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...e 1.1(b) illustrates these invariant manifolds for a type-0 subcritical Hopf equilibrium point in R3. x y z Wu(p) Wc(p) (a) Manifolds W c(p) and W s(p) for a type-1 subcritical Hopf equilibrium point p of system (2.1) in R3. W c(p) is not unique. Three choices of W c(p) are displayed in this figure. x y z Ws(p) Wc(p (b) Manifolds W c(p) and W u (p) for a type-0 subcritical Hopf equilibrium point p of system (2.1) in R3. In this case, W c(p) is unique. Figure 1 The stable and unstable manifolds of a hyperbolic equilibrium point are defined by extending the local manifolds through the flow, see [16]. Often, this technique to define the global manifolds cannot be applied to general non-hyperbolic equilibrium points. Even though, in the particular case of subcritical Hopf equilibrium points, one can also define the global manifolds W s(p), W u(p), W c(p) and W cu(p) by extending the local manifolds W sloc(p), W u loc(p), W c loc(p) and W culoc(p) through the flow. 4 SUBCRITICAL HOPF EQUILIBRIUM POINT ON THE STABILITY BOUNDARY In this section, results of characterization of equilibrium points on the boundary of the stability region are presented. The characterization of the boundary of stab...

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...zed as follows. In Section 2, a review of the characterization of the boundary of the stability region of nonlinear autonomous dynamic systems is presented. In Section 3, the subcritical Hopf equilibrium points are studied and the local dynamics on the neighborhood of these points is reviewed. The main contribution of this paper is presented in Section 4. 2 PRELIMINARIES In this section, we review some classic concepts related to the theory of dynamical systems, which are essential for the further developments of this work. More details on the contents explored in this section can be found at [18, 15]. Consider the nonlinear autonomous dynamic system: x = f (x) (2.1) where x ∈ Rn and f : Rn → Rn is a smooth vector field. We use the term smooth to refer to a field whose differentiability class is large enough, namely a vector field of class Cr with r ≥ 1. The solution of (2.1) starting at x at time t = 0 is denoted by ϕ(t, x). Suppose that xs is an asymptotically stable equilibrium point of system (2.1). The stability region (or region of attraction) of xs is the set A(xs ) = {x ∈ Rn |ϕ(t, x) → xs as t → +∞}, of all initial conditions x ∈ Rn whose trajectories converge to xs when t tends t...

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...zed as follows. In Section 2, a review of the characterization of the boundary of the stability region of nonlinear autonomous dynamic systems is presented. In Section 3, the subcritical Hopf equilibrium points are studied and the local dynamics on the neighborhood of these points is reviewed. The main contribution of this paper is presented in Section 4. 2 PRELIMINARIES In this section, we review some classic concepts related to the theory of dynamical systems, which are essential for the further developments of this work. More details on the contents explored in this section can be found at [18, 15]. Consider the nonlinear autonomous dynamic system: x = f (x) (2.1) where x ∈ Rn and f : Rn → Rn is a smooth vector field. We use the term smooth to refer to a field whose differentiability class is large enough, namely a vector field of class Cr with r ≥ 1. The solution of (2.1) starting at x at time t = 0 is denoted by ϕ(t, x). Suppose that xs is an asymptotically stable equilibrium point of system (2.1). The stability region (or region of attraction) of xs is the set A(xs ) = {x ∈ Rn |ϕ(t, x) → xs as t → +∞}, of all initial conditions x ∈ Rn whose trajectories converge to xs when t tends t...

