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This paper aims to determine the maximum number of limit cycles that can bifurcate from the linear center x ˙ = y {\dot{x}=y} , y ˙ = - x {\dot{y}=-x} , when we perturb it inside a class of piecewise smooth differential systems, which are separated by the straight line y = 0 {y=0} . Using the averaging theory of first order for piecewise smooth differential systems, we prove that if the degree of the polynomial that appears in the perturbation term is equal to n, then the maximum number of limit cycles bifurcating from the period annulus around the linear center is at most n or n - 1 {n-1} , and this number can be reached. [ABSTRACT FROM AUTHOR]

In this paper we investigate a particular case of a zero-Hopf bifurcation of a four dimensional analytic differential system. We prove that at most five periodic orbits bifurcate from the zero-Hopf equilibrium using the averaging theory of first order and give a specific example to illustrate this conclusion. Moreover we prove the non-existence of C 1 first integrals in a neighbourhood of these periodic orbits. [ABSTRACT FROM AUTHOR]

In this article, the integrability and linearizability of a class of cubic Z 2 -equivariant systems x ˙ = - 1 2 x - a 21 y + 1 2 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 , y ˙ = (- q - b 21) y + b 21 x 2 y + b 12 x y 2 + b 03 y 3 , are studied. For any positive integer q, we obtain the first three saddle quantities of the above systems by theoretical analysis. Moreover, for any positive integer q, we derive the necessary and sufficient conditions for the linearizability of the above systems under some assumptions. [ABSTRACT FROM AUTHOR]

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center. [ABSTRACT FROM AUTHOR]

We study dynamics of a fast–slow Leslie–Gower predator–prey system with Allee effect and Holling Type II functional response. More specifically, we show some sufficient conditions to guarantee the existence of two positive equilibria of the system and their location, and then we further fully determine their dynamics. Based on geometric singular perturbation theory and the slow–fast normal form, we determine the associated bifurcation curve and observe canard explosion. Besides, we also find a homoclinic orbit to a saddle with slow and fast segments, in which, the stable and unstable manifolds of the saddle are connected under explicit parameters conditions. [ABSTRACT FROM AUTHOR]

Effective management of predator–prey systems is crucial for sustaining ecological balance and preserving biodiversity, which requires full understanding the dynamics of such systems with harvesting and stocking. This paper aims to investigate the global dynamics of a Rosenzweig–MacArthur model considering the interplay of these intervention practices. We reveal that this model undergoes a sequence of bifurcations, including cusp of codimensions 2 and 3, saddle-node bifurcation, Bogdanov–Takens (BT) bifurcation of codimensions 2 and 3, and degenerate Hopf bifurcation of codimension 2. In particular, a codimension-2 cusp of limit cycles is found, which indicates the coexistence of three limit cycles. An interesting and novel scenario is discovered: two distinct homoclinic cycle curves connect their respective BT bifurcation points. This differs from most models where a single homoclinic cycle curve may connect both BT bifurcation points. Moreover, we find that two families of limit cycles converge toward a heteroclinic cycle, signaling the risk of overexploitation. From a biological perspective, the prey population may undergo extinction for all initial states under large constant harvesting rate. Further, the simultaneous stocking of both populations is not conducive to the coexistence of both species; the stocking of one population and the harvesting of the other will promote the coexistence of two populations; while the simultaneous harvesting of two populations may result in multiple limit cycles, which effectively underscore the positive effect of harvesting and stocking. Identifying the optimal timing to harvest or stock predators and prey is crucial to prevent system collapse. This work promotes to a deeper understanding of the dynamics of ecosystems when harvesting and stocking occurs simultaneously. Further, it reveals the important roles of harvesting and stocking, contributing to the effective management of predator–prey systems. [ABSTRACT FROM AUTHOR]

Characterizing existence or not of periodic orbit is a classical problem, and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system x ˙ = y , y ˙ = - g (x) - f (x , y) y are obtained and by examples shows that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by Andreev (Am Math Soc Transl 8:183–207, 1958) on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters. [ABSTRACT FROM AUTHOR]

A global center for a vector field in the plane is a singular point p having R 2 filled of periodic orbits with the exception of the singular point p. Polynomial differential systems of degree 2 have no global centers. In this paper we classify the global nilpotent centers of planar cubic polynomial Hamiltonian systems symmetric with respect to the y-axis. [ABSTRACT FROM AUTHOR]

