �=�0, we define

U−�0

:= {p∈U : ��0(t, p)→ x�0 as t →∞}

U+�0

:= U −U−�0

and for �∈ (�0 −�,�0), we define

U−� := {p∈U : ��(t, p)→ ys

� as t →∞}U+

� := U −U−�

3. STABILITY BOUNDARY CHARACTERIZATION

In this section, an overview of the existing body of theory about the characterization of the stabilityboundary of nonlinear dynamical systems is presented.

Suppose x s is an asymptotically stable equilibrium point of (1). The stability region (or basinof attraction) of x s is the set

A(x s)={x ∈Rn : �(t, x)→ x s as t →∞}of all initial conditions x ∈Rn whose trajectories converge to x s when t tends to infinite.

The stability region A(x s) is an open and invariant set. Its closure A(x s) is invariant and thestability boundary �A(x s) is a closed and invariant set. If A(x s) is not dense in Rn , then �A(x s)is of dimension n−1 [26].

The unstable equilibrium points that lie on the stability boundary �A(x s) play an essential rolein the stability boundary characterization. Let x s be a hyperbolic asymptotically stable equilibriumpoint of (1) and consider the following assumptions:

(A1) All the equilibrium points on �A(x s) are hyperbolic.(A2) The stable and unstable manifolds of equilibrium points on �A(x s) satisfy the transversality

condition.(A3) Every trajectory on �A(x s) approaches one of the equilibrium points as t →∞.



Assumptions (A1) and (A2) are generic properties of dynamical systems in the form of (1).In other words, they are satisfied for almost all dynamical systems in the form of (1) and in practicedo not need to be verified. On the contrary, assumption (A3) is not a generic property of dynamicalsystems and has to be verified. The existence of an energy function is a sufficient condition for thesatisfaction of assumption (A3). We refer the reader to [15] for more details regarding this issue.

Under assumptions (A1)–(A3), next theorem provides necessary and sufficient conditions toguarantee that an equilibrium point lies on the stability boundary in terms of properties of its stableand unstable manifolds. Its proof can be found in [15].

Theorem 3.1 (Hyperbolic equilibrium points on the stability boundary)Let x s be a hyperbolic asymptotically stable equilibrium point of (1) and A(x s) be its stabilityregion. If assumptions (A1)–(A3) are satisfied and x∗ is a hyperbolic equilibrium point of (1),then:

(i) x∗ ∈�A(x s) if and only if W u(x∗)∩ A(x s) �=∅.(ii) x∗ ∈�A(x s) if and only if W s(x∗)⊆�A(x s).

Theorem 3.1 offers insight into how to numerically check if a hyperbolic equilibrium point lieson the stability boundary. A computational algorithm to check it was proposed in [15].

Next theorem explores Theorem 3.1 to provide a complete characterization of the stabilityboundary �A(x s) in terms of the unstable equilibrium points lying on the stability boundary.It asserts the stability boundary �A(x s) is the union of the stable manifolds of the equilibriumpoints on �A(x s).

Theorem 3.2 (Stability boundary characterization, [15])Let x s be a hyperbolic asymptotically stable equilibrium point of (1) and A(x s) be its stabilityregion. If assumptions (A1)–(A3) are satisfied, then:

�A(x s)=⋃i

W s(xi )

where xi , i =1,2, . . . are the equilibrium points on �A(x s).

Consider system (3) and suppose x s�0

is a hyperbolic asymptotically stable equilibrium point of(3) for �=�0. If system (3) satisfies assumptions (A1)–(A3) for �=�0, then the stability boundary�A�0 (x s

�0) is characterized, according to Theorem 3.2, as the union of the stable manifolds of the

unstable equilibrium points that lie on the stability boundary. More precisely, if xi�0

, i =1,2, . . .,

are the equilibrium points on �A�0 (x s�0

) then

�A�0 (x s�0

)=⋃i

W s�0

(xi�0

)

The main aim of this paper is to study the behavior of the stability region and its boundaryunder parameter variation.

The Implicit Function Theorem [27] guarantees that the hyperbolic equilibrium points persistunder small perturbations of the vector field. In other words, if x∗

�0is a hyperbolic equilibrium

point of system (3) for �=�0, then there exists �>0 and a neighborhood U of x∗�0

such that system(3) possesses an unique hyperbolic equilibrium point x∗

� on U for all �∈ (�0 −�,�0 +�). Moreover,using the continuity of the eigenvalues with respect to parameter �, we can affirm that the pertubedequilibrium point x∗

� has the same type of stability as that of x∗�0

. In particular, if x s�0

is a hyperbolicasymptotically stable equilibrium point of (3) for a fixed parameter �=�0, then there exists �>0such that system (3) possesses a unique hyperbolic asymptotically stable equilibrium point x s

� inthe neighborhood of x s

�0for all �∈ (�0 −�,�0 +�). Therefore, it makes sense to study the stability

region A�(x s�) of the perturbed system and how it behaves under parameter variation.

If the number of equilibrium points on the stability boundary �A�0 (x s�0

) is finite and assumptions(A1)–(A3) are satisfied for every �∈ (�0 −�,�0 +�), with �>0, then there exists �>0, ��, such that



if xi�0

, i =1,2, . . . ,k, belongs to the stability boundary �A�0 (x s�0

), then the perturbed equilibrium

point xi� belongs to the stability boundary �A�(x s

�) of the perturbed system for all �∈ (�0 −�,�0 +�)[28]. Moreover, �A�(x s

�) contains the same number of equilibrium points of �A�0 (x s�0

). As aconclusion, the perturbed stability boundary is composed of the union of the stable manifolds ofthe perturbed equilibrium points xi

�, i =1,2, . . . ,k, that is,

�A�(x s�)=⋃

iW s

�(x�i )

where x�i , i =1,2, . . . ,k are the perturbed equilibrium points on �A�(x s�) for all �∈ (�0 −�,�0 +�).

In a certain sense, under assumptions (A1)–(A3), the stability boundary does not suffer drasticchanges with small parameter variation. In this paper, we study the behavior of the stabilityboundary under parameter variation for a particular case of violation of assumption (A1), that is,when a type-zero saddle-node equilibrium point lies on the stability boundary. In this case, drasticchanges in the stability boundary and stability region are verified.

4. SADDLE-NODE EQUILIBRIUM POINT ON THE STABILITY BOUNDARY

In this section, a complete characterization of the stability boundary in the presence of a type-zerosaddle-node equilibrium point is developed.

Next lemma and Theorem 4.1 offer necessary and sufficient conditions to guarantee that atype-zero saddle-node equilibrium point lies on the stability boundary in terms of the propertiesof its stable and center manifolds. They also provide insight into how to develop a computationalprocedure to check if a type-zero saddle-node equilibrium point lies on the stability boundary.

