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In this paper, we revisit a piecewise smooth fast-slow predator–prey model with Holling type I functional response and predator harvesting, where the harvesting rate is sufficiently small compared to the intrinsic growth rate of prey. The model undergoes two bifurcation mechanisms: (i) singular slow-fast cycle bifurcation, through which the model can have a unique and stable relaxation oscillation or homoclinic orbit; (ii) boundary equilibrium bifurcation, from which a unique and unstable limit cycle occurs. Additionally, we show the coexistence of a large-amplitude relaxation oscillation (or homoclinic orbit) and a small-amplitude limit cycle. [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]

We provide a detailed bifurcation analysis in a three-dimensional system describing interaction between tumor cells, healthy tissue cells, and cells of the immune system. As is well known from previous studies, the most interesting dynamical regimes in this model are associated with the spiral chaos arising due to the Shilnikov homoclinic loop to a saddle-focus equilibrium [1-2, 3]. We explain how this equilibrium appears and how it gives rise to Shilnikov attractors. The main part of this work is devoted to the study of codimension-two bifurcations which, as we show, are the organizing centers in the system. In particular, we describe bifurcation unfoldings for an equilibrium state when: (1) it has a pair of zero eigenvalues (Bogdanov – Takens bifurcation) and (2) zero and a pair of purely imaginary eigenvalues (zero-Hopf bifurcation). It is shown how these bifurcations are related to the emergence of the observed chaotic attractors. [ABSTRACT FROM AUTHOR]

In this paper we study bifurcations in mass-action networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, Andronov–Hopf, Bogdanov–Takens and Bautin bifurcations, and prove the non-occurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that mass-action networks without conservation laws admit no bifurcations of codimension greater than m - 2 , where m is the number of reactions; we fully characterise quadratic, rank-one mass-action networks admitting fold bifurcations; and we write down some necessary conditions for Andronov–Hopf and cusp bifurcations in mass-action networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in mass-action networks. [ABSTRACT FROM AUTHOR]

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

Singular cycles (homoclinic orbits and heteroclinic cycles) play an important role in the study of chaotic dynamics of dynamical systems. This paper provides the coexistence of singular cycles that intersect the switching manifold transversely at two points in a class of three-dimensional two-zone piecewise affine systems. Moreover, the switching manifold of the systems is constructed by two perpendicular planes. Different to the three-dimensional piecewise affine systems with a switching plane, the system can ensure the coexistence of two homoclinic orbits to the same one equilibrium point and two heteroclinic cycles constructing by three heteroclinic orbits. In addition, three examples with simulations of the singular cycles are provided to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model of fluid flows in a convective thermal system is developed, and a five-dimensional dynamical system is developed for the investigation of the convective fluid dynamics. The analytical solutions of periodic motions to chaos of the convective fluid flows are developed for steady-state vortex flows, and the corresponding stability and bifurcations of periodic motions in the five-dimensional dynamical system are studied. The harmonic frequency-amplitude characteristics for periodic flows are obtained, which provide energy distribution in the parameter space. Analytical homoclinic orbits for the convective fluid flow systems are developed for the asymptotic convection through the infinite-many homoclinic orbits in the five-dimensional dynamical system. The dynamics of fluid flows in the convective thermal systems are revealed, and one can use such methodology to predict atmospheric and oceanic phenomena through thermal convections. [ABSTRACT FROM AUTHOR]

We propose a mathematical model for two species competing for a limited resource associated to polyhydroxyalkanoate (PHA) production, which possesses regime of stable states: stable equilibrium, and periodic and aperiodic oscillations. Such regimes of stable oscillations are absent in the model without taking into account PHA production but is known to exist in experimental model associated to the production of PHA. It explains the capacity of the system to sustain itself at the lowest value of resource. Thus, the proposed system provides a simpler four-dimensional model containing monod functions with such behaviours. Using analytical tools and numerical bifurcation analysis, we describe parameter regions and bifurcation structures leading to the existence and the coexistence between stable, unstable equilibrium and limit cycle. These explain critical parameter sensitivity impact on the process. Considering the effects on the proposed system under a frequent alternation of the input resource, we investigate how the increase of the length of the feast period in the Feast-Famine conditions, increases the PHA-production or decreases the lowest value of the resource at the equilibrium. [ABSTRACT FROM AUTHOR]

The shallow-water wave equations with different forms of delays are presented in this work, such as no delay, local delay and nonlocal delay, which are described in the form of convolutions with different kernels. These shallow-wave equations satisfy the asymptotic integrability condition and include the Korteweg-de Vries equation, Camassa-Holm equation and Degasperis-Procesi equation as particular cases. The existence and non-existence of solitary wave are established by the invariant manifold theory and geometric singular perturbation theory. It is found that different delays have various effects on the existence of solitary waves. In particular, the Melnikov functions with divergence free or not are derived for different delays to measure the separation of stable and unstable manifolds, so that the existence of solitary waves could be justified. [ABSTRACT FROM AUTHOR]