This paper aims to study the sufficient and necessary conditions for a generalized polynomial Liénard system with a unique equilibrium and one periodic annulus. In such a case, there are two types of equilibrium: a centre or a neighbourhood of the equilibrium consisting of a hyperbolic sector and an elliptical sector. The sufficient and necessary conditions for both cases are established and global phase portraits in the Poincaré disc are presented under certain conditions. [ABSTRACT FROM AUTHOR]

In the existing literatures, there exist at least four limit cycles for 3D Lotka-Volterra competitive systems in Zeeman's classes 26 and 27 with no explicit critical parameter values, and three limit cycles in Zeeman's classes 27 and 29 with explicit critical parameter values. Giving explicit critical parameter values in Zeeman's classes 26 - 31 and accurate focus values at them except class 30, we rigorously prove that Zeeman's classes 27 - 30 have at least three limit cycles, and Zeeman's classes 26 and 31 have at least two small amplitude limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of non-autonomous piecewise differential equations defined as follows: d x / d t = a 0 (t) + ∑ i = 1 n a i (t) | x | i , where n ∈ N + and each a i (t) is real, 1-periodic, and smooth function. We deal with two basic problems related to their limit cycles (isolated solutions satisfying x (0) = x (1)) . First, we prove that, for any given n ∈ N + , there is no upper bound on the number of limit cycles of such equations. Second, we demonstrate that if a 1 (t) , … , a n (t) do not change sign and have the same sign in the interval [0, 1], then the equation has at most two limit cycles. We provide a comprehensive analysis of all possible configurations of these limit cycles. In addition, we extend the result of at most two limit cycles to a broader class of general non-autonomous piecewise polynomial differential equations and offer a criterion for determining the uniqueness of the limit cycle within this class of equations. [ABSTRACT FROM AUTHOR]

We study the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous cubic centre of discontinuous and continuous piecewise differential systems with three zones formed by the discontinuity set Σ = { (x , y) ∈ R 2 : (y = 0) ∨ (x = 0 ∧ y ≥ 0) } . More precisely, we consider the following perturbed systems x ˙ = - y + x 2 y + ϵ p i (x , y) , y ˙ = x + x y 2 + ϵ q i (x , y) , i = 1 , 2 , 3 , where p i and q i are polynomials of degree m. Using the averaging theory of first order, we prove that for m = 1 , 2 , 3 , at most 3, 9 and 15 limit cycles bifurcate from the periodic annulus of the isochronous cubic centre in the discontinuous case, respectively. While for the continuous case, it can appear 1, 5 and 6 limit cycles from the periodic orbits of these centres, respectively. Furthermore, we extend our study when p i and q i are homogeneous polynomials with 1 ≤ m ≤ 3 , obtaining respectively for m = 1 , 2 , 3 at most one, seven and at least twelve limit cycles, which bifurcate from the periodic orbits of the isochronous cubic centre. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of a closed trajectory of an autonomous planar system with constraints. In contrast to previously known results, we do not require the constraint to be convex. [ABSTRACT FROM AUTHOR]

In our previous work, we have studied the limit cycles for a class of discontinuous piecewise quadratic polynomial differential systems with a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle α = π / 2 . The unperturbed system is the quadratic uniform isochronous center x ˙ = - y + x y , y ˙ = x + y 2 with a family of periodic orbits surrounding the origin. In this paper, we continue to investigate this kind of piecewise differential systems, but now the angle between the two rays is α ∈ (0 , π / 2) ∪ [ 3 π / 2 , 2 π) . Using the Chebyshev theory, we prove that the maximum number of hyperbolic limit cycles that can bifurcate from these periodic orbits using the averaging theory of first order is exactly 8 for α ∈ (0 , π / 2) ∪ [ 3 π / 2 , 2 π) . Together with our previous work, which concerns on the case of α = π / 2 , we can conclude that using the averaging theory of first order the maximum number of hyperbolic limit cycles is exactly 8, when this quadratic center is perturbed inside the above-mentioned classes separated by a non-regular line of discontinuity with α ∈ (0 , π / 2 ] ∪ [ 3 π / 2 , 2 π) . [ABSTRACT FROM AUTHOR]

