Semiflows Strongly Focusing Monotone with Respect to High-Rank Cones: I. Generic Dynamics

Abstract

We consider a smooth semiflow strongly focusing monotone with respect to a cone of rank k on a Banach space. We obtain its generic dynamics, that is, semiorbits with initial data from an open and dense subset of any open bounded set either are pseudo-ordered or convergent to an equilibrium. For the case $k=1$, it is the celebrated Hirsch’s Generic Convergence Theorem. For the case $k=2$, we obtain the generic Poincaré-Bendixson Theorem.

1 Introduction

We investigate the generic dynamics of semiflows strongly focusing monotone with respect to a cone C of rank k on an infinite dimensional Banach space X. Roughly speaking, a cone of rank k (abbr. k-cone) is a closed subset of X containing a subspace of dimension k but no subspace of higher dimension, which is introduced by Krasnosel’skij, Lifshits and Sobolev [16] to obtain a Krein–Rutman theory on a Banach space, and also in the poineering works of Fusco and Oliva [5, 6] on the finite dimensional space. A convex cone K gives rise to a 1-cone, $K\cup (-K)$. Therefore, the class of semiflows strongly monotone with respect to k-cones includes the classical monotone semiflows originating from the groundbreaking works of Hirsch (see [8,9,10,11,12,13,14]). Due to the lack of convexity in $k (k\geqslant 2)$-cones, it is a very challenging task to study the behaviors of this general class of systems strongly monotone with respect to k-cones. Despite some progress being made (see [1,2,3,4]), their dynamics is far from being understood. Here, we introduce a slightly stronger property than strong monotonicity with respect to k-cones, that is, strongly focusing monotonicity with respect to k-cones, and study typical behaviors of most semiorbits in the topological sense in this paper.

A strongly focusing operator with respect to a k-cone C originated from Krasnosel’skij et.al [16] to prove a Krein–Rutman type theorem with respect to k-cones for a single operator, and also from Lian and Wang [19] to investigate the relationship between Multiplicative Ergodic Theorem and Krein–Rutman type Theorem for random linear dynamical systems. Roughly speaking, the image of C for a strongly focusing operator is a subset of C such that unite vectors contained in it are uniformly separated from the boundary of C (see Definition 2.1(ii)). We should point out that any strongly positive operator R with respect to C (see Definition 2.1(i)) on a finite dimensional space is strongly focusing. Therefore, the smooth semiflows strongly focusing monotone with respect to the k-cone C on an infinite dimensional space is a kind of natural extension of the smooth flows with respect to k-cones on a finite dimensional space (refer to the flows in [2]). More precisely, this class of semiflows strongly monotone with respect to the k-cone C satisfies that for each compact invariant set $\Sigma$, one can find constants $\delta ,T,\kappa >0$ such that there is a strongly focusing operator $T_{(x,y)}$ with separation index greater than $\kappa$ such that $T_{(x,y)}(x-y)=\Phi _T(x)-\Phi _T(y)$ for any $z\in \Sigma$ and $x,y\in B_{\delta }(z)$, where $B_{\delta }(z)$ is the ball centred at z with radius $\delta$ (see also Definition 2.2(ii)). This class of semiflows would have significant potential applications to the study of dynamics of nonlinear evolution equations.

There are exactly two types of nontrivial positive semiorbits for the semiflow $\Phi _t$ monotone with respect to the k-cone C (or a convex cone K): pseudo-ordered semiorbits and unordered semiorbits. A nontrivial positive semiorbit $O^+(x)=\{\Phi _t(x):\,t\ge 0\}$ is pseudo-ordered if it contains a pair of different ordered points $\Phi _{\tau }(x)$ and $\Phi _s(x)$, i.e. $\Phi _{\tau }(x)-\Phi _{s}(x)\in C\setminus \{0\}$ (or $\Phi _{\tau }(x)-\Phi _{s}(x)\in K\setminus \{0\}$); otherwise, it is called unordered.

For the classical (in the sense of Hirsch’s) monotone systems, the order reduced by a convex cone is a partial order relationship. Base on this fact, Monotone Convergence Criterion, the first key building block in Hirsch’s theory, can be established. It is to say that every precompact pseudo-ordered semiorbit converges to an equilibrium. The partial order also plays an important role in the further developments from Monotone Convergence Criterion, that includes Nonordering of Limit Sets and Limit Set Dichotomy. These results consist of the key building blocks (see [24, Theorem 2.1, p.491]) for establishing Hirsch’s Generic Convergence Theorem.

Compared with classical monotone systems, the order reduced by k-cones is a symmetric relationship, that causes the structure of the omega-limit set $\omega (x)$ of a pseudo-ordered semiorbit $O^+(x)$ is more complicated and new techniques are needed to analyze dynamics of the semiflows monotone with respect to high-rank cones. Sanchez [22] firstly treated the problem on the structure of the omega-limit set $\omega (x)$ of a pseudo-ordered orbit for flows on $\mathbb {R}^n$ strongly monotone with respect to a $k(k\ge 2)$-cone C. He used the $C^1$-closed lemma to prove that any orbit in $\omega (x)$ of a pseudo-ordered orbit is ordered, that is, the difference of any two points in any given orbit in $\omega (x)$ is in C; and further obtain a Poincaré-Bendixson theorem, that is, the omega-limit set $\omega (x)$ of a pseudo-ordered orbit containing no equilibrium is a closed orbit. For the total-ordering property of the entire set $\omega (x)$, he [22, p.1984] posed it as an open problem. In our previous work [1], we creatively utilized topological properties of continuous semiflows to study the total-ordering property for continuous semiflows strongly monotone with respect to a k-cone in a general Banach space and obtained the Order-Trichotomy (see [1, Theorem B]) for the omega-limit set $\omega (x)$ of a pseudo-ordered semiorbit. More precisely, we proved that either (a) $\omega (x)$ is ordered; or (b) $\omega (x)$ is an unordered set consisting of equilibria; or otherwise, (c) $\omega (x)$ possesses a certain ordered homoclinic property. In our previous work [1, Theorem A and C], we extended Sanchez’s results to semiflows only continuous on an infinite dimensional space.