We study the codimension-two bifurcations exhibited by a recently-developed SIR-type mathematical model for the spread of COVID-19, as its two main parameters -the susceptible individuals' cautiousness level and the hospitals' bed-occupancy rate- vary over their domains. We use AUTO to generate the model's bifurcation diagrams near the relevant bifurcation points: two Bogdanov-Takens points and two generalised Hopf points, as well as a number of phase portraits describing the model's orbital behaviours for various pairs of parameter values near each bifurcation point. The analysis shows that, when a backward bifurcation occurs at the basic reproduction threshold, the transition of the model's asymptotic behaviour from endemic to disease-free takes place via an unexpectedly complex sequence of topological changes, involving the births and disappearances of not only equilibria but also limit cycles and homoclinic orbits. Epidemiologically, the analysis confirms the importance of a proper control of the values of the aforementioned parameters for a successful eradication of COVID-19. We recommend a number of strategies by which such a control may be achieved.

This paper develops an anticontrol approach to design a 3D continuous-time autonomous chaotic system with saddle-focus homoclinic orbit, based on two chaotification criterions for all orbits to be globally bounded with positive Lyapunov exponents. By using the Shil'nikov theorem, a Poincaré return map near the origin is found in the designed controlled system, confirming the existence of chaos in sense of the Smale horseshoe. [ABSTRACT FROM AUTHOR]

Intrinsic bursting neurons represent a common neuronal type that displays bursting patterns upon depolarization stimulation. These neurons can be described by a system of seven-dimensional equations, which pose a challenge for dynamical analysis. To overcome this limitation, we employed the projection reduction method to reduce the dimensionality of the model. Our approach demonstrated that the reduced model retained the inherent bursting characteristics of the original model. Following reduction, we investigated the bi-parameter bifurcation of the equilibrium point in the reduced model. Specifically, we analyzed the Bogdanov-Takens bifurcation that arises in the reduced system. Notably, the topological structure of the neuronal model near the bifurcation point can be effectively revealed with our proposed method. By leveraging the proposed projection reduction method, we could explore the bursting mechanism in the reduced Pospischil model with greater precision. Our approach offers an effective foundation for generating theories and hypotheses that can be tested experimentally. Furthermore, it enables links to be drawn between neuronal morphology and function, thereby facilitating a deeper understanding of the complex dynamical behaviors that underlie intrinsic bursting neurons.

Motivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system: x˙=a(y−x)\dot{x}=a(y-x), y˙=cx3+dy−x3z\dot{y}=c\sqrt[3]{x}+{\rm{d}}y-\sqrt[3]{x}z and z˙=−bz+x3y\dot{z}=-bz+\sqrt[3]{x}y. Instead, the unstable manifolds of nontrivial symmetrical equilibria tending to the stable manifold of the origin creates a pair of heteroclinic orbits. This drives one to further investigate it and reveal its other hidden dynamics: Hopf bifurcation, invariant algebraic surfaces, ultimate bound sets, globally exponentially attractive sets, existence of homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and so on. The main contributions of this work are summarized as follows: First, the ultimate boundedness of that system yields the globally exponentially attractive sets of it. Second, the existence of another heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, on the invariant algebraic surface z=34ax43z=\frac{3}{4a}\sqrt[3]{{x}^{4}}, the existence of a pair of homoclinic orbits to the origin, and two pairs of heteroclinic orbits to two pairs of nontrivial symmetrical equilibria is also proved by utilizing a Hamiltonian function. In addition, the correctness of the theoretical results is illustrated via numerical examples.

In this paper, we investigate the generalized Radhakrishnan–Kundu–Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. Then, we discuss the dynamical behavior of the associated regular system and we obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, and the peakon, homoclinic and heteroclinic solutions. [ABSTRACT FROM AUTHOR]

Bifurcations and chaos of a three-dimensional single-loop feedback system with a discontinuous piecewise linear feedback function are examined. Chaotic attractors are generated at the same time of the destabilization of foci accompanied with grazing. Multiple periodic solutions are connected with homoclinic orbits based at a pseudo saddle-focus, which satisfies the condition of Shil'nikov chaos formally. The generation of chaotic oscillations is shown in a circuit experiment on a linear ring oscillator with a comparator. The homoclinic bifurcations and chaos are also shown in a ring neural network with a nonmonotonic neuron. [ABSTRACT FROM AUTHOR]