We study planar discontinuous piecewise differential systems formed by three linear Hamiltonian saddles separated by the non-regular line $ \Sigma =\{(x,y)\in \mathbb {R}^2: (y=0) \vee (x=0 \wedge y\geq 0)\} $ Σ={(x,y)∈R2:(y=0)∨(x=0∧y≥0)}. We prove that when the linear Hamiltonian saddles are homogeneous they have no limit cycles, and when they are non homogeneous they can have at most three limit cycles having exactly one point on each branch of Σ, and at most one limit cycle having four intersection points on Σ. Moreover we show that they can have at most one limit cycle having four intersection points on Σ and three limit cycles having three intersection points on Σ, simultaneously. Thus we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems. Furthermore we show that for the three types of combinations of the limit cycles here studied in two of them the upper bound is sharp by providing examples of these systems with the maximum number of possible limit cycles. For the remaining type of limit cycles the upper bound is four and we have examples with three limit cycles. So at this moment it is an open problem to know if the sharp upper bound is either four or three. [ABSTRACT FROM AUTHOR]

This article discusses the rotations (windings) of solutions to the basic system of equations y ′ = F 1 ⁢ ( x , y ) {y^{\prime}=F_{1}(x,y)} and x ′ = F 2 ⁢ ( x , y ) {x^{\prime}=F_{2}(x,y)} . This allows us to return to the topic of known limit cycles from a much broader point of view, in particular, it makes it possible to describe the conditions for the existence of limit cycles. [ABSTRACT FROM AUTHOR]

Consider the polynomial differential system of degree m of the form x ˙ = − y (1 + μ (a 2 x − a 1 y)) + x (ν (a 1 x + a 2 y) + Ω m − 1 (x , y)) , y ˙ = x (1 + μ (a 2 x − a 1 y)) + y (ν (a 1 x + a 2 y) + Ω m − 1 (x , y)) , where μ and ν are real numbers such that (μ 2 + ν 2) (μ + ν (m − 2)) (a 1 2 + a 2 2) ≠ 0 , m > 2 and Ωm−1(x,y) is a homogenous polynomial of degree m − 1. A conjecture, stated in J. Differential Equations 2019, suggests that when ν = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (x,y) → (X,Y) the system is invariant under the transformation (X,Y,t) → (−X,Y, −t). For every degree m we prove the extension of this conjecture to any value of ν except for a finite set of values of μ. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems. We first establish the Poincaré map near a periodic orbit, and deduce the first order derivative of the map for general piecewise smooth systems on the plane. Then, we obtain a sufficient condition for determining the stability of limit cycles for these systems. [ABSTRACT FROM AUTHOR]

In this work, we study the monotonicity of the ratio of two line integrals I i (h) = ∫ Γ h + f i (x) y d x + ∫ Γ h - f i (x) y d x , i = 0 , 1 , with Γ h + = { (x , y) ∈ R 2 | H + (x , y) = h , x > 0 } and Γ h - = { (x , y) ∈ R 2 | H - (x , y) = h ~ , x ≤ 0 } , where H + (x , y) and H - (x , y) have the form 1 2 y 2 + Ψ 1 (x) and 1 2 y 2 + Ψ 2 (x) , respectively. We first present a criterion function defined directly by the functions which appear in the above integrals and prove the monotonicity of this criterion function implying the monotonicity of the ratio of these line integrals which is very important to find the number of limit cycles. Then we give several examples to illustrate the application of this criterion. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation of limit cycles, centers, and isochronous centers for a class of three-dimensional Kukles systems of degree 3. Through calculating the singular point quantities, we obtain a necessary condition for the origin to be a center, then the Darboux integrability theory is used to prove that the necessary condition is also sufficient. Then, we demonstrate that the origin is an isochronous center under the obtained center condition. Finally, we determine that the highest order of the origin to be a weak focus is ten, while the maximum number of small-amplitude limit cycles bifurcating from this weak focus is nine. Moreover, we show that this maximum number of 9 can be realized. It is worthwhile to say that this number is also a new lower bound on the number of limit cycles bifurcating from the single weak focus for three-dimensional cubic polynomial systems. [ABSTRACT FROM AUTHOR]