For semiflows strongly monotone with respect to high-rank cones, the symmetry of the order and the complexity of an omega-limit set $\Omega$ cause that it is difficult to reappear the key building blocks in Hirsch’s theory. To treat the generic dynamics of flows $\Phi _t$ strongly monotone with respect to a k-cone on $\mathbb {R}^n$, we turned to analyze the local dynamics for each type of omega-limit set $\Omega$ in our previous works [2], where the types are classified by our approach of smooth ergodic arguments. More precisely, the linear skew-product flow $(\Phi _t, D\Phi _t)$ admits k-exponential separation along $\Omega$ associated with the k-cone C provided by the strong positivity of $D_x\Phi _t$ for any $x\in \Omega$ and $t>0$. Roughly speaking, this property describes that there exist k-dimensional invariant subbundle $\Omega \times (E_x)$ and k-codimensional invariant subbundle $\Omega \times (F_x)$ with respect to $(\Phi _t, D\Phi _t)$ such that $\Omega \times \mathbb {R}^n=\Omega \times (E_x)\oplus \Omega \times (F_x)$; and more, the action of $(\Phi _t, D\Phi _t)$ on $\Omega \times (E_x)$ dominates the one on $\Omega \times (F_x)$ as $t\rightarrow \infty$ (see Definition 2.3 or its versions for random dynamics in [18, 19]). The related crucial tool is the k-Lyapunov exponent $\lambda _{kx}$ of $x\in \Omega$ (defined as $\lambda _{kx}=\limsup \limits _{t\rightarrow +\infty }\frac{\log m(D_x\Phi _t|_{E_x})}{t}$, see also Definition 2.4 and (2.1)), which describes the action’s growth rate of $(\Phi _t, D\Phi _t)$ on the k-dimensional subbundle $\Omega \times (E_x)$. The theory on Lyapunov exponents and Multiplicative Ergodic Theorem ensures [7, 17,18,19,20,21, 26] that $\lambda _{kx}$ is actually the limit for “most" points $x\in \Omega$; such points for which $\lambda _{kx}$ is the limit are said to be regular and other points are said to be irregular. According to the sign of the k-Lyapunov exponents of the regular/irregular points on any given omega-limit set $\Omega$, three are three types of $\Omega$: (i) $\lambda _{kx}>0$ for any point x in $\Omega$; (ii) $\lambda _{kx}>0$ for any regular point $x\in \Omega$ and $\lambda _{kz}\le 0$ for some irregular point $z\in \Omega$; (iii) $\lambda _{kx}\le 0$ for some regular point $x\in \Omega$. By discussing the local behaviors of each type of omega-limit sets, we obtain the finite dimensional version of Generic dynamics theorem (see [2, Theorem A]); and further combinate it with the Poincaré-Bendixson theorem (see Lemma 2.6 and also [1, Theorem C]) of the omega-limit set of a pseudo-ordered semiorbit to get the generic Poincaré-Bendixson theorem (see [2, Theorem B]) on $\mathbb {R}^n$.

Our purpose in this paper is to investigate the infinite dimensional version of generic dynamics of semiflow $\Phi _t$ strongly focusing monotone with respect to a k-cone C on a Banach space. We prove that

• For generic (open and dense) positive semiorbits either are pseudo-ordered or converge to an equilibrium.

• Whenever $k=2$, for generic points, the omega-limit set containing no equilibrium is a periodic orbit.

By the strong positivity of $D_x\Phi _t$ in Definition 2.2(i), for each compact invariant set $\Sigma$ on which $\Phi _t$ admits a flow extension, the linear skew-product semiflow $(\Phi _t,D\Phi _t)$ admits k-exponential separation $\Sigma \times X=\Sigma \times (E_x)\oplus \Sigma \times (F_x)$ along $\Sigma$ associated with C. Here, $\Phi _t$ is said to admit a flow extension on $\Sigma$, if there is a flow $\tilde{\Phi }_t$ such that $\tilde{\Phi }_t(x)=\Phi _t(x)$ for any $x\in \Sigma$ and $t\ge 0.$ Since the unite ball in an infinite dimensional Banach space is lack of compactness, unite vectors in the k-codimensional invariant subbundle $\Sigma \times (F_x)$ with respect to $(\Phi _t,D\Phi _t)$ are not uniformly far away from the boundary $\partial C$ of the k-cone C. The method in [2] is not effective to estimate the proportion of the projection onto the fibres $E_x$ and $F_x$ with $x\in \Sigma$ for a nonzero vector $v\in C$ in an infinite dimensional space. We turn to estimate the proportion of the projection onto the fibres $E_x$ and $F_x$ with $x\in \Sigma$ for the difference $\Phi _t(\tilde{x})-\Phi _t(\tilde{y})$ of a pair of ordered distinct points $\Phi _t(\tilde{x})$ and $\Phi _t(\tilde{y})$ by utilizing the strongly focusing monotonicity in Definition 2.2(ii). By this novel approach, we analyze the local dynamical features for each type of omega-limit sets and furthermore deduce the infinite dimensional version of generic dynamics. For case $k=2$, the generic Poincaré-Bendixson theorem are also obtained.

The paper is organized as follows. In Sect. 2, we give some notations and summarize the preliminary results. In Sect. 3, we present the main results on the infinite dimensional version of Generic dynamics and generic Poincaré-Bendixson theorem for semiflows strongly focusing monotone with respect to the k-cone C. In Sect. 4, we discuss the local behaviors for each type of omega-limit sets. In Sect. 5, we prove our main results.

2 Notations and Preliminary Results

In this section, we give some preliminary knowledge to be used in the next sections. We start with basic nations and definitions on semiflows strongly focusing monotone with respect to high-rank cones. We then introduce the k-exponential separation and k-Lyapunov exponents with some crucial properties of them.

2.1 Semiflows Strongly Monotone with Respect to High-Rank Cones

Let $(X,\Vert \cdot \Vert )$ be a Banach space equipped with a norm $\Vert \cdot \Vert$. A semiflow on X is a continuous map $\Phi :\mathbb {R}^+\times X\rightarrow X$ with $\Phi _0=\textrm{Id}$ and $\Phi _t\circ \Phi _s=\Phi _{t+s}$ for $t,s\ge 0$. Here, $\mathbb {R}^+=[0,+\infty )$, $\Phi _t(\cdot )=\Phi (t,\cdot )$ for $t\ge 0$, and $\textrm{Id}$ is the identity map on X. A semiflow $\Phi _t$ on X is called $C^{1,\alpha }$-smooth if $\Phi |_{\mathbb {R}^+\times X}$ is a $C^{1,\alpha }$-map (a $C^1$-map with a locally $\alpha$-Hölder derivative) with $\alpha \in (0,1]$. The derivative of $\Phi _t$ with respect to x, at (t, x), is denoted by $D_x\Phi _t$.

Let $x\in X$, the positive semiorbit of x is denoted by $O^+(x)=\{\Phi _t(x):t\ge 0\}$. A negative semiorbit (resp. full-orbit) of x is a continuous function $\psi :\mathbb {R}^-=\{t\in \mathbb {R}|t\le 0\}\rightarrow X$ (resp. $\psi :\mathbb {R}\rightarrow X$) such that $\psi (0)=x$ and, for any $s\le 0$ (resp. $s\in \mathbb {R}$), $\Phi _t(\psi (s))=\psi (t+s)$ holds for $0\le t\le -s$ (resp. $0\le t$). Clearly, if $\psi$ is a negative semiorbit of x, then $\psi$ can be extended to a full-orbit $\tilde{\psi }(t)$ such that $\tilde{\psi }(t)= \psi (t)$ for $t\le 0$ and $\tilde{\psi }(t)=\Phi _t(x)$ for $t\ge 0$. On the other hand, any full orbit of x when restricted on $\mathbb {R}^-$ is a negative semiorbit of x. Since $\Phi _t$ is just a semiflow, a negative semiorbit of x may not exist, and it is not necessary to be unique even if one exists.