Lemma 4.1 (Type-zero saddle-node equilibrium point on the stability boundary)Let x s

�0be an asymptotically stable equilibrium point of (3) for �=�0 and A�0 (x s

�0) be its stability

region. If x�0 is a type-zero saddle-node equilibrium point of (3) for �=�0, then:

(i) x�0 ∈�A�0 (x s�0

) if and only if W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅.

(ii) x�0 ∈�A�0 (x s�0

) if and only if (W s�0

(x�0 )−{x�0})∩�A�0 (x s�0

) �=∅.

The proof of Lemma 4.1 is given in Appendix A.A stronger version of the previous lemma can be proven under some additional assumptions. Let

x s�0

be an asymptotically stable equilibrium point and x�0 be a type-zero saddle-node equilibriumpoint of (3), for a fixed parameter �=�0, and consider the following assumptions:

(A1′) All the equilibrium points on �A�0 (x s

�0) are hyperbolic, except possibly for x�0 .

(A4) The stable manifold of equilibrium points on �A�0 (x s�0

) and the manifold W c+�0

(x�0 ) of the

type-zero saddle-node equilibrium point on �A�0 (x s�0

) satisfy the transversality condition.

Assumption (A1′) is weaker than (A1). It allows the presence of a non-hyperbolic equilibrium

point x�0 on the stability boundary. Assumption (A4) is an additional assumption of transversality.Under assumptions (A1

′), (A3) and (A4), next theorem offers necessary and sufficient conditions,

which are sharper and more useful than conditions of Lemma 4.1, to guarantee that a type-zerosaddle-node equilibrium point is on the stability boundary of nonlinear autonomous dynamicalsystems.

Theorem 4.1 (Further characterization of the type-zero saddle-node equilibrium point on thestability boundary)Let A(x s

�0) be the stability region of the asymptotically stable equilibrium point x s

�0of (3) for �=�0.

If assumptions (A1′), (A3) and (A4) are satisfied and x�0 is a type-zero saddle-node equilibrium



Figure 3. A type-zero saddle-node equilibrium point on the stability boundary of the asymptoticallystable equilibrium point x s

�0.

point of (3) for �=�0, then:

(i) x�0 ∈�A�0 (x s�0

) if and only if W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅.

(ii) x�0 ∈�A�0 (x s�0

) if and only if W s�0

(x�0 )⊂�A�0 (x s�0

).

The proof of Theorem 4.1 is given in Appendix A.Figure 3 illustrates Theorem 4.1. It shows a type-zero saddle-node equilibrium point on the

stability boundary of the asymptotically stable equilibrium point x s�0

. The unstable component of

the center manifold W c+�0

(x�0 ) intercepts the stability region A�0 (x s�0

) while the stable manifoldW s

�0(x�0 ) belongs to the stability boundary.

Under assumptions (A1′), (A2)–(A4) and exploring the results of Theorems 3.1 and 4.1, we

obtain the next corollary whose proof is analogous to the proof of Theorem 4.1.

Corollary 4.1 (Hyperbolic equilibrium points on the stability boundary)Let A�0 (x s


�0of (3) for

�=�0 and suppose that x�0 is a type-zero saddle-node equilibrium point lying on the stabilityboundary �A�0 (x s

�0). If assumptions (A1

′), (A2)–(A4) are satisfied and x∗

�0is a hyperbolic equi-

librium point of (3) for �=�0, then:

(i) x∗�0

∈�A�0 (x�0 ) if and only if W u�0

(x∗�0

)∩ A�0 (x s�0

) �=∅.

(ii) x∗�0

∈�A�0 (x s�0

) if and only if W s�0

(x∗�0

)⊂�A�0 (x s�0

).

Remark 4.1Corollary 4.1 is a more general result than Theorem 3.1, since the hyperbolicity assumption (A1)used in the proof of Theorem 3.1 is relaxed. It provides the same necessary and sufficient conditionsof Theorem 3.1 to check if a hyperbolic equilibrium point lies on the stability boundary.

Using the results of Theorem 4.1 and Corollary 4.1, next theorem provides a complete charac-terization of the stability boundary if a type-zero saddle-node equilibrium point lies on �A�0 (x s

�0).

Theorem 4.2 (Stability boundary characterization)Let A�0 (x s


�0of (3) for

�=�0 and suppose that x�0 is a type-zero saddle-node equilibrium point lying on the stability



(a) (b)

Figure 4. Examples of dynamical systems in which assumption (A4) is violated.

boundary �A�0 (x s�0

). If assumptions (A1′), (A2)–(A4) are satisfied then:

�A�0 (x s�0

)=⋃i

W s�0

(x�i0)⋃

W s�0

(x�0 )

where x�i0, i =1,2, . . . are the hyperbolic equilibrium points on �A�0 (x s

�0).

The proof of Theorem 4.2 is given in Appendix A.Theorem 4.2 is a generalization of Theorem 3.2. It shows that the stability boundary is composed

of the union of the stable manifolds of all equilibrium points on the stability boundary, includingthe stable manifold of the type-zero saddle-node equilibrium point on the stability boundary.

The importance of assumptions (A1)–(A3) in the establishment of a characterization of thestability boundary has already been discussed in [15]. Figures 4(a) and (b) illustrate the importanceof assumption (A4) in the statement of Theorems 4.1 and 4.2. Figure 4(a) shows an example of adynamical system in which the unstable component W c+

�0(x�0 ) of the central manifold of a type-

zero saddle-node equilibrium point x�0 has a non-transversal intersection with the stable manifold

W s�0

(x�0 ). In this case, although x�0 lies on the stability boundary, W c+�0

(x�0 ) does not intersect thestability region, the gray region in Figure 4(a). Moreover, the stability boundary is not composedof the stable manifold of the type-zero saddle-node equilibrium point x�0 . Figure 4(b) shows an

example of a dynamical system in which the unstable component W c+�0

(x�0 ) of the central manifoldof a type-zero saddle-node equilibrium point x�0 has a non-transversal intersection with the stablemanifold W s

�0(x∗

�0) of a type-one hyperbolic equilibrium point. In this case, although x�0 and x∗

�0

lie on the stability boundary, W c+�0

(x�0 ) does not intersect the stability region, the gray region inFigure 4(b). Moreover, the stability boundary is not composed of the union of the stable manifoldsof the equilibrium points x�0 and x∗

�0.

Example 4.1Consider the system of differential equations

x1 = x21 +x2

2 −1

x2 = x21 −x2 +�

(4)

with (x1, x2)∈R2 and �∈R.