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...oundary of the stability region is comprised of the stable manifolds of all equilibrium points on the boundary of the stability region, including the stable manifolds of the subcritical Hopf equilibrium points of type k, with 0 ≤ k ≤ n − 2, which belong to the boundary of the stability region. Keywords: dynamical systems, nonlinear systems, stability region, boundary of the stability region, subcritical Hopf equilibrium point. 1 INTRODUCTION Dynamic and topological characterizations of the boundary of the stability regions of autonomous nonlinear dynamic systems were developed, for example in [3, 5]. Those characterizations were derived under some assumptions over the vector field, including hyperbolicity of equilibrium points on the boundary of the stability region and transversality conditions. Although the hyperbolicity of equilibrium points of a dynamical system is a generic property, that is, it is satisfied for almost all dynamic systems, violation of the hyperbolicity condition of equilibrium points on the boundary of the stability region commonly occurs when the system is subject to variations of parameters. With this variation of parameters, the occurrence of local bifurcations ...

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...oundary of the stability region is comprised of the stable manifolds of all equilibrium points on the boundary of the stability region, including the stable manifolds of the subcritical Hopf equilibrium points of type k, with 0 ≤ k ≤ n − 2, which belong to the boundary of the stability region. Keywords: dynamical systems, nonlinear systems, stability region, boundary of the stability region, subcritical Hopf equilibrium point. 1 INTRODUCTION Dynamic and topological characterizations of the boundary of the stability regions of autonomous nonlinear dynamic systems were developed, for example in [3, 5]. Those characterizations were derived under some assumptions over the vector field, including hyperbolicity of equilibrium points on the boundary of the stability region and transversality conditions. Although the hyperbolicity of equilibrium points of a dynamical system is a generic property, that is, it is satisfied for almost all dynamic systems, violation of the hyperbolicity condition of equilibrium points on the boundary of the stability region commonly occurs when the system is subject to variations of parameters. With this variation of parameters, the occurrence of local bifurcations ...

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...nce A(xs ) is closed. Since p /∈ A(xs ), we have that p ∈ Rn \ A(xs ). Therefore, p ∈ ∂ A(xs ). Now suppose that W sloc(p) ∩ ∂ A(xs ) = ∅. Therefore, there exists q ∈ W sloc(p) ∩ ∂ A(xs ). Note that ϕ(t, q) −→ p as t −→ +∞. Since set ∂ A(xs ) is invariant and q ∈ ∂ A(xs ), thus ϕ(t, q) ∈ ∂ A(xs ) for all t ≥ 0. Since ∂ A(xs ) is closed, thus p ∈ ∂ A(xs ). (=⇒) Suppose that p ∈ ∂ A(xs ). Let Nc be a fundamental domain of W c(p), that is,⋃ t∈R ϕ(t, Nc) = W c(p) \ {p}. Let Ncε be a fundamental neighborhood of radius ε of Nc, namely Ncε = {x ∈ Rn : d(x, Nc) < ε}. As a consequence of λ-lemma, see [6], there exists a neighborhood U of p such that ⋃ t≤0 ϕ(t, Ncε ) ⊃ U \ W sloc(p). Since p ∈ ∂ A(xs ), then U ∩ A(xs ) = ∅. On the other hand, W sloc(p) ∩ A(xs ) = ∅. Thus, {U \ W sloc(p)} ∩ A(xs ) = ∅. Consequently, there is a point z ∈ Ncε and a time t such that ϕ(t , z) ∈ A(xs ). Since A(xs ) is invariant, then z ∈ A(xs ). As ε can be chosen arbitrarily small, we can find a sequence of points {zi } with zi ∈ A(xs ) for all i = 1, 2, . . . such that d(zi , Nu) → 0 when i → +∞. By construction, the sequence {zi } is bounded and therefore has a convergent subsequence. Let {zik } be a convergen...

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... closed and invariant set. Tend. Mat. Apl. Comput., 17, N. 2 (2016) “main” — 2016/9/4 — 23:01 — page 213 — #3 GOUVEIA JR, AMARAL and ALBERTO 213 With the motivation of better understanding the boundary of the stability region and getting better estimates of the stability region, characterizations of the boundary of the stability region were developed. The first characterization of the boundary of the stability region of an asymptotically stable equilibrium point xs of system (2.1) was developed in [14]. A generalization of the characterization proposed in [14] was developed in [4], under the following assumptions: (A1) All the equilibrium points on ∂ A(xs ) are hyperbolic; (A2) The stable and unstable manifolds of equilibrium points on ∂ A(xs ) satisfy the transversality condition; (A3) Trajectories on ∂ A(xs ) approach one of the equilibrium points as t → ∞. The boundary of the stability region of an asymptotically stable equilibrium point xs of system (2.1), satisfying assumptions (A1), (A2) and (A3), is the union of all stable manifolds of the equilibrium points on the boundary, in other words ∂ A(xs ) = ⋃i W s(xi ), where xi , i = 1, 2, . . . are the hyperbolic equ...