An equilibrium (also called a trivial orbit) is a point x for which $O^+(x)=\{x\}$. Let E be the set of all equilibria w.r.t. $\Phi _t$. A nontrivial positive semiorbit $O^+(x)$ is said to be a periodic orbit if $\Phi _T(x)=x$ for a $T>0$. The nontrivial semiorbit $O^+(x)$ is said to be a T-periodic orbit if there is a $T>0$ such that $\Phi _T(x)=x$ and $\Phi _t(x)\ne x$ for any $t\in (0,T)$, where T is called the minimal period of $O^+(x)$.

A subset $\Sigma \subset X$ is called positively invariant with respect to $\Phi _t$ (for short, positively invariant) if $\Phi _{t}(\Sigma )\subset \Sigma$ for any $t\in \mathbb {R}^+$, and is called invariant if $\Phi _{t}(\Sigma )=\Sigma$ for any $t\in \mathbb {R}^+$. Clearly, for any $x\in \Sigma$, there exists a negative semiorbit of x, provided that $\Sigma$ is invariant. Let $\Sigma \subset X$ be an invariant set. $\Phi _t$ is said to admit a flow extension on $\Sigma$, if there is a flow $\tilde{\Phi }_t$ such that $\tilde{\Phi }_t(x)=\Phi _t(x)$ for any $x\in \Sigma$ and $t\ge 0.$

The omega-limit (abbr. $\omega$-limit) set $\omega (x)$ of $x\in X$ is defined by $\omega (x)=\cap _{s\ge 0}\overline{\cup _{t\ge s}\Phi _t(x)}$. If $O^+(x)$ is precompact, then $\omega (x)$ is nonempty, compact, connected and invariant. Given a subset $D\subset X$, the positive semiorbit $O^+(D)$ of D is defined as $O^+(D)=\bigcup \limits _{x\in D}O^+(x)$. A subset D is called $\omega$-compact if $O^+(x)$ is precompact for each $x\in D$ and $\bigcup \limits _{x\in D}\omega (x)$ is precompact. Clearly, D is $\omega$-compact provided by the compactness of $\overline{O^+(D)}$.

A closed set $C\subset X$ is called a cone of rank-k (abbr. k-cone) if

• (i) For any $v\in C$ and $l\in \mathbb {R},$ $lv\in C$;

• (ii) $\max \{\dim W:C\supset W \text { linear subspace}\}=k.$

Moreover, the integer $k(\ge 1)$ is called the rank of C. A k-cone $C\subset X$ is said to be solid if its interior $\text {Int}C\ne \emptyset$; and C is called k-solid if there is a k-dimensional linear subspace W such that $W\setminus \{0\}\subset \text {Int}C$. Given a k-cone $C\subset X$, we say that C is complemented if there exists a k-codimensional subspace $H^{c}\subset X$ such that $H^{c}\cap C=\{0\}$. For two points $x,y\in X$, we call that x and y are ordered, denoted by $x\thicksim y$, if $x-y\in C$. Otherwise, x, y are called to be unordered, denoted by $x\rightharpoondown y$. The pair $x,y\in X$ are said to be strongly ordered, denoted by $x\thickapprox y$, if $x-y\in \text {Int}C$. A nonempty set $W\subset X$ is called ordered if $x\thicksim y$ for any $x,y\in W$ and it is called (resp. strongly ordered) unordered if it is not a singleton and (resp. $x\thickapprox y$) $x\rightharpoondown y$ for any two distinct points $x,y\in W$.

Let $d(x,y)=\Vert x-y\Vert$ for any $x,y\in X$ and $d(x,B)=\inf \limits _{y\in B}d(x,y)$ for any $x\in X, B\subset X$.

Throughout this paper, we assume C is a complemented k-solid cone and $\Phi _t$ with compact x-derivative $D_x\Phi _t$ for any $x\in X$ and $t>0$ admits a flow extension on each nonempty omega-limit set $\omega (x)$.

Definition 2.1

(i) A linear operator $R\in L(X)$ is called strongly positive with respect to C if $R\,\big (C\setminus \{0\}\big )\subset \text {Int} C$.

(ii) A linear operator R $\in L(X)$ is called strongly focusing with respect to C if $0 \notin R(C\setminus \{0\})$ and there is a $\kappa >0$ such that

$$\begin{aligned} \underline{\text {dist}}(RC, X\setminus C)=\kappa , \end{aligned}$$

where $\underline{\text {dist}}(L_1,L_2)$ is the separation index between set $L_1$ and $L_2$ defined by

$$\begin{aligned} \underline{\text {dist}}(L_1,L_2)=\inf \limits _{v\in L_1, \Vert v\Vert =1}\{\inf \limits _{u\in L_2}\Vert v-u\Vert \}. \end{aligned}$$

Here, $\kappa$ is also called the separation index of R.

Remark 2.1

(i) A strongly focusing operator is automatically a strongly positive operator.

(ii) Let R be a strongly positive operator w.r.t. C on $\mathbb {R}^n$. Then, R is also strongly focusing w.r.t. C.

A semiflow $\Phi _t$ on X is called monotone with respect to C if

$$\begin{aligned} \Phi _t(x)\thicksim \Phi _t(y)\,\text { whenever } x\thicksim y \text { and } t\ge 0; \end{aligned}$$

and $\Phi _t$ is called strongly monotone with respect to C if $\Phi _t$ is monotone with respect to C and

$$\begin{aligned} \Phi _t(x)\approx \Phi _t(y)\,\text { whenever } x\ne y, x\thicksim y \text { and } t>0. \end{aligned}$$

A nontrivial positive semiorbit $O^+(x)$ is called pseudo-ordered (also called of Type-I), if there exist two distinct points $\Phi _{t_1}(x),\Phi _{t_2}(x)$ in $O^+(x)$ such that $\Phi _{t_1}(x)\thicksim \Phi _{t_2}(x)$. Otherwise, $O^+(x)$ is called unordered (also called of Type-II). Hereafter, we let

$$\begin{aligned} Q=\{x\in X: O^+(x) \text { is pseudo-ordered}\}. \end{aligned}$$

Definition 2.2

A semiflow $\Phi _t$ is called strongly focusing monotone with respect to C, if it satisfies:

(i) It is $C^1$-smooth and strongly monotone with respect to C such that the x-derivative $D_x\Phi _t$ of $\Phi _t \,(t>0)$ is strongly positive with respect to C for any $x\in X$;

(ii) For each compact invariant set $\Sigma$ with respect to $\Phi _t$, one can find constants $\delta ,T,\kappa >0$ such that there exists a strongly focusing operator $T_{(x,y)}$ with separation index greater than $\kappa$ such that $T_{(x,y)}(x-y)=\Phi _T(x)-\Phi _T(y)$ for any $z\in \Sigma$ and $x, y\in B_{\delta }(z)$, where $B_{\delta }(z)$ is the closed ball centred at z with radius $\delta$.