System (4) possesses, for �0 =−1, three equilibrium points; they are x�0 = (0,−1), a type-zero saddle-node equilibrium point, x s

�0= (−1,0), a hyperbolic asymptotically stable equilibrium

point and x∗�0

= (1,0), a type-one hyperbolic equilibrium point. Both, the type-zero saddle-node



Figure 5. The phase portait of system (4) for �0 =−1.

equilibrium point and the type-one equilibrium point x∗�0

= (1,0) belong to the stability boundary

of x s�0

= (−1,0). The stability boundary �A�0 (−1,0) is formed according to Theorem 4.2, as theunion of the stable manifold of the type-one hyperbolic equilibrium point (1,0) and the stablemanifold of the type-zero saddle-node equilibrium point (0,−1), see Figure 5.

Exploring the complete characterizations of the stability boundary provided in Theorems 3.2and 4.2, Stability Region Bifurcations are studied in the next section, that is, the behavior of A�(x s

�)under parameter variation is investigated.

5. STABILITY REGION BIFURCATION

In this section, we develop results that describe the behavior of stability region and stabilityboundary in the neighborhood of a type-zero saddle-node bifurcation value.

We start this section proving that hyperbolic equilibrium points on the stability boundary persistson the stability boundary under small changes in parameters. This is a more general result thanthe results of persistence that have been proven in [1, 28].

Theorem 5.1 (Persistence of hyperbolic equilibrium points on the stability boundary)Let x�0 be a type-zero saddle-node equilibrium point lying on the stability boundary �A�0 (x s

�0)

of the hyperbolic asymptotically stable equilibrium point x s�0

of (3) for �=�0. If assumptions(A1)–(A3) are satisfied in an open interval containing �0, except at the type-zero saddle-nodebifurcation value �0 where assumptions (A1

′), (A2)–(A4) are satisfied, and x∗

�0is a hyperbolic

equilibrium point lying on �A�0 (x s�0

), then there is �>0 such that the perturbed equilibrium point

x∗� ∈�A�(x s

�) for all �∈ (�0 −�,�0 +�).

The proof of Theorem 5.1 is given in Appendix A.Theorem 5.1 guarantees that the hyperbolic equilibrium points that lie on the stability boundary

at the type-zero bifurcation value �0 persist on the stability boundary under small variation ofparameters. On the contrary, the non-hyperbolic type-zero saddle-node equilibrium point does notpersist under parameter variation. The local behavior of the stability boundary in the neighborhoodof a type-zero saddle-node equilibrium point is studied in the next theorem.

Theorem 5.2 (Stability boundary behavior in the neighborhood of a type-zero saddle-node equi-librium point)Let x�0 be a type-zero saddle-node equilibrium point lying on the stability boundary �A�0 (x s

�0)


of (3) for �=�0. If assumptions



(A1)–(A3) are satisfied in an open interval containing �0, except at the type-zero saddle-nodebifurcation value �0 where assumptions (A1

′), (A2)–(A4) are satisfied, and the number of equi-

librium points on �A�0 (x s�0

) is finite, then there is a neighborhood U of x�0 , �′>0 and �>0

such that:

(i) There are only two hyperbolic equilibrium points ys� and yu

� on U of type-zero and type-

one, respectively, for all �∈ (�0 −�′,�0) and there is no equilibrium point on U for all

�∈ (�0,�0 +�′). Moreover, the stable manifold of the equilibrium point of type-zero and the

unstable manifold of the equilibrium point of type-one intersect along a one-dimensionalmanifold.

(ii) yu� ∈�A�(x s

�)∩�A�(ys�) for all �∈ (�0 −�,�0).

(iii) U ⊂ A�(x s�) for all �∈ (�0,�0 +�).

The proof of Theorem 5.2 is given in Appendix A.Theorem 5.2 studies the local behavior of the stability boundary in a neighborhood U of the

type-zero saddle-node equilibrium point. For �<�0 the type-one hyperbolic equilibrium point yu� on

U lies on the stability boundary of x s�. As � increases, a stable equilibrium point ys

� approaches thetype-one equilibrium point yu

� . They coalesce inside U at �=�0 in a single type-zero saddle-nodeequilibrium point x�0 . At �=�0, the type-zero saddle-node equilibrium point lies on the stabilityboundary of x s

�0. As � continues to increase, the equilibrium x�0 disappears and the neighborhood

U of the saddle-node equilibrium point now is contained in the stability region of x s�. Theorem 5.2

shows that the stability region and the stability boundary suffer drastic changes at �=�0. Thestability boundary intercepts the neighborhood U for ��0 while U is totally contained in thestability region for �>�0.

Let us now study the global behavior of the stability boundary under parameter variation.Given x�0 a saddle-node equilibrium point of type-zero of (3), for a fixed parameter �=�0, we

define the following concept of weak stability region of x�0 .

Definition 5.1Let x�0 be a saddle-node equilibrium point of type 0 of (3) for a fixed parameter �=�0. The weakstability region of x�0 is the set S�0 (x�0 ) of points p∈Rn whose trajectories converge to x�0 ast −→+∞:

S�0 (x�0 )={p∈Rn : ��0(t, p)−→ x�0 as t −→+∞}

Remark 5.1For any neighborhood U of x�0 the set U−

�0⊂ S�0 (x�0 ). The weak stability region does not persist

under small perturbations of the vector field f�0 , that is, both the type-zero saddle-node equilibriumpoint and S�0 (x�0 ) disappear with the variation of parameter �. In spite of that, Theorems 3.1 isstill valid by replacing �A(x s

�0) for �S�0 (x�0 ).

We explore the concept of weak stability region to prove the next theorem.

Theorem 5.3 (The inheritance of equilibrium points on the stability boundary)Let x�0 be a type-zero saddle-node equilibrium point lying on the stability boundary �A�0 (x s

�0)


of (3) for �=�0. If assumptions(A1)–(A3) are satisfied in an open interval containing �0, except at the type-zero saddle-nodebifurcation value �0 where assumptions (A1

′), (A2)–(A4) are satisfied, and the number of equi-

librium points on �A�0 (x s�0

) is finite, then there is a neighborhood U of x�0 , �′>0, �>0 and >0

such that:

(i) There are only two hyperbolic equilibrium points ys� and yu

� on U of type-zero and type-

one, respectively, for all �∈ (�0 −�′,�0) and there is no equilibrium point on U for all

�∈ (�0,�0 +�′).