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...icient l1 < 0 and is called a subcritical Hopf equilibrium point if the first Lyapunov coefficient l1 > 0. Hopf equilibrium points can be also classified in types according to the number of eigenvalues of Dx f (p) with positive real part. A Hopf equilibrium point p of (2.1) is called a type-k Hopf equilibrium point if Dx f (p) has k (k ≤ n − 2) eigenvalues with positive real part and n − k − 2 with negative real part. In this paper, we are primarily concerned with subcritical Hopf equilibium points. If p is a subcritical Hopf equilibrium point, then the following properties are satisfied, see [18, 10]: (1) p is a type-0 subcritical Hopf equilibrium point of (2.1): (i) The (n − 2)-dimensional local stable manifold W sloc(p) of p exists, is unique, and if q ∈ W sloc(p) then ϕ(t, q) −→ p as t −→ +∞. (ii) The bidimensional local center manifold W cloc(p) of p exists, is unique, and if q ∈ W cloc(p) then ϕ(t, q) −→ p as t −→ −∞. (2) p is a type-k subcritical Hopf equilibrium point of (2.1), with 1 ≤ k ≤ n − 3: (i) The k-dimensional local unstable manifold W uloc(p) of p exists, is unique, and if q ∈ W uloc(p) then ϕ(t, q) −→ p as t −→ −∞. (ii) The (n − k − 2)-dimensional local stable manifold W...

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...f the equilibrium points on the boundary, in other words ∂ A(xs ) = ⋃i W s(xi ), where xi , i = 1, 2, . . . are the hyperbolic equilibrium points on the stability boundary ∂ A(xs ). Assumption (A3) is not a generic property of dynamical systems and needs to be checked [3]. Sufficient conditions for the satisfaction of assumption (A3) were given in [3]. The existence of an energy function is a sufficient condition to guarantee the fulfilment of assumption (A3), and, consequently, a fairly large class of dynamical systems satisfy this condition, see [3]. Although assumption (A1) is generic, see [11], studying the characterization of the stability boundary in the presence of non-hyperbolic equilibrium points is important to understand how the stability region changes as a consequence of parameter variations. These changes were already investigated in the occurrence of type-zero saddle-node bifurcations on the stability boundary [1], [2] and in the occurrence of type-k supercritical Hopf equilibrium points, with 1 ≤ k ≤ n − 2, [8, 9]. In this paper, we also study the characterization of the boundary of the stability region when assumption (A1) is violated. More specifically, we study the c...

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...\ {x}) ∩ A(xs ) = ∅. Consequently, (W u(x) \ {x}) ∩ A(xs ) = ∅, because W uloc(x) ⊂ W u(x). Let us show, under assumptions (A1”), (A2”) and (A3) that (W u(x)\{x})∩ A(xs ) = ∅ implies W u(x) ∩ A(xs ) = ∅. Let q ∈ (W u(x) \ {x}) ∩ A(xs ). If q ∈ A(xs ), then there is nothing to be proved. Suppose that q ∈ ∂ A(xs ). From condition (A3), there is an equilibrium point p ∈ ∂ A(xs ) such that ϕ(t, q) → p as t → +∞. By supposition (A1”), p is a hyperbolic equilibrium point or a subcritical Hopf equilibrium point. By the dimension of the unstable manifold of the equilibrium point, see [7], we conclude that dim W cu( p) < dim W u(x) or dim W c( p) < dim W u(x) if p is a subcritical Hopf equilibrium point or dim W u( p) < dim W u(x) if p is a hyperbolic equilibrium point. Let x be a type-1 hyperbolic equilibrium point. Consequently, dim W u( p) < 1. Hence dim W u( p) = 0 and consequently p is a type-zero hyperbolic equilibrium point. This leads us to a contradiction, because these type-zero equilibrium points cannot belong to ∂ A(xs ). Hence, q ∈ A(xs ) and therefore, W u(x) ∩ A(xs ) = ∅. Let x be a type-2 hyperbolic equilibrium point. If p is a subcritical Hopf ...