Remark 2.2

Let $\Phi _t$ be a $C^1$-smooth flow strongly monotone w.r.t. C on $\mathbb {R}^n$, whose x-derivative $D_x\Phi _t$ is strongly positive w.r.t. C for any $x\in \mathbb {R}^n$ and $t>0$. Then, $\Phi _t$ is strongly focusing monotone w.r.t. C.

Remark 2.3

The condition “for each compact invariant set $\Sigma$" in the strongly focusing monotonicity in Definition 2.2(ii) can be relaxed and becomes “for each omega-limit set $\omega (x)$" in the proof of the results in this paper.

Remark 2.4

Let $\tilde{\Sigma }=\text {Co}\{B_{\delta }(\Sigma )\}\times \text {Co}\{B_{\delta }(\Sigma )\}$. Here, $B_{\delta }(\Sigma )=\{v\in X: d(v,\Sigma )\le \delta \}$ and $\text {Co}\{B_{\delta }(\Sigma )\}$ is the convex hull of $B_{\delta }(\Sigma )$. Let $T_{(x,y)}=\int _{0}^{1}D_{y+s(x-y)}\Phi _Tds$ for any $(x,y)\in \tilde{\Sigma }$. Then, one has $T_{(x,y)}(x-y)=\Phi _{T}(x)-\Phi _{T}(y)$. Let $\kappa >0$. Compared with Definition 2.2(ii), the following condition has more restriction.

$$\begin{aligned} \{T_{(x,y)}\}_{(x,y)\in \tilde{\Sigma }}\,\,\text {{consists of} strongly focusing operators with separation index greater than}\,\,\kappa . \end{aligned}$$

(*)

Now, we give several useful results on semiflows strongly monotone with respect to C.

Lemma 2.5

Assume that $\Phi _t$ is strongly monotone with respect to C. If $x\thicksim y$ and there is a sequence $t_n\rightarrow \infty$ such that $\Phi _{t_n}(x)\rightarrow z$ and $\Phi _{t_n}(y)\rightarrow z$, then O(z) is nontrivial and ordered, or z is an equilibrium.

Proof

See [1, Lemma 4.3]. $\square$

Lemma 2.6

Assume that $\Phi _t$ is strongly monotone with respect to the k-cone C with $k=2$ and $O^+(x)$ be a precompact pseudo-ordered semiorbit. If $\omega (x)\cap E=\emptyset$, $\omega (x)$ is a periodic orbit.

Proof

See [1, Theorem C]. $\square$

2.2 k-Exponential Separation and k-Lyapunov Exponents

Let G(k, X) be the Grassmanian of k-dimensional linear subspaces of X, which consists of all k-dimensional linear subspaces in X. G(k, X) is a completed metric space by endowing the gap metric (see, for example, [15, 17]). More precisely, for any nontrivial closed subspaces $L_1,L_2\subset X$, define that

$$\begin{aligned} d(L_1,L_2)=\max \left\{ \sup _{v\in L_1\cap S}\inf _{u\in L_2\cap S}\Vert v-u\Vert , \sup _{v\in L_2\cap S}\inf _{u\in L_1\cap S}\Vert v-u\Vert \right\} , \end{aligned}$$

where $S=\{v\in X:\Vert v\Vert =1\}$ is the unit sphere. For a solid k-cone $C\subset X$, we denote by $\Gamma _k(C)$ the set of k-dimensional subspaces inside C, that is,

$$\begin{aligned} \Gamma _k(C)=\{L\in G(k,X):\,L\subset C\}. \end{aligned}$$

Let $\Sigma \subset X$ be a compact invariant subset w.r.t. $\Phi _t$ on which $\Phi _t$ admits a flow extension. We consider the linear skew-product semiflow $(\Phi _t,\,D\Phi _t)$ on $\Sigma \times X$, which is defined as $(\Phi _t, D\Phi _t)(x,v)=(\Phi _t(x),D_x\Phi _t v)$ for any $(x,v)\in \Sigma \times X$ and $t\in \mathbb {R}^+$. Here, $D_x\Phi _t$ is the Fréchet derivative of $\Phi _t$ at $x\in \Sigma$. Let $\{E_x\}_{x\in \Sigma }$ be a family of k-dimensional subspaces of X. We call $\Sigma \times (E_x)$ a k-dimensional continuous vector bundle on $\Sigma$ if the map $\Sigma \mapsto G(k,\,X): x\mapsto E_x$ is continuous. Let $\{F_x\}_{x\in \Sigma }$ be a family of k-codimensional closed vector subspaces of X. We call $\Sigma \times (F_x)$ a k-codimensional continuous vector bundle on $\Sigma$ if there is a k-dimensional continuous vector bundle $\Sigma \times (L_x)\subset \Sigma \times X^*$ such that the kernel $\textrm{Ker}(L_x)=F_x$ for each $x\in \Sigma$. Here, $X^*$ is the dual space of X.

Let $\Sigma \times (E_x)$ be a k-dimensional continuous vector bundle on $\Sigma$ and $\Sigma \times (F_x)$ be a k-codimensional continuous vector bundle on $\Sigma$ such that $X=E_x\oplus F_x$ for all $x\in \Sigma$. We define the family of projections associated with the decomposition $X=E_x\oplus F_x$ as $\{\Pi ^{E_x}\}_{x\in \Sigma }$ where $\Pi ^{E_x}$ is the linear projection of X onto $E_x$ along $F_x$ for each $x\in \Sigma$. Write $\Pi ^{F_x}=I-\Pi ^{E_x}$ for each $x\in \Sigma$. Clearly, $\Pi ^{F_x}$ is the linear projection of X onto $F_x$ along $E_x$. Moreover, both $\Pi ^{E_x}$ and $\Pi ^{F_x}$ are continuous with respect to $x\in \Sigma$. We say that the decomposition $X=E_x\oplus F_x$ is invariant with respect to $(\Phi _t,\,D\Phi _t)$ if $D_x\Phi _tE_x=E_{\Phi _t(x)}$, $D_x\Phi _tF_{x}\subset F_{\Phi _t(x)}$ for each $x\in \Sigma$ and $t\ge 0$.