Moreover, if x∗�0

is a hyperbolic equilibrium point lying on the boundary of the weak

stability region �S�0 (x�0 ) for �=�0, then:(ii) The perturbed equilibrium point x∗

� ∈�A�(ys�)∪�A�(x s

�) for all �∈ (�0 −�,�0).(iii) The perturbed equilibrium point x∗

� ∈�A�(x s�) for all �∈ (�0,�0 +).

The proof of Theorem 5.3 is given in Appendix A.Roughly speaking, Theorem 5.3 shows that the stability region of x s

� takes over the stabilityregion of the stable equilibrium point ys

� that disappears in a saddle-node bifurcation at �=�0.Technically, it shows that hyperbolic equilibrium points on the stability boundary of ys

� for �<�0are inherited by the stability boundary of x s

� as � increases and passes through �0.Using Theorems 4.2, 5.2 and 5.3, we prove the following corollary regarding the complete

characterization of the stability boundary in the neighborhood of a saddle-node bifurcation value.

Corollary 5.1 (Stability boundary characterization in the neighborhood of a type-zero saddle-nodebifurcation value)Let x�0 be a type-zero saddle-node equilibrium point lying on the stability boundary �A�0 (x s

�0)


of (3), for a fixed parameter �=�0.

Consider also the existence of the hyperbolic equilibrium points x∗� j

0

, j =1, . . . ,m on �S�0 (x�0 ).

If assumptions (A1)–(A3) are satisfied in an open interval containing �0, except at the type-zerosaddle-node bifurcation value �0 where assumptions (A1

′), (A2)–(A4) are satisfied, and the number

of equilibrium points on �A�0 (x s�0

) is finite. Then:

(i) For �=�0 we have

�A�0 (x s�0

)=⋃i

W s�0

(x�i0)⋃

W s�0

(x�0 )

where x�i0, i =1,2, . . . ,k are the hyperbolic equilibrium points on �A�0 (x s

�0).

(ii) There is �>0 such that, for all �∈ (�0 −�,�0),

�A�(x s�)=⋃

iW s

�(x�i )⋃

W s�(yu

� )

where x�i , i =1,2, . . . ,k are the perturbed hyperbolic equilibrium points on �A�(x s�) and yu

�is an additional type-one equilibrium point, which originated from the type-zero saddle-nodebifurcation, that also belongs to �A�(x s

�).(iii) There is �>0 such that, for all �∈ (�0,�0 +�),

�A�(x s�)=⋃

iW s

�(x�i )⋃

jW s

�(x∗� j )

where x�i , i =1,2, . . . ,k and x∗� j , j =1, . . . ,m are the perturbed hyperbolic equilibrium

points on �A�(x s�).

Figures 6–8 illustrate Theorems 5.2 and 5.3. Figure 6 shows stability regions of system (3),for �<�0. There are two hyperbolic asymptotically stable equilibrium points x s

� and ys� in this

figure. The type-one equilibrium point yu� belongs to the stability boundary of both asymptotically

stable equilibrium points, while x∗� lies on the stability boundary of ys

�. As � increases, the systemundergoes a type-zero saddle-node bifurcation at �=�0. Figure 7 depicts stability regions ofsystem (3), for �=�0. The equilibrium points yu

� and ys� of Figure 6 are coalesced into a single

equilibrium point x�0 in Figure 7. The equilibrium point x�0 is a type-zero saddle-node equilibriumpoint that belongs to the stability boundary of the asymptotically stable equilibrium point x s

�0. The

equilibrium point x∗� of Figure 6 persists. The equilibrium point x∗

�0, which was on the stability

boundary of the asymptotically stable equilibrium point that undergoes a bifurcation, belongs nowto the boundary of the weak stability region of the type-zero saddle-node equilibrium point x�0 . As �continues to increase, the type-zero saddle-node equilibrium point disappears and the perturbed



Figure 6. Stability regions of system (3) for �<�0. The darkest shaded area is the stability region of theasymptotically stable equilibrium point ys

� while the lightest shaded area is the stability region of theasymptotically stable equilibrium point x s

�.

Figure 7. Stability regions of system (3) for �=�0. The asymptotically stable equilibrium point ys� coalesced

with the type-one equilibrium point yu� into a single type-zero saddle-node equilibrium point x�0 . The

type-zero saddle-node equilibrium point x�0 lies on the stability boundary of x s�0

and the darkest shadedarea is the weak stability region of x�0 .

Figure 8. Stability regions of system (3) for �>�0. The type-one hyperbolic equilibrium point x∗� which

belonged to the stability boundary of ys� for �<�0 belongs now to the stability boundary of x s

�. Theequilibrium point x s

� takes over the stability region of the equilibrium point ys�, which disappeared.



type-one equilibrium point x∗� , which belonged to the stability boundary of ys

� for �<�0, belongsnow to the stability boundary of x s

�, Figure 8. In other words, the stability region of the perturbedequilibrium point x s

� ‘inherits’ the stability region of the asymptotically stable equilibrium pointthat disappeared in the type-zero saddle-node bifurcation.

6. EXAMPLES AND APPLICATIONS

Example 6.1Consider the same system (4) of Example 4.1. For �0 =−1, we have seen in Figure 5 thatthe stability boundary �A�0 (−1,0) is formed as the union of the stable manifold of the type-one hyperbolic equilibrium point (1,0) and the stable manifold of the type-zero saddle-nodeequilibrium point (0,−1). For �=−1.02, system (4) possesses four equilibrium points; theyare x s

� = (−0.99,−0.02) a hyperbolic asymptotically stable equilibrium point, x∗� = (0.99,−0.02)

a type-one hyperbolic equilibrium point, yu� = (−0.2,−0.97) a type-one hyperbolic equilibrium

point and ys� = (0.2,−0.97) a type-zero hyperbolic equilibrium point. The equilibrium points

yu� and ys

� originated from the type-zero saddle-node equilibrium point in a type-zero saddle-node bifurcation. Moreover, yu

� ∈�A�(0.2,−0.97)∩�A�(−0.99,−0.02), confirming the results ofTheorem 5.2, and x∗

� = (0.99,−0.02)∈�A�(−0.99,−0.02), confirming the results of Theorem 5.1,see Figure 9(a). For �=−0.98, system (4) possesses two equilibrium points; they are x s

� =(−0.99,0.01), a hyperbolic asymptotically stable equilibrium point and x∗

� = (0.99,0.01), a type-one equilibrium point, which belongs to the stability boundary �A�(−0.99,0.01) and confirms theresults of Theorem 5.1, see Figure 9(b).

Example 6.2Consider the system of differential equations

x1 = x41 −1.25x2

1 −x2 +0.25

x2 = −x2 +�(5)

with (x1, x2)∈R2 and �∈R.