In this paper, we study limit cycle bifurcations near homoclinic and heteroclinic loops in piecewise smooth systems with three zones separated by two parallel straight lines. By introducing suitable Poincaré map near a homoclinic loop, we derive some stability criteria and establish bifurcation theory of limit cycles via stability-changing method near homoclinic and heteroclinic loops, respectively. As applications, we give two examples, which consider a class of Liénard piecewise linear differential systems. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyzing the codimension-2 bifurcations of a generalized three-dimensional (3D) jerk system which is jerk function with all quadratic and cubic terms. First, the locally stability of Bagdanov–Takens (double-zero) equilibrium is systematically investigated using the center manifold and normal form theory combined with some bifurcation theories, and under some certain parameter conditions, the Bagdanov–Takens equilibrium of this jerk system could be a saddle, a focus, a degenerate equilibrium or an equilibrium which neighborhood is formed from an elliptic sector and a hyperbolic sector on its two-dimensional center manifold. Second, some adequate parameter conditions for guaranteeing the occurrence of Hopf bifurcation, Bautin (degenerate Hopf) bifurcation and the stability of the Hopf equilibrium of this jerk system are given. Finally, numerical simulation is used to verify the above theoretical analyses. [ABSTRACT FROM AUTHOR]

In this paper we study the periodic orbits of a class of 2n-dimensional control dynamical systems with a perturbation of continuous piecewise smooth quadratic polynomial and a perturbation of discontinuous piecewise smooth polynomial of an arbitrarily given degree, respectively. By applying the averaging theory for continuous and discontinuous piecewise smooth systems, the number of isolated zeros of the averaging function can be determined, which provides a lower bound of the maximum number of isolated periodic orbits. At last, we give examples to show that the lower bound of the maximum number is accessible by the properties of algebraic closure and transcendental extension. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a wide family of dynamical systems whose characteristic feature is a polynomial right-hand side containing coprime forms of the phase variables of the system. One of the equations of the system contains a third-degree polynomial (cubic form), the other equation contains a quadratic form. We consider the problem of constructing all possible phase portraits in the Poincaré disk for systems from the family considered and establish criteria for the implementation of each portrait that are close to coefficient criteria. This problem is solved by using the central and orthogonal Poincaré methods of sequential mappings and a number of other methods developed by the authors for the purposes of this study. We obtained rigorous qualitative and quantitative results. More than 250 topologically distinct phase portraits of various systems were constructed. The absence of limit cycles of systems of this family is proved. Methods developed can be useful for the further study of systems with polynomial right-hand sides of other forms. [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Liénard differential system of arbitrary degree: x ˙ = y , y ˙ = a 1 x + a 2 x 2 m + (a 3 + a 4 x 2 n) y applying to a Duffing–Van der Pol oscillator. We prove that it has abundant dynamics, such as the generalized transcritical bifurcation, Hopf bifurcation, heteroclinic bifurcation, homoclinic bifurcation and double limit cycle bifurcation. The associated global bifurcation diagram in the parameter space and the corresponding global phase portraits in the Poincaré disc are presented. These analytic results are also demonstrated by concrete examples via numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first-order Melnikov functions and limit cycle bifurcations of a near-Hamiltonian system near a cuspidal loop. By establishing relations between the coefficients in the expansions of the two Melnikov functions, we give a general method to obtain the number of limit cycles near the cuspidal loop. As an application, we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles. [ABSTRACT FROM AUTHOR]

This paper studies limit cycle bifurcation of an (n + 1) th order generalized Liénard differential system with an elliptic Hamiltonian H (x , y) = 1 2 x 2 + 1 2 y 2 - 1 4 x 4 , which bifurcates at least [ n 2 ] limit cycles near either a centre or a heteroclinic loop with two hyperbolic saddles for n ≤ 150 , where about two thirds of the limit cycles are large amplitude ones. To achieve this goal, we provide the precise expression of the third coefficient in the asymptotic expansion of the first order Melnikov function of an analytic near-Hamiltonian system near a polycycle with m (≥ 2) hyperbolic saddles, without any prescribed condition, whereas all the known results need. As well as we also characterize its higher order coefficients. Here [a] is the integer part function of a real number a. [ABSTRACT FROM AUTHOR]

We give a proof of the distribution property of limit cycles in so-called quadratic systems. We prove that the possible limit cycle distributions are either (n,0)\left(n,0) or (n,1)\left(n,1) (where n∈{0}∪Nn\in \left\{0\right\}\cup {\mathbb{N}}). The aim of this article is to simplify and fill gaps in the original proof by Zhang (On the distribution and number of limit cycles for quadratic systems with two foci, Qual. Theory Dyn. Sys. 3 (2002), 437–463). The sixteenth Hilbert problem asks for an upper bound for the number of limit cycles in polynomial systems H(n)H\left(n), where nn is the maximum degree of the polynomial defining the system. A consequence of the distribution property is that it reduces the study of H(2)H\left(2) to the study of the maximum number of limit cycles surrounding one singularity.