Definition 2.3

Let $\Sigma \subset X$ be a compact invariant subset w.r.t. $\Phi _t$ on which $\Phi _t$ admits a flow extension. The linear skew-product semiflow $(\Phi _t,\,D\Phi _t)$ admits a k-exponential separation along $\Sigma$ (for short, k-exponential separation), if there are k-dimensional continuous vector bundle $\Sigma \times (E_x)$ and k-codimensional continuous vector bundle $\Sigma \times (F_x)$ such that

• (i) $X=E_x\oplus F_x$ for any $x\in \Sigma$;

• (ii) $D_x\Phi _tE_x=E_{\Phi _t(x)}$, $D_x\Phi _tF_{x}\subset F_{\Phi _t(x)}$ for any $x\in \Sigma$ and $t>0$;

• (iii) there are constants $M>0$ and $0<\gamma <1$ such that

  $$\begin{aligned} \Vert D_x\Phi _tw\Vert \le M\gamma ^{t}\Vert D_x\Phi _tv\Vert \end{aligned}$$

  for all $x\in \Sigma$, $w\in F_x\cap S$, $v\in E_{x}\cap S$ and $t\ge 0$, where $S=\{v\in X:\Vert v\Vert =1\}$. Let $C\subset X$ be a complemented k-solid cone. If, in addition,

• (iv) $E_x\subset \textrm{Int}C\cup \{0\}$ and $F_x\cap C=\{0\}$ for any $x\in \Sigma$,

then $(\Phi _t,\,D\Phi _t)$ is said to admit a k-exponential separation along $\Sigma$ associated with C.

Since $E_x$ is k dimensional for any $x\in \Sigma$, one can define the infimum norm $m(D_x\Phi _t|_{_{E_x}})$ of $D_x\Phi _t$ restricted on $E_x$ for each $x\in \Sigma$ and $t\ge 0$ as follows:

$$\begin{aligned} m(D_x\Phi _t|_{_{E_x}})=\inf \limits _{v\in E_x\cap S}\Vert D_x\Phi _tv\Vert , \end{aligned}$$

(2.1)

where $S=\{v\in X: \Vert v\Vert =1\}$.

Definition 2.4

For each $x\in \Sigma$, the k-Lyapunov exponent is defined as

$$\begin{aligned} \lambda _{kx}=\limsup _{t\rightarrow +\infty }\dfrac{\log m(D_x\Phi _t|_{_{E_x}})}{t}. \end{aligned}$$

(2.2)

A point $x\in \Sigma$ is called a regular point if $\lambda _{kx}=\lim \limits _{t\rightarrow +\infty }\dfrac{\log m(D_x\Phi _t|_{_{E_x}})}{t}$.

Lemma 2.7

Assume that $\Sigma \subset X$ be a compact invariant subset with respect to $\Phi _t$ on which $\Phi _t$ admits a flow extension. Assume that $\Phi _t$ is $C^1$-smooth such that $D_x\Phi _t (C\setminus \{0\})\subset \text {Int}\,C$ for any $x\in \Sigma$ and $t>0$. Then, $(\Phi _t,D\Phi _t)$ admits a k-exponential separation along $\Sigma$ associated with C.

Proof

See Tere$\check{s}\check{c}$ák [25, Corollary 2.2]. One may also refer to Tere$\check{s}\check{c}$ák [25, Theorem 4.1]. $\square$

Now, we give some crucial lemmas for $(\Phi _t,D\Phi _t)$ admitting a k-exponential separation $X=E_x\oplus F_x$ along a compact invariant set $\Sigma$ associated with C, which satisfies (i)-(iv) in Definition 2.3.

Lemma 2.8

There exists a constant $\delta ^{\prime }>0$ such that

$$\begin{aligned} \{v\in X: d(v,\,E_x\cap S)\le \delta ^{\prime }\}\subset \text {Int}\,C \,\,\,\text {for any}\,\,x\in \Sigma . \end{aligned}$$

Proof

See [2, Lemma 3.3]. We here point out that all arguments in [2, Lemma 3.3] still remain valid for $C^{1}$-smooth semiflow $\Phi _t$ on a Banach space. $\square$

Lemma 2.9

• (i) The projections $\Pi ^{E_x}$ and $\Pi ^{F_x}$ are bounded uniformly for $x\in \Sigma$.

• (ii) There exists a constant $C_1>0$ such that, if $v\in X\setminus \{0\}$ satisfies $\Vert \Pi ^{E_x}(v)\Vert \ge C_1 \Vert \Pi ^{F_x}(v)\Vert$ for some $x\in \Sigma$, then $v\in \text {Int}\,C$.

Proof

See [2, Lemma 3.5(i) and (ii)]. The arguments in [2, Lemma 3.5(i) and (ii)] are also effective for the semiflow $\Phi _t$ on a Banach space. $\square$

Lemma 2.10

Let $x\in \Sigma$. Then

• (i) If $w\in F_x\setminus \{0\}$, then $\lambda (x,w)\le \lambda _{kx}+\log (\gamma )$, where $\lambda (x,w)=\limsup \limits _{t\rightarrow \infty } \frac{\log \Vert D_x\Phi _t w\Vert }{t}.$

• (ii) Let x be a regular point. If $\lambda _{kx}\le 0$, then there exists a number $\beta \in (\gamma ,1)$ such that for any $\epsilon >0$, there is a constant $C_{\epsilon }>0$ such that

$$\begin{aligned} \Vert D_{\Phi _{t_1} (x)}\Phi _{t_2}w\Vert \le C_{\epsilon }e^{\epsilon t_1}\beta ^{t_2}\Vert w\Vert \end{aligned}$$

for any $w\in F_{\Phi _{t_1}(x)}\setminus \{0\}$ and $t_1,t_2>0$.

Proof

The results are directly implied by repeating all arguments in [2, Lemma 3.6] $\square$

Remark 2.11

Lemma 2.8–2.10 are the infinite demensional version of [2, Lemma 3.3, 3.5(i)–(ii), 3.6]. Lemma 2.5 and 2.8–2.10 are crucial tools for the arguments of Theorem 4.4 and Lemma 4.5.

3 Main Results

Let $C_E=\{x\in X:\,\,\omega (x)\,\,\text {is a singleton}\}.$

Theorem A

(Generic dynamics thoerem) Assume that $\Phi _t$ is a $C^{1,\alpha }$-smooth semiflow strongly focusing monotone with respect the k-cone C. Let $\mathcal {D}\subset X$ be an open bounded set such that $\mathcal {O^+}(\mathcal {D})$ is precompact. Then $\textrm{Int}(Q\cup C_E)$ (interior in X) is dense in $\mathcal {D}$.

Remark 3.1

Theorem A states that, for smooth semiflow $\Phi _t$ strongly focusing monotone with respect to the k-cone C, generic (open and dense) positive semiorbits either are pseudo-ordered or convergent to equilibria. If the rank $k=1$, Theorem A automatically implies Hirsch’s Generic Convergence Theorem due to the Monotone Convergence Criterion.