System (5) passes through �0 =0.25, three equilibrium points; they are x�0 = (0,0.25) a type-zerosaddle-node equilibrium point, x s

�0= (−1.11,0.25) a hyperbolic asymptotically stable equilibrium

point and x∗�0

= (1.11,0.25) a type-one hyperbolic equilibrium point. The type-zero saddle-nodeequilibrium point belongs to the stability boundary of x s

�0= (−1.11,0.25) and the type-one equi-

librium point x∗�0

= (1.11,0.25) belongs to the boundary of the weak stability region of (0,0.25).

The stability boundary of �A�0 (−1.11,0.25) is formed according to Theorem 4.2, as the stable

(b)(a)

Figure 9. Phase portaits of system (4): (a) the phase portait of system (4) for �=−1.02 and(b) the phase portait of system (4) for �=−0.98.



(c)

(a) (b)

Figure 10. Phase portaits of system (5): (a) the phase portait of system (5) for �=0.2; (b) the phaseportait of system (5) for �0 =0.25; and (c) the phase portait of system (5) for �=0.3.

manifold of the type-zero saddle-node equilibrium point (0,0.25), see Figure 10(b). For �=0.2,system (5) possesses four equilibrium points; they are x s

� = (−1.09,0.2) a hyperbolic asymptoticallystable equilibrium point, x∗

� = (1.09,0.2) a type-one hyperbolic equilibrium point, yu� = (−0.2,0.2) a

type-one hyperbolic equilibrium point and ys� = (0.2,0.2) a type-zero hyperbolic equilibrium point.

The equilibrium points yu� and ys

� originated from the type-zero saddle-node equilibrium point in atype-zero saddle-node bifurcation. Moreover, yu

� ∈�A�(−1.09,0.2)∩�A�(0.2,0.02), according toTheorem 5.2, and x∗

� = (1.09,0.02)∈�A�(0.2,0.02), confirming the results of Theorem 5.3, seeFigure 10(a). For �=0.3, system (5) possesses two equilibrium points; they are x s

� = (−1.13,0.3)a hyperbolic asymptotically stable equilibrium point and x∗

� = (1.13,0.3) a type-one equilibriumpoint, which belongs to the stability boundary �A�(−1.13,0.3) according to Theorem 5.3, seeFigure 10(c).

Example 6.3Consider the system of differential equations that models a Hopfield Artificial Neural Network

xi = 1

Ci

(2∑

j=1Ti j g j (x j )− 1

Rixi + Ii

)(6)

where i =1,2, xi ∈R, gi (xi )= (2/) tan−1(�xi/2). The constant �>0 is the slope of gi (xi ) at xi =0while Ci>0, Tij and Ri are physical constants. The inputs Ii of the network will be considered asparameters of the system.

Considering T11 =4.5, T12 =T21 =T22 =C1 =C2 = R1 = R2 =1, �=0.3, I2 =0.02 and varyingparameter I1, a type-zero saddle-node bifurcation occurs at the bifurcation value I1 =0.418.More precisely, system (6) possesses, for I1 =0.418, two equilibrium points; they are



(c)

(a) (b)

Figure 11. Phase portaits of system (6): (a) the phase portait of system (6) for I1 =0.4; (b) the phaseportait of system (6) for I1 =0.418; and (c) the phase portait of system (6) for I1 =0.43.

x�0 = (−1.44,−0.515) a type-zero saddle-node equilibrium point, and x s�0

= (3.6943,0.95804)a hyperbolic asymptotically stable equilibrium point. The type-zero saddle-node equilibriumpoint belongs to the stability boundary of x s

�0= (3.6943,0.95804). The stability boundary of

�A�0 (3.6943,0.95804) is formed, according to Theorem 4.2, as the stable manifold of thetype-zero saddle-node equilibrium point (−1.44,−0.515), see Figure 11(b). For �=0.4, system(6) possesses three equilibrium points; they are x s

� = (3.6661,0.95523) a hyperbolic asymptoticallystable equilibrium point, yu

� = (−1.17,−0.42728) a type-one hyperbolic equilibrium point andys� = (−1.7408,−0.59001) a type-zero hyperbolic equilibrium point. The equilibrium points yu

� andys� originated from the type-zero saddle-node equilibrium point in a type-zero saddle-node bifurca-

tion. Moreover, yu� ∈�A�(3.6661,0.95523)∩�A�(−1.7408,−0.59001), according to Theorem 5.2,

see Figure 11(a). For �=0.43, system (6) possesses one equilibrium point x s� = (3.713,0.95989),

a hyperbolic globally asymptotically stable equilibrium point, see Figure 11(c). This exampleshows that the relationship between inputs and outputs of a Hopfield Artificial Neural Networkmay not be unique. Depending on the history, the output of the network may be different for thesame input. Figure 12 shows a bifurcation diagram. It shows the coordinate x1 of equilibriumpoints x = (x1, x2) as a function of the input value I1. For I1 =0.4 the output of the networksis (−1.7408,−0.59001). As I1 increases, this equilibrium slowly changes until the bifurcationvalue I1 =0.418 is reached. If � continues to increase, then the output of the network jumps from(−1.44,−0.515) to (3.6943,0.95804) as shown in Figure 12. For I1 =0.5, for example, the outputof the network is (3.8205,0.97023). If now the input I1 decreases back to the initial value 0.4,the output will be (3.6661,0.95523), which is different from the initial output for the same inputvalue. This behavior may have important consequences when the network is used for example forpattern recognition.



Figure 12. Bifurcation diagram of system (6). This graph depicts the first x1-component of equilibriumpoints of (6) a function of the input I1. Dashed lines correspond to the coordinates of the type-oneequilibrium point while continuous lines indicate asymptotically stable equilibrium points. A type-zerosaddle-node bifurcation occurs at I1 =0.418. If parameter I1 is slowly increased, then at the value 0.418, theoutput of the network suddenly jumps from the type-zero saddle-node equilibrium point (−1.44,−0.515)

to the asymptotically stable equilibrium point (3.6943,0.95804).

7. CONCLUSION

In this paper, we studied the behavior of the stability region of nonlinear autonomous dynamicalsystems under parameter variation. In particular, we studied the behavior of the stability boundarywhen a type-zero saddle-node bifurcation occurs in the stability boundary.

By studying the characterization of the stability boundary at the type-zero saddle-node bifurcationvalue �0, we have shown that the stability region suffers drastic changes as the parameter passesthrough �0. In a certain sense, we have shown that the stability region of the asymptotically stableequilibrium point that undergoes a type-zero saddle-node bifurcation is ‘inherited’ by anotherasymptotically stable equilibrium point. In other words, the stability region of the asymptoticallystable equilibrium point that persists is enlarged as the parameter passes through �0. It inheritsthe stability region of the asymptotically stable equilibrium point that disappears in a type-zerosaddle-node bifurcation.