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...en in [3]. The existence of an energy function is a sufficient condition to guarantee the fulfilment of assumption (A3), and, consequently, a fairly large class of dynamical systems satisfy this condition, see [3]. Although assumption (A1) is generic, see [11], studying the characterization of the stability boundary in the presence of non-hyperbolic equilibrium points is important to understand how the stability region changes as a consequence of parameter variations. These changes were already investigated in the occurrence of type-zero saddle-node bifurcations on the stability boundary [1], [2] and in the occurrence of type-k supercritical Hopf equilibrium points, with 1 ≤ k ≤ n − 2, [8, 9]. In this paper, we also study the characterization of the boundary of the stability region when assumption (A1) is violated. More specifically, we study the characterization of the stability boundary when a subcritical Hopf non-hyperbolic equilibrium point is found on the stability boundary. 3 SUBCRITICAL HOPF EQUILIBRIUM POINT In this section, a particular type of non-hyperbolic equilibrium point, namely the subcritical Hopf equilibrium point, is studied. Particularly, the dynamic behavior in a ...

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...rto@usp.br “main” — 2016/9/4 — 23:01 — page 212 — #2 212 SUBCRITICAL HOPF EQUILIBRIUM POINTS IN THE BOUNDARY OF THE STABILITY REGION In this paper, we are interested in studying the characterization of the stability region and its boundary when the hyperbolicity condition on the boundary is violated due to the presence of nonhyperbolic equilibrium points. Some advances in this direction have already been obtained and reported in the literature. A complete characterization of the boundary of the stability region in the presence of saddle-node equilibrium points was developed in [1]. A complete characterization was also developed considering type-k supercritical Hopf equilibrium points, with k ≥ 1, on the boundary [9]. In this paper, a complete characterization of the stability boundary is developed admitting the existence of type-k subcritical Hopf nonhyperbolic equilibrium points, with k ≥ 1, on the boundary. More precisely, if xs is an asymptotically stable equilibrium point and A(xs ) is its stability region, it is proven in this paper, under mild assumptions, that: ∂ A(xs ) = ⋃ i W s(xi ) ⋃ j W s(p j ), that is, the stability boundary ∂ A(xs ) is comprised of the un...

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...losure A(xs ) is invariant and the boundary of the stability region ∂ A(xs ) is a closed and invariant set. Tend. Mat. Apl. Comput., 17, N. 2 (2016) “main” — 2016/9/4 — 23:01 — page 213 — #3 GOUVEIA JR, AMARAL and ALBERTO 213 With the motivation of better understanding the boundary of the stability region and getting better estimates of the stability region, characterizations of the boundary of the stability region were developed. The first characterization of the boundary of the stability region of an asymptotically stable equilibrium point xs of system (2.1) was developed in [14]. A generalization of the characterization proposed in [14] was developed in [4], under the following assumptions: (A1) All the equilibrium points on ∂ A(xs ) are hyperbolic; (A2) The stable and unstable manifolds of equilibrium points on ∂ A(xs ) satisfy the transversality condition; (A3) Trajectories on ∂ A(xs ) approach one of the equilibrium points as t → ∞. The boundary of the stability region of an asymptotically stable equilibrium point xs of system (2.1), satisfying assumptions (A1), (A2) and (A3), is the union of all stable manifolds of the equilibrium points on the boundary, in other...