Theorem B

(Generic Poincaré-Bendixson theorem) Assume that $\Phi _t$ is a $C^{1,\alpha }$-smooth semiflow strongly focusing monotone with respect to the k-cone C. Let $k=2$ and $\mathcal {D}\subset X$ be an open bounded set such that $\mathcal {O}^+(\mathcal {D})$ is precomact. Then, for generic (open and dense) points $x\in \mathcal {D}$, the omega-limit set $\omega (x)$ containing no equilibria is a periodic orbit.

4 Local Behaviors of Omega-Limit Sets

Due to Lemma 2.7, we hereafter always assume that for any compact invariant set $\Sigma$ on which $\Phi _t$ admits a flow extension, the linear skew-product semiflow $(\Phi _t,D\Phi _t)$ admits a k-exponential separation along $\Sigma$ such that $X=E_x\oplus F_x$ for any $x\in \Sigma$. $\Sigma \times (E_x)$ and $\Sigma \times (F_x)$ are the corresponding k-dimensional and k-codimensional continuous invariant vector subbundles. In this paper, we attempt to extend the works on generic dynamics from classical monotone systems w.r.t. convex cones (see [23]) and flows strongly monotone w.r.t. k-cones (see [2]) to the semiflows strongly focusing monotone w.r.t. the k-cone C on an infinite dimensional Banach space.

We define the set of regular points on nonempty compact $\omega (x)$ as:

$$\begin{aligned} \omega _0(x)=\{z\in \omega (x):\,z\,\, \text {is a regular point}\}.\end{aligned}$$

(4.1)

Due to the Multiplicative Ergodic Theorem (cf. [20, Theorem A]) and the similiar arguments for [26, Proposition 4.11], $\omega _0(x)$ is non-empty. Moreover, it is easy to see that any equilibrium in $\omega (x)$ is regular and hence, is contained in $\omega _0(x)$. By utilizing the k-Lyapunov exponents on $\omega (x)$, we classify the omega-limit sets into three types and obtain their local behaviors.

Firstly, we prove that if $\lambda _{kz}>0$ for any $z\in \omega (x)$, then x is highly unstable (see Lemma 4.2), and meanwhile, it belongs to the closure $\overline{Q}$ (see Theorem 4.3). We secondly show that if $\lambda _{k\tilde{z}}>0$ for any regular point $\tilde{z}\in \omega _0(x)$ and there is an irregular point z with $\lambda _{kz}\le 0$, then $x\in \overline{Q}$ (see Theorem 4.4). We finally show that if $\omega (x)$ contains a regular point z such that $\lambda _{kz}\le 0$, then either $x\in Q$ or $\omega (x)$ is a singleton (see Theorem 4.6).

We start with discussion on the case that $\lambda _{kz}>0$ for any point $z\in \omega (x)$. Before going further, we give two technical lemmas.

Lemma 4.1

If $\lambda _{kz}>0$ for any $z\in \omega (x)$, then for any constant $\kappa >0$, there is a local constant (hence bounded) function $\nu _{\kappa }(z)$ on $\omega (x)$ (depending on $\kappa$) such that

$$\begin{aligned} \begin{aligned}&\frac{\Vert D_z\Phi _{\nu _{\kappa }(z)}w_F\Vert }{\Vert D_z\Phi _{\nu _{\kappa }(z)}w_E\Vert } <\frac{\kappa }{2(1+\kappa )},\\ {}&\Vert D_z\Phi _{\nu _{\kappa }(z)}w_E\Vert >\frac{4}{\kappa } \end{aligned} \end{aligned}$$

(4.2)

for any $z\in \omega (x)$ and $w_E\in E_z\cap S$ and $w_F\in F_z\cap S$, where $S=\{v\in X: \Vert v\Vert =1\}$.

Proof

By the definition of $\lambda _{kz}$, for each $z\in \omega (x)$, there is a sequence $t_n\rightarrow +\infty$ such that

$$\begin{aligned} \Vert D_z\Phi _{t_n}w_E\Vert >e^{\frac{\lambda _{kz}}{2}t_n} \end{aligned}$$

for any $w_E\in E_z\cap S$. Furthermore, the definition of k-exponential separation along $\Sigma$ indicates that there exist $M>0$ and $\gamma \in (0,1)$ such that

$$\begin{aligned} \frac{\Vert D_z\Phi _{t}w_F\Vert }{\Vert D_z\Phi _{t}w_E\Vert }< M\gamma ^t \end{aligned}$$

for any $t>0$ and $w_E\in E_z\cap S,\,w_F\in F_z\cap S$. Since $\lambda _{kz}>0$, one can find a $N_{\kappa }(z)>0$ such that

$$\begin{aligned} \frac{\Vert D_z\Phi _{t_n}w_F\Vert }{\Vert D_z\Phi _{t_n}w_E\Vert }<\frac{\kappa }{2(1+\kappa )} \end{aligned}$$

and

$$\begin{aligned} \Vert D_z\Phi _{t_n}w_E\Vert >\frac{4}{\kappa } \end{aligned}$$

for any $t_n>N_{\kappa }(z)$ and $w_E\in E_z\cap S$.

Therefore, for each $z\in \omega (x)$, one can associate with a number $\nu _{\kappa }(z)\ge N_{\kappa }(z)$ such that (4.2) holds for any $w_E\in E_z\cap S$ and $w_F\in F_z\cap S$. Moreover, together with the compactness of $\omega (x)$ and the smoothness of $\Phi _t$, one can further take such $\nu _{\kappa }(z)$ as a local constant (hence bounded) function. We have completed the proof. $\square$

Lemma 4.2

Assume that $\lambda _{kz}>0$ for any $z\in \omega (x)$. There exists a constant $\delta ^{\prime \prime }>0$ such that

$$\begin{aligned} \limsup _{t\rightarrow +\infty }\Vert \Phi _t(y)-\Phi _t(x)\Vert \ge \delta ^{\prime \prime }, \end{aligned}$$

whenever y satisfies $y\ne x$ and $y\thicksim x$.

Proof

Since $\Phi _t$ is strongly focusing monotone w.r.t. C, one can find constants $\delta ,T,\kappa >0$ such that there exists a strongly focusing operator $T_{(x,y)}$ with separation index greater than $\kappa$ such that $T_{(x,y)}(x-y)=\Phi _T(x)-\Phi _T(y)$ for any $z\in \omega (x)$ and $x,y\in B_{\delta }(z)$, where $B_{\delta }(z)$ is the closed ball centred at z with radius $\delta$.