Studying saddle-node bifurcations on the stability boundary was the first step in the study of thebehavior of stability region under parameter variation. Future work in this area includes the analysisof other types of bifurcation on the stability boundary such as type-k saddle-node bifurcations withk higher or equal to 1 and hopf bifurcations. Applications of these results include stability analysisof power systems and artificial neural network theory.

APPENDIX A

This appendix contains the proof of lemmas and main theorems that are proposed in this paper.

Proof of Lemma 4.1

(i) (⇐�) Suppose that W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅. Then there exists x ∈W c+�0

(x�0 )∩ A�0 (x s�0

).

Note that ��0(t, x)−→ x�0 as t −→−∞. On the other hand, set A�0 (x s

�0) is invariant, thus



�(t, x)∈ A�0 (x s�0

) for all t�0. As a consequence, x�0 ∈ A�0 (x s�0

). Since x�0 /∈ A�0 (x s�0

), we have that

x�0 ∈{Rn − A�0 (x s�0

)}. Therefore, x�0 ∈�A�0 (x s�0

).

(i) (�⇒) Suppose that x�0 ∈�A�0 (x s�0

). Let B(q,�) be a ball of radius �>0 centered at q for

some q ∈W c+�0

(x�0 ). Consider a disk D of dimension n−1 contained in B(q,�) and transversal to

W c+�0

(x�0 ) at q . As a consequence of �-lema for non-hyperbolic equilibrium points [21], we can

affirm that ∪t�0��0(t, B(q,�))⊃U+

�0where U is a neighborhood of x�0 . Since x�0 ∈�A�0 (x s

�0),

we have that U ∩ A�0 (x s�0

) �=∅. On the other hand, U−�0

∩ A�0 (x s�0

)=∅, thus U+�0

∩ A�0 (x s�0

) �=∅.Thus, there exists a point p∈ B(q,�) and a time t∗ such that ��0

(t∗, p)∈ A�0 (x s�0

). Since A�0 (x s�0

)is invariant, we have that p∈ A�0 (x s

�0). Since � can be chosen arbitrarily small, we can find a

sequence of points {pi } with pi ∈ A�0 (x s�0

) for all i =1,2, . . . such that pi −→q as i −→∞, that

is, q ∈ A�0 (x s�0

). Since q ∈W c+�0

(x�0 ), we have that W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅.The proof of (ii) is similar to the proof of (i) and will be omitted. �

Proof of Theorem 4.1

(i) (⇐�) Suppose that W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅. Since W c+�0

(x�0 )∩ A�0 (x s�0

)⊂W c+�0

(x�0 )∩A�0 (x s

�0) we have that W c+

�0(x�0 )∩ A�0 (x s

�0) �=∅. Thus, from Lemma 4.1, one concludes that

x�0 ∈�A�0 (x s�0

).

(i)(�⇒) Suppose that x�0 ∈�A�0 (x s�0

). From Lemma 4.1, we can affirm that W c+�0

(x�0 )∩A�0 (x s

�0) �=∅. We are going to show, under assumptions (A1

′), (A3) and (A4) that W c+

�0(x�0 )∩

A�0 (x s�0

) �=∅ implies W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅. Let p∈W c+�0

(x�0 )∩ A�0 (x s�0

). If p∈ A�0 (x s�0

), then

there is nothing to be proved . Suppose that p∈�A�0 (x s�0

). Assumption (A3) asserts the existence

of an equilibrium point x∗�0

∈�A�0 (x s�0

) such that ��(t, p)−→ x∗�0

as t −→∞. From Lemmas B2and B3 given in Appendix B, we can affirm that x∗

�0�= x�0 . As a consequence of assumption

(A1′), x∗

�0is a hyperbolic equilibrium point. Since p∈W c+

�0(x�0 )∩W s

�0(x∗

�0), assumption (A4) and

Lemma B1 given in Appendix B guarantee that x∗�0

is a type-zero hyperbolic equilibrium point.

But this fact leads to an absurd, since x∗�0

∈�A�0 (x s�0

). Therefore, W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅.

(ii) (⇐�) Suppose that W s�0

(x�0 )⊂�A�0 (x s�0

). Since x�0 ∈W s�0

(x�0 ), then x�0 ∈�A�0 (x s�0

).

(ii) (�⇒) Suppose that x�0 ∈�A�0 (x s�0

). From item (i), we have that W c+�0

(x�0 )∩ A�0 (x s�0

) �=∅.

Let w∈W c+�0

(x�0 )∩ A�0 (x s�0

) and B(w,�) be an open ball with an arbitrarily small radius � centeredat w. Radius � can be chosen arbitrarily small such that B(w,�)⊂ A�0 (x s

�0). Let p be an arbitrary

point of W s�0

(x�0 ) and consider a disk D transversal to W s�0

(x�0 ) at p of dimension 1. As aconsequence of �-lemma for non-hyperbolic equilibrium points [21], there exists an element z ∈ Dand a time t∗>0 such that ��0

(t∗, z)∈ B(w,�). Since A�0 (x s�0

) is invariant, we have that z ∈A�0 (x s

�0). Since � and the disk D can be chosen arbitrarily small, then there exist points of A�0 (x s

�0)

arbitrarily close to p. Therefore p∈ A�0 (x s�0

). Since W s�0

(x�0 ) cannot contain points on A�0 (x s�0

),

p∈�A�0 (x s�0

). Exploring the fact that p was arbitrarily taken in W s�0

(x�0 ), we can affirm that

W s�0

(x�0 )⊂�A�0 (x s�0

). �

Proof of Theorem 4.2If the hyperbolic equilibrium point x�i

0∈�A�0 (x s

�0), then, from Corollary 4.1, we have that

W s�0

(x�i0)⊂�A�0 (x s

�0). Since x�0 ∈�A�0 (x s

�0), we have that W s

�0(x�0 )⊂�A�0 (x s

�0) from Theorem 4.1.

Therefore, ∪i W s�0

(x�i0)∪W s

�0(x�0 )⊂�A�0 (x s

�0). On the other hand, from assumption (A3), if

p∈�A(x s�0

), then ��0(t, p)−→ x�i

0or ��0

(t, p)−→ x�0 as t −→∞. As a consequence of Lemma B3



given in Appendix B, the point p /∈W c−(x�0 ). Thus, we can affirm that p∈W s�0

(x�i0) or p∈W s

�0(x�0 ).