Poincaré–Bendixson index theorem showed that: if planar smooth differential systems have only finitely number of singular points inside a limit cycle, then the sum of the indices at these singular points is 1. Hence there is at least one singular point lie inside a limit cycle. These results are useful in practice since its provide information about the existence and location of limit cycles. It is also well known that planar smooth linear system cannot have limit cycles. While in a recent paper Llibre and Teixeira (Nonl Dyn 88:157–164, 2017), Llibre and Teixeira constructed a planar piecewise linear differential systems formed by two linear differential systems separated by the straight line Σ : { (x , y) | x = 0 } , such that both linear differential have no singular points, neither real nor virtual, but it can have a limit cycle. This paper revisit these piecewise linear differential systems by regularization process. Our results show that there are three Σ - singular points inside the limit cycle, and the sum of the indices at these singular points is 1. Thus the Poincaré-Bendixson index theorem is also valid for such piecewise linear differential systems. [ABSTRACT FROM AUTHOR]

In this paper periodic trajectories of dynamical systems presenting discontinuities are studied. The considered model consists of two distinct linear differential systems, each one containing a single equilibrium point of centre type. Each system is defined on disjoint regions of the plane, the separation line is a union of two half-straight lines contained on the coordinate axes. The obtained differential system is non-smooth, so we apply Filippov's theory to study the transitions from one dynamical system to another. The combination of the two linear plus the Filippov system acting on the separation line generates a nonlinear regime observed by the presence of limit cycles, sliding and tangential periodic trajectories as well as the coexistence of such objects. In theorem we establish the location, stability and hyperbolicity of limit cycles for certain classes of the considered model. In theorem we perform the global analysis of a representative model through bifurcation theory to analyse the birth of limit cycles, sliding periodic trajectories and tangential ones. We also provide some results addressing the coexistence of periodic trajectories and two potential physical interpretations of the model considered in the paper, one addressing nonlinear oscillations and the other considering slow-fast systems of neuron models. The main techniques employed to obtain the results are first integrals, Poincaré half return maps, and elements of bifurcation theory. [ABSTRACT FROM AUTHOR]

The topological index, or simply the index, of an equilibrium point of a vector field is an integer which saves important information about the local phase portrait of the equilibrium point. There are mainly two ways to calculate the index of an isolated equilibrium point of a smooth vector field. First Poincaré and Bendixson proved that the index of an equilibrium point can be obtained from the number of hyperbolic and elliptic sectors that there are in a neighborhood of the equilibrium point, which is known as Poincaré–Bendixson formula for the topological index of an equilibrium point. Second several works contributed to the algebraic method of Cauchy’s index for computing the index of an equilibrium point. In this paper, we extend the Poincaré–Bendixson formula to planar piecewise smooth vector fields. Applying this formula, we provide the index of generic codimension-0 and codimension-1 equilibrium points for piecewise smooth vector fields. [ABSTRACT FROM AUTHOR]

We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve y = x n with n ≥ 2 . We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of n, proving the extended 16th Hilbert problem in this case. In particular, we show that for n = 2 this bound can be reached. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the existence of zero-Hopf bifurcations of a new hyperchaotic system, using the averaging theory of dynamical systems of second order. Furthermore, at most one periodic orbit can bifurcate from the origin of coordinates.

We obtain lower bounds for the maximum number of limit cycles bifurcating from periodic annuli of discontinuous planar piecewise linear Hamiltonian differential systems with three zones separated by two parallel straight lines, assuming that the linear differential subsystem in the region between the two straight lines, called of central subsystem, has a saddle at a point equidistant from these lines. (Obviously, the other subsystems have saddles or centers.) We prove that at least six limit cycles bifurcate from the periodic annuli of these kind of piecewise Hamiltonian differential systems, by linear perturbations. Normal forms and Melnikov functions, defined in two and three zones, are the main techniques used in the proof of the results. [ABSTRACT FROM AUTHOR]