For any given $y\in X$ such that $y\ne x$ and $y\thicksim x$, let $\tilde{y}_{t}=\Phi _{t}(y)$ and $\tilde{x}_{t}=\Phi _{t}(x)$ for $t>0$. By the strongly focusing monotonicity of $\Phi _t$, one has $\tilde{y}_{t}\ne \tilde{x}_{t}$ for any $t\in \mathbb {R}^+$. Since $\omega (x)$ attracts x, one can take a curve $\{z_t\}_{t>0}\subset \omega (x)$ such that

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Vert \tilde{x}_t-z_t\Vert =0. \end{aligned}$$

(4.3)

Hence, there exists a $\tilde{T}_y>T$ such that $\Vert \tilde{x}_{t-T}-z_{t-T}\Vert <\frac{\delta }{2}$ for any $t>\tilde{T}_y$. Moreover, if $\tilde{y}_{t-T}\in {B_{\delta }(z_{t-T})}$ for some $t>\tilde{T}_y$, then there is a strongly focusing operator $T_{\tilde{y}_{t-T},\tilde{x}_{t-T}}$ with separation index greater than $\kappa$ such that

$$\begin{aligned} \tilde{y}_t-\tilde{x}_t=T_{\tilde{y}_{t-T},\tilde{x}_{t-T}}(\tilde{y}_{t-T}-\tilde{x}_{t-T}). \end{aligned}$$

By Lemma 2.7, $(\Phi _t, D\Phi _t)$ admits a k-exponential separation along $\omega (x)$ associated with C. Denoted by $\omega (x)\times (E_z)$ and $\omega (x)\times (F_z)$ the corresponding k-dimensional invariant subbundle k-codimensional invariant subbundle respectively. Recall that $\Pi ^{E_z}$ is the projection onto $E_z$ along $F_z$ and $\Pi ^{F_z}=I-\Pi ^{E_z}$. Clearly, for $t>\tilde{T}_y$ and $\tilde{y}_{t-T}\in B_{\delta }(z_{t-T})$, one has $d(\frac{\tilde{y}_t-\tilde{x}_t}{\Vert \tilde{y}_t-\tilde{x}_t\Vert },X\setminus C)\ge \kappa$ and hence,

$$\begin{aligned} \begin{aligned}&\frac{\Vert \Pi ^{E_{z_t}}(\tilde{y}_t-\tilde{x}_t)\Vert }{\Vert \tilde{y}_t-\tilde{x}_t\Vert }\ge \kappa ,\\&\frac{\Vert \Pi ^{F_{z_t}}(\tilde{y}_t-\tilde{x}_t)\Vert }{\Vert \Pi ^{E_{z_t}}(\tilde{y}_t-\tilde{x}_t)\Vert }\le \frac{1+\kappa }{\kappa }. \end{aligned} \end{aligned}$$

(4.4)

Let $\nu _{\kappa }$ be the local constant function mentioned in Lemma 4.1 and $m_{\kappa }=\max \limits _{z\in \omega (x)}\{\nu _{\kappa }(z)\}$. By the smoothness of $\Phi _t$ and the compactness of $\omega (x)$, there is a $\delta ^{''}\in (0,\frac{\delta }{2})$ such that

$$\begin{aligned} \Vert D_{y_1}\Phi _t-D_{y_2}\Phi _t\Vert <\frac{1}{2} \end{aligned}$$

(4.5)

for any $t\in [0,m_{\kappa }]$ and $y_1,y_2\in \mathcal {B}_{2\delta ^{''}}(\omega (x))$ satisfying $\Vert y_1-y_2\Vert <2\delta ^{''}$, where $\mathcal {B}_{2\delta ^{''}}(\omega (x))=\{v\in X:\,d(v,\omega (x))\le 2\delta ^{''}\}$.

Now, we will prove that $\delta ^{''}$ is the desired constant. Prove by contrary. Suppose that

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }\Vert \tilde{y}_t-\tilde{x}_t\Vert <\delta ^{''} \end{aligned}$$

for some $y\in X$ satisfying $y\ne x$ and $y\thicksim x$. Then, one can find a $N_1>\tilde{T}_y$ such that

$$\begin{aligned} \Vert \tilde{y}_{t-T}-\tilde{x}_{t-T}\Vert<\delta ^{''}\,\,\text {and}\,\,\Vert \tilde{x}_{t-T}-z_{t-T}\Vert <\delta ^{''} \end{aligned}$$

for any $t\ge N_1$. Hence, $\tilde{y}_{t-T},\tilde{x}_{t-T}\in \mathcal {B}_{\delta }(z_{t-T})$ and (4.4) hold for any $t>N_1$. Take $\tau _1\ge N_1$ and $\tau _{n+1}=\tau _{n}+\nu _{\kappa }(z_{\tau _n})$ for $n=1,2,\cdots$. Denoted by $y_{\tau _n}=\Phi _{\tau _n}(y)$, $x_{\tau _n}=\Phi _{\tau _n}(x)$ and $\Pi ^{E_{\tau _n}}=\Pi ^{E_{z_{\tau _n}}}$, $\Pi ^{F_{\tau _n}}=\Pi ^{F_{z_{\tau _n}}}$. Then, one has

$$\begin{aligned} \begin{aligned} \Vert D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}(y_{\tau _n}-x_{\tau _n})\Vert&\ge \Vert D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}\Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert \cdot \big [1-\frac{\Vert D_{z_{\tau _n}} \Phi _{\nu _{\kappa }(z_{\tau _n})}\Pi ^{F_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert }{\Vert D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})} \Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert }\big ]\nonumber \\&\overset{(4.2)}{\ge }\Vert D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}\Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert \cdot [1-\frac{\kappa }{2(1+\kappa )}\cdot \frac{\Vert \Pi ^{F_{\tau _n}}(y_{\tau _n}-x_{\tau _{n}})\Vert }{\Vert \Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _{n}})\Vert }]\nonumber \\&\overset{(4.4)}{\ge }\Vert D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}\Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert \cdot [1-\frac{\kappa }{2(1+\kappa )}\cdot \frac{1+\kappa }{\kappa }]\nonumber \\&=\frac{1}{2}\Vert D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}\Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert \nonumber \\&\overset{(4.2)}{\ge }\frac{2}{\kappa }\Vert \Pi ^{E_{\tau _n}}(y_{\tau _n}-x_{\tau _n})\Vert \nonumber \\&\overset{(4.4)}{\ge }2\Vert y_{\tau _n}-x_{\tau _n}\Vert \end{aligned} \end{aligned}$$

for any $n>0$. Notice that

$$\begin{aligned}{} & {} y_{\tau _{n+1}}-x_{\tau _{n+1}}=D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}(y_{\tau _n}-x_{\tau _n}) +\int _{0}^{1}[D_{x_{\tau _n}+s(y_{\tau _n}-x_{\tau _n})}\Phi _{\nu _{\kappa }(z_{\tau _n})} -D_{z_{\tau _n}}\Phi _{\nu _{\kappa }(z_{\tau _n})}]ds \\ {}{} & {} \quad \cdot (y_{\tau _n}-x_{\tau _n}) \end{aligned}$$

for any $n>0$. Together with (4.5), one has

$$\begin{aligned} \Vert y_{\tau _{n+1}}-x_{\tau _{n+1}}\Vert \ge \frac{3}{2}\cdot \Vert y_{\tau _{n}}-x_{\tau _{n}}\Vert \end{aligned}$$

for any $n>0$. Hence, $\lim \limits _{n\rightarrow \infty }\Vert y_{\tau _n}-x_{\tau _n})\Vert =\infty$, a contradiction.