Therefore �A�0 (x s�0

)⊂∪i W s�0

(x�i0)∪W s

�0(x�0 ) and the theorem is proven. �

Proof of Theorem 5.1Given a hyperbolic equilibrium point x∗

�0and a hyperbolic asymptotically stable equilibrium

point x s�0

, the Implicit Function Theorem [27] guarantees the persistence of them under smallvariation of parameter �. In other words, there exist neighborhoods U and V of x∗

�0and x s

�0,

respectively, and �1>0 such that there exist a unique hyperbolic equilibrium point x∗� in U and

a unique hyperbolic asymptotically stable equilibrium point x s� in V for all �∈ (�0 −�1,�0 +�1).

Since x∗�0

∈�A�0 (x s�0

), Corollary 4.1 guarantees that W u�0

(x∗�0

)∩ A�0 (x s�0

) �=∅, in particular there isan open set D contained in A�0 (x s

�0) such that W u

�0(x∗

�0)∩{A�0 (x s

�0)∩ D} �=∅ and this intersection is

transversal. Since transversal intersections persist under small pertubations [25], there exists �2>0such that W u

� (x∗� )∩{A�(x s

�)∩ D} �=∅ for all �∈ (�0 −�2,�0 +�2). Since assumptions (A1)–(A3) aresatisfied for all � close to �0, Theorem 3.1 guarantees that x∗

� ∈�A�(x s�) for all �∈ (�0 −�,�0 +�)

with �=min{�1,�2} and the theorem is proven. �


(i) The existence of neighborhood U and scalar �′

with the desired properties is a directconsequence of Theorem 2.1.

(ii) First of all, we will show the existence �>0 such that yu� ∈�A�(x s

�) for all �∈ (�0 −�,�0).Let xi

�0, i =,1 . . . ,k, be the equilibrium points on �A�0 (x s

�0). Since the type-zero saddle-node

equilibrium point x�0 ∈�A�0 (x s�0

) and the number of equilibrium points on �A�0 (x s�0

) is finite, then

the neighborhood U can be chosen arbitrarily small such that W s�0

(xi�0

)∩U =∅ for all i =,1 . . . ,k.Moreover, U ∩ A�0 (x s

�0) �=∅, and in particular, there is an open set D contained in A�0 (x s

�0) such that

U ∩{A�0 (x s�0

)∩ D} �=∅ and this intersection is transversal. Since empty intersections are transversaland transversal intersections persit under small pertubations [25], there exists �2>0 such thatU ∩{A�(x s

�)∩ D} �=∅ and W s�(xi

�)∩U =∅ for all i =,1 . . . ,k and �∈ (�0 −�2,�0 +�2). Since xi�0

∈�A�0 (x s

�0) for i =1, . . . ,k, Theorem 5.1 guarantees the existence of �3>0 such that xi

� ∈�A�(x s�) for

all i =1, . . . ,k and �∈ (�0 −�3,�0 +�3) . Taking �=min{�2,�3,�′ }, we have that U ∩ A�(x s

�) �=∅,xi� ∈�A�(x s

�), W s�(xi

�)∩U =∅ ∀i =1, . . . ,k and there are two hyperbolic equilibrium points ys� and

yu� on U for all �∈ (�0 −�,�0). From the facts that U ∩ A�(x s

�) �=∅, the neighborhood U is connectedand U is not entirely contained on A(x s

�), we can affirm that U ∩�A�(x s�) �=∅ for all �∈ (�0 −�,�0).

Since assumptions (A1)–(A3) are satisfied for all � close to �0, Theorem 3.2 guarantees thatthe stability boundary �A�(x s

�) is the union of the stable manifolds of the equilibrium points on�A�(x s

�). As a consequence, one concludes that U intersects the stable manifold of at least oneequilibrium point on �A�(x s

�). Since W s�(xi

�)∩U =∅ ∀i =1, . . . ,k and ys� /∈�A�(x s

�), because ys� is

of type-zero, necessarily we have that yu� ∈�A�(x s

�) for all �∈ (�0 −�,�0).We will also show that yu

� ∈�A�(ys�) for all �∈ (�0 −�,�0). From item (i), one has that W s

�(ys�)∩

W u� (yu

�) �=∅, that is, A�(ys�)∩W u

� (yu� ) �=∅, therefore Theorem 3.1 guarantees that yu

� ∈�A�(ys�) for

all �∈ (�0 −�,�0).(iii) Considering the same �>0 of item (ii), one has that U ∩ A�(x s

�) �=∅, xi� ∈�A�(x s

�),W s

�(xi�)∩U =∅ for all i =1, . . . ,k and �∈ (�0 −�,�0 +�). In particular, according to item

(i), also there is no equilibrium point on U for all �∈ (�0,�0 +�). From the facts thatU ∩ A�(x s

�) �=∅, the stability boundary �A�(x s�) is the union of the stable manifolds of the

equilibrium points on �A�(x s�) and the neighborhood U is connected, we can affirm that

U ⊂ A�(x s�) for all �∈ (�0,�0 +�). Suppose on the contrary that U ∩�A�(x s

�) �=∅, then U wouldintersect the stable manifold of at least one equilibrium point on �A�(x s

�), leading to anabsurd. �




(i) The existence of neighborhood U and scalar �′

with the desired properties is a directconsequence of Theorem 2.1.

(ii) Let x∗�0

∈�S�0 (x�0 ). If x∗�0

∈�S�0 (x�0 )∩�A�0 (x s�0

), Theorem 5.1 guarantees the existence of

�>0 such that x∗� ∈�A�(x s

�) for all �∈ (�0 −�,�0). Suppose that x∗�0

∈�S�0 (x�0 ) and x∗�0

/∈�A�0 (x s�0

),

in particular x∗�0

/∈W s�0

(x�0 ). From the fact that x∗�0

∈�S�0 (x�0 ), Remark 5.1 and Theorem 3.1

guarantee that W u�0

(x∗�0

)∩U−�0

�=∅. As x∗�0

/∈�A�0 (x s�0

), we can affirm that W u�0

(x∗�0

) intersects U−�0

only in points of U−�0

−W s�0

(x�0 ). Consider q ∈W u�0

(x∗�0

)∩{U−�0

−W s�0

(x�0 )} and r>0 such that

B(q;r )⊂{U−�0

−W s�0

(x�0 )}. Moreover, given �>0 sufficiently small, the number r>0 can be chosen

arbitrarily small such that B(q;r )⊂U−� for all �∈ (�0 −�,�0). On the other hand, as the unstable

manifold W u�0

(x∗�0

) intersects B(q;r ) transversally and transversal intersections persist under smallpertubations [25], there exists �2>0 such that W u

�0(x∗

�0)∩ B(q;r ) �=∅ for all �∈ (�0 −�2,�0 +�2).