From the beginning of this century many articles have been published on the continuous and discontinuous piecewise differential systems specially in the plane. The big interest on these piecewise differential systems mainly comes from their increasing number of applications for modelling many natural phenomena. As it is usual in the planar differential systems one of the main difficulties for understanding their dynamics consists in controlling their limit cycles. The major part of papers studying continuous piecewise differential systems has an straight line as the line of separation between the differential systems forming the continuous piecewise differential systems. In this work we consider continuous piecewise differential systems separated by a circle and formed by one linear differential center and one quadratic differential center. We study the maximum number of limit cycles that such kind of continuos piecewise differential system can exhibit. [ABSTRACT FROM AUTHOR]

In this paper we investigate the number of small amplitude limit cycles bifurcated from a class of planar piecewise smooth quadratic systems with focus-parabolic type critical point having exactly one switching line. By considering the Lyapunov constants of the system, we obtain some conditions under which (0, 0) is a center or a focus of order eleven. We prove that at least ten limit cycles can bifurcate from (0, 0), which is a new lower bound of the cyclicity of piecewise smooth quadratic systems with focus-parabolic type critical point having exactly one switching line. For four of the center conditions, we prove that at least six limit cycles can bifurcate from (0, 0) without destroying the singularity by higher order perturbations techniques, one more than that obtained by considering the linear parts of the perturbed Lyapunov constants. Finally, the center-focus problem for a subclass of the system is completely solved. [ABSTRACT FROM AUTHOR]

In this paper, we first give criteria on the existence and uniqueness of limit cycles for x ¨ + f (x , x ˙) x ˙ + g (x) = 0 , which is widely used to model the hybrid mechanical oscillators such as Van der Pol–Duffing–Rayleigh oscillator, Van der Pol–Rayleigh oscillator and so on. By applying the criteria, we completely prove the uniqueness of limit cycles for Van der Pol–Duffing–Rayleigh oscillator and an asymmetric and nonsmooth Van der Pol–Rayleigh oscillator. Moreover, we provide the location of the limit cycle when it exists. [ABSTRACT FROM AUTHOR]

In 1960 Markus and Yamabe made the conjecture that if a C 1 differential system x ˙ = F (x) in R n has a unique equilibrium point and DF(x) is Hurwitz for all x ∈ R n , then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true in R 2 and false in R n for all n ≥ 3 . In (The Markus–Yamabe conjecture for continuous and discontinuous piecewise linear differential systems, 2020) the authors extended the Markus–Yamabe conjecture to continuous and discontinuous piecewise linear differential systems in R n separated by a hyperplane, they proved for the continuous systems that the extended conjecture is true in R 2 and false in R n for all n ≥ 3 , but for discontinuous systems they proved that the conjecture is false in R n for all n ≥ 2 . In this paper first we show that there are no continuous piecewise linear differential systems separated by a conic × R n - 2 except the linear differential systems in R n . And after we prove that the extended Markus–Yamabe conjecture to discontinuous piecewise linear differential systems in R n separated by a conic × R n - 2 is false in R n for all n ≥ 2 . [ABSTRACT FROM AUTHOR]

The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R2{{\mathbb{R}}}^{2}, particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.

We consider the dynamics of the IS–LM economic model, which is a fundamental macroeconomic model describing the relationship between interest rates and assets market. There is generally a long time lag between changes in the interest rate and the consequential effects on the economy. Therefore, most central banks change the interest rates slowly, which yields a slow–fast IS–LM model. Using geometric singular perturbation theory, we establish the sufficient and necessary conditions on the existence of relaxation oscillations for the slow–fast IS–LM model. We also apply our abstract genericity results to study some concrete particular slow–fast IS–LM models. Numerical simulations are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Our work is concerned with the number of limit cycles and isochronous center conditions for a class of three-dimensional cubic Kolmogorov systems with an equilibrium point in the positive octant where the system has biological meaning. A formal series method of computing singular point values (equivalent to focal values) is applied to investigate the Hopf bifurcation and center problem on center manifolds for the Kolmogorov system. Using this we derive two sets of conditions for the equilibrium point to be a center on a center manifold for the system, and prove that at most seven small-amplitude limit cycles can be bifurcated from an isolated positive equilibrium point. We prove that seven limit cycles can be created in this way, obtaining a new result on the number of limit cycles in three-dimensional cubic Kolmogorov systems. Moreover, two sets of necessary conditions for the existence of an isochronous center on the center manifold for such systems are obtained by the computation of period constants. The Darboux theory of linearizability is applied to show the sufficiency of the conditions. [ABSTRACT FROM AUTHOR]