Therefore, we have completed the proof. $\square$

Theorem 4.3

Let $\mathcal {D}$ be an open $\omega$-compact set and $x\in \mathcal {D}$. If $\lambda _{kz}>0$ for any $z\in \omega (x)$, then one has $x\in \overline{Q}$.

Proof

See [2, Theorem 4.5]. We here point out that all arguments in [2, Theorem 4.5] still remain valid for semiflow $\Phi _t$ strongly focusing monotone w.r.t. C on an infinite dimensional Banach space. $\square$

We now consider the case that $\lambda _{k\tilde{z}}>0$ for any $\tilde{z}\in \omega _0(x)$ and $\lambda _{kz}\le 0$ for some $z\in \omega (x)\setminus \omega _0(x)$.

Theorem 4.4

Let $\mathcal {D}$ be an open $\omega$-compact set and $x\in \mathcal {D}$. Assume that $\lambda _{k\tilde{z}}>0$ for any $\tilde{z}\in \omega _0(x)$. If there exists some $z\in \omega (x)\setminus \omega _0(x)$ such that $\lambda _{kz}\le 0$, then $x\in \overline{Q}$.

Proof

Since $\Phi _t$ admits a flow extension on $\omega (x)$, one can define a vector $v_y= \frac{d}{dt}\mid _{t=0}\Phi _t(y)\in X$ for any $y\in \omega (x)$. The smoothness of $\Phi _t$ implies that the map $y:\mapsto v_y$ is continuous. Hence, all arguments in [2, Theorem 4.6] are still effective for semiflow $\Phi _t$ strongly focusing monotone with respect to C on an infinite dimensional Banach space. $\square$

Before discussing the case that $\lambda _{kz}\le 0$ for a regular point $z\in \omega _0(x)$, we present the following lemma, which describes the nonlinear dynamics nearby a regular point.

Lemma 4.5

Let $x\in \Sigma$ be a regular point. If $\lambda _{kx}\le 0$, then there exists an open neighborhood $\mathcal {V}$ of x such that for any $y\in \mathcal {V}$, one of two following properties holds:

(a) $\Vert \Phi _{t}(x)-\Phi _{t}(y)\Vert \rightarrow 0$ as $t\rightarrow +\infty$;

(b) There exists a $T>0$ such that $\Phi _{T}(x)-\Phi _{T}(y)\in C\setminus \{0\}$; and hence, $\Phi _{t}(x)-\Phi _{t}(y)\in \text {Int}\, C$ for any $t>T$.

Proof

By repeating all arguments in [2, Lemma 4.1], the conclusion in this lemma can be obtained for $\Phi _t$ strongly focusing monotone w.r.t. C. $\square$

Theorem 4.6

Assume that there exists a regular point $z\in \omega (x)$ satisfying $\lambda _{kz}\le 0$. Then either $x\in Q$, or $\omega (x)=\{z\}$ consists of a singleton.

Proof

The result is implied by repeating all arguments in [2, Theorem 4.2]. $\square$

5 Proofs of Theorem A and B

Due to the local behaviors in the last section, we can describe the generic dynamics of the semiflow $\Phi _t$ strongly focusing monotone w.r.t. C (see Theorem A) on a general Banach space, which concludes that generic (open and dense) positive semiorbits either are pseudo-ordered or convergent to an equilibrium. When $k=2$, together with the results in Lemma 2.6, we will further show the Poincaré-Bendixson Theorem (see Theorem B), that is to say, for generic (open and dense) points, its $\omega$-limit set containing no equilibria is a periodic orbit.

Proof of Theorem A

We note that $\mathcal {D}$ is $\omega$-compact because $\mathcal {O}^+(\mathcal {D})$ is precompact. Then, for any $x\in \mathcal {D}$, one of the following three alternatives holds:

• (a) $\lambda _{kz}>0$ for any $z\in \omega (x)$;

• (b) $\lambda _{k\tilde{z}}>0$ for any regular point $\tilde{z}\in \omega _0(x)$ and $\lambda _{kz}\le 0$ for some irregular point $z\in \omega (x)\setminus \omega _0(x)$;

• (c) $\lambda _{kz}\le 0$ for some regular point $z\in \omega (x)$.

By virtue of Theorem 4.3, 4.4 and 4.6, one has $x\in \overline{Q}\cup C_E$ for any $x\in \mathcal {D}$. Thus, $Q\cup C_{E}$ is dense in $\mathcal {D}$.

To prove that $\text {Int} (Q\cup C_{E})$ is dense in $\mathcal {D}$ by contrary, we suppose that there is an open subset U of $\mathcal {D}$ such that $U\cap \text {Int} (Q\cup C_{E})=\emptyset$. By strong monotonicity of $\Phi _t$, Q is open. Together with Theorem 4.3 and 4.4, case (a) and (b) will not occur for any point in U. Then, only case (c) can occur for any point in U. Moreover, by virtue of Theorem 4.6, one has that $U\subset C_E$. Recall that U is open. Thus, $U\subset \text {Int} (Q\cup C_{E})$, a contradiction.

Therefore, we have completed the proof.

Proof of Theorem B

By Theorem A, $\textrm{Int}(Q\cup C_E)$ is open and dense in $\mathcal {D}$. Now, given any $x\in \textrm{Int}(Q\cup C_E)\cap \mathcal {D}$, if $\omega (x)\cap E=\emptyset$, then $x\in Q$. It then follows from Lemma 2.6, $\omega (x)$ is a periodic orbit. Therefore, we have completed the proof.

Remark 5.1

In this paper, we introduce the concept of strongly focusing monotonicity and extend our previous works on the generic dynamics of flows strongly monotone with respect to high-rank cones on $\mathbb {R}^n$ (see [2]) to the class of general semiflows strongly focusing monotone with respect to the k-cone C on an infinite dimensional Banach space. In the sequel, we intend to investigate the “typical” behaviors of this class of semiflows in the measure theoretic sense, and establish the total-ordering property and Poincaré-Bendixson property of omega-limit sets. In future work, we will extend the strongly focusing monotonicity from semiflows to discrete systems, cocycles and skew-product semiflows.