Taking �=min{�2,�}, we have that W u�0

(x∗�0

)∩ B(q;r ) �=∅ and B(q;r )⊂U−� for all �∈ (�0 −�,�0).

Since assumptions (A1)–(A3) are satisfied for � close to �0 and U−� ⊂ A�(ys

�), Theorem 3.1guarantees that x∗

� ∈�A�(ys�) for all �∈ (�0 −�,�0). Therefore, x∗

� ∈�A�(ys�)∪�A�(x s

�) for all �∈(�0 −�,�0) with �=min{�,�}.

(iii) According to item (ii), W u� (x∗

� )∩U �=∅ for all �∈ (�0,�0 +�2). Taking =min{�2,�} where�>0 is defined in item (iii) of Theorem 5.2, we have that W u

� (x∗� )∩U �=∅ and U ⊂ A�(x s

�) for all�∈ (�0,�0 +). Since assumptions (A1)–(A3) are satisfied for � close to �0, Theorem 3.1 guaranteesthat x∗

� ∈�A�(x s�) for all �∈ (�0,�0 +). �

APPENDIX B

In this appendix, the proof of auxiliary lemmas are given.

Lemma B1Let x�0 be a type-zero saddle-node equilibrium point and x∗

�0a hyperbolic equilibrium point of (3)

for �=�0. If W c+�0

(x�0 ) and W s�0

(x∗�0

) satisfy the transversality conditions and W c+�0

(x�0 )∩W s�0

(x∗�0

) �=∅. Then x∗

�0is of type-zero.

ProofSince W c+

�0(x�0 ) and W s

�0(x∗

�0) satisfy the transversality condition and W c+

�0(x�0 )∩W s

�0(x∗

�0) �=∅, we

have that

dim W c+�0

(x�0 )+dim W s�0

(x∗�0

)=n+dim(W c+�0

(x�0 )∩W s�0

(x∗�0

))

Given x ∈W c+�0

(x�0 )∩W s�0

(x∗�0

) and as W c+�0

(x�0 ) and W s�0

(x∗�0

) are invariant sets, we have that the

orbit that passes through x is entirely contained on W c+�0

(x�0 )∩W s�0

(x∗�0

), thus we can affirm that

dim(W c+�0

(x�0 )∩W s�0

(x∗�0

))�1, that is,

dim W c+�0

(x�0 )+dim W s�0

(x∗�0

)�n+1

or equivalently,

dim W c+�0

(x�0 )�n+1−dim W s�0

(x∗�0

)

Since x∗�0

is a hyperbolic equilibrium point we have that

dim W s�0

(x∗�0

)+dim W u�0

(x∗�0

)=n



thus the previous inequality becomes

dim W c+�0

(x�0 )�(dim W s�0

(x∗�0

)+dim W u�0

(x∗�0

))+1−dim W s�0

(x∗�0

)=dim W u�0

(x∗�0

)+1

that is,

dim W c+�0

(x�0 )�dim W u�0

(x∗�0

)+1>dim W u�0

(x∗�0

)

Since dim W c+�0

(x�0 )=1, the previous inequality guarantees that dim W u�0

(x∗�0

)=0. Since the typeof x∗

�0is equal to the dimension of the unstable manifold W u

�0(x∗

�0), the lemma is proven. �

Lemma B2Let x�0 be a type-zero saddle-node equilibrium point of (3) for �=�0. If W c+

�0(x�0 ) and W s

�0(x�0 )

satisfy the transversality condition then W c+�0

(x�0 )∩W s�0

(x�0 )=∅.

ProofSuppose on the contrary that W c+

�0(x�0 )∩W s

�0(x�0 ) �=∅. Since W c+

�0(x�0 ) and W s

�0(x�0 ) satisfy the

transversality condition, we have that

dim W c+�0

(x�0 )+dim W s�0

(x�0 )−n =dim(W c+�0

(x�0 )∩W s�0

(x�0 ))

Since dim W c+�0

(x�0 )=1 and dim W s�0

(x�0 )=n−1 the previous inequality becomes

dim(W c+�0

(x�0 )∩W s�0

(x�0 ))=1+n−1−n =0

that is, dim(W c+�0

(x�0 )∩W s�0

(x�0 ))=0. On the other hand, if there exists x ∈W c+�0

(x�0 )∩W s�0

(x�0 ),

then the orbit that passes through x is entirely contained on W c+�0

(x�0 )∩W s�0

(x�0 ), thus we have that

dim(W c+�0

(x�0 )∩W s�0

(x�0 ))�1, leading to an absurd. Therefore, the intersection W c+�0

(x�0 )∩W s�0

(x�0 )is empty and the lemma is proved. �

Lemma B3Let x�0 be a type-zero saddle-node equilibrium point of (3) for �=�0 lying on the stability boundary

�A�0 (x s�0

) of the asymptotically stable equilibrium point x s�0

. Then W c−�0

(x�0 )∩�A�0 (x s�0

)=∅.

ProofSuppose, on the contrary, the existence of p∈W c−

�0(x�0 )∩�A�0 (x s

�0). In particular ��0

(t, p)−→ x�0

as t −→∞, thus there exists T >0 such that ��0(T, p)∈{U−

�0−W s

�0(x�0 )} where U is a neighbor-

hood of x�0 . Consider �>0 arbitrarily small such that B(��0(T, p),�)⊂{U−

�0−W s

�0(x�0 )}. From

the continuity of solutions with respect to the initial conditions, there exists �>0 such thatd(��0

(T,q),��0(T, p))<� for all q , p satisfying d(q, p)<�. Since p∈�A�0 (x s

�0), there exists

q0 ∈ A�0 (x s�0

) such that d(q0, p)<�. Thus, we can affirm that ��0(T,q0)∈ B(��0

(T, p),�)⊂{U−�0

−W s

�0(x�0 )}⊂U−

�0. Exploring the property of convergence of set U−

�0and the property of flows ��0

we get

��0(t,q0)−→ x�0 as t −→∞

leading to an absurd, since q0 ∈ A�0 (x s�0

). Therefore, W c−�0

(x�0 )∩�A�0 (x s�0

)=∅ and the lemma isproved. �

ACKNOWLEDGEMENTS

This class file was developed by Sunrise Setting Ltd., Torquay, Devon, U.K. John Wiley & Sons, Ltd. areindebted especially to Alistair Smith for his work on both the class file and the present document.



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Stability region bifurcations of nonlinear autonomous dynamical systems: Type-zero saddle-node bifurcations