Intrinsic bursting neurons represent a common neuronal type that displays bursting patterns upon depolarization stimulation. These neurons can be described by a system of seven-dimensional equations, which pose a challenge for dynamical analysis. To overcome this limitation, we employed the projection reduction method to reduce the dimensionality of the model. Our approach demonstrated that the reduced model retained the inherent bursting characteristics of the original model. Following reduction, we investigated the bi-parameter bifurcation of the equilibrium point in the reduced model. Specifically, we analyzed the Bogdanov-Takens bifurcation that arises in the reduced system. Notably, the topological structure of the neuronal model near the bifurcation point can be effectively revealed with our proposed method. By leveraging the proposed projection reduction method, we could explore the bursting mechanism in the reduced Pospischil model with greater precision. Our approach offers an effective foundation for generating theories and hypotheses that can be tested experimentally. Furthermore, it enables links to be drawn between neuronal morphology and function, thereby facilitating a deeper understanding of the complex dynamical behaviors that underlie intrinsic bursting neurons. [ABSTRACT FROM AUTHOR]

The bifurcations and chaos of a ring of three unidirectionally coupled neuron-like elements are examined as a minimal chaotic neural network. The output function of one neuron is nonmonotonic and piecewise constant while those of the other two neurons are linear. Two kinds of nonmonotonic output functions are considered and it is shown that periodic solutions undergo grazing bifurcations owing to discontinuity in the nonmonotonic functions. Chaotic attractors are created directly through a grazing bifurcation and homoclinic orbits based at pseudo steady states are generated. It is shown that homoclinic/heteroclinic orbits satisfying the condition of Shil'nikov chaos are caused by overshoot in the nonmonotonic functions. [ABSTRACT FROM AUTHOR]

We consider the non-semisimple 0:1 resonance (i.e. the unperturbed equilibrium has two purely imaginary eigenvalues ± i α ( α ∈ R and α > 0) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432], this bifurcation essentially triggered by the reversible universal unfolding M = 1 2 p 2 + 1 4 q 4 + (λ + I 1) q 2 with respect to BCKV-restricted morphisms of the planar non-semisimple singularity 1 2 p 2 + 1 4 q 4 (the I 1 is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the S 1 -symmetry generated by the additional pair of imaginary eigenvalues ± i α to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g. 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432]. An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation. [ABSTRACT FROM AUTHOR]

We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of three-dimensional flows and maps. The presented results are illustrated with various specific examples. [ABSTRACT FROM AUTHOR]

We obtain explicit formulas for the separatrix map of a multidimensional slow–fast Hamiltonian system. This map is used to partly extend Neishtadt's results on the jumps of adiabatic invariants at the separatrix to the multidimensional case. [ABSTRACT FROM AUTHOR]

Many physical and engineering systems have certain symmetric properties. Homoclinic orbits play an important role in studying the global dynamics of dynamical systems. This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincaré maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant D to study the homoclinic bifurcations to limit cycles for the case | λ 1 | = λ 3 . Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map. [ABSTRACT FROM AUTHOR]

Chaotic oscillations induced by single rectification in networks of linear neuron-like elements are examined on three prototype models: one nonautonomous system and two autonomous systems. The first is a system of coupled neurons with periodic input; the second is a system of three coupled neurons with six couplings; the third is a ring of four unidirectionally coupled neurons with one reverse coupling. In each system, the output function of one neuron is ramp and that of the others is linear. Each system is piecewise linear and the phase space is separated into two domains by a single border. Steady states, periodic solutions and homoclinic orbits are derived rigorously and their stability is evaluated with the eigenvalues of the Jacobian matrices. The bifurcation analysis of the three systems shows that chaotic attractors could be generated through cascades of period-doubling bifurcations of periodic solutions. [ABSTRACT FROM AUTHOR]

Following Part I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under some conditions. We next extend the Melnikov method of Part I to reversible systems and obtain some theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones. [ABSTRACT FROM AUTHOR]

Chaotic dynamics of an impulse Duffing-van der Pol system is studied in this paper. With the Melnikov method, the existence condition of transversal homoclinic point is obtained, and chaos threshold is presented. In addition, numerical simulations including phase portraits and time histories are carried out to verify the analytical results. Bifurcation diagrams are also given, from which it can be seen that the system may undergo chaotic motions through period doubling bifurcations. [ABSTRACT FROM AUTHOR]

Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting. [ABSTRACT FROM AUTHOR]

The existence of homoclinic orbits is discussed analytically for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, that is, under the same parameter setting, homoclinic orbits and chaos appear simultaneously in the system. In addition, homoclinic chaos can be suppressed to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

In this paper, we investigate the generalized Radhakrishnan–Kundu–Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. Then, we discuss the dynamical behavior of the associated regular system and we obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, and the peakon, homoclinic and heteroclinic solutions.

Let ⊂ ℝ n be a chaotic attractor generated by a quadratic system of ordinary differential equations ẋ = f (x). A method for constructing new chaotic attractors based on the attractor is proposed. The idea of the method is to replace the state vector x = (x 1 , ... , x n) T located on the right side of the original system with new vector u (x) ; where u (x) = K ⋅ (h 1 (x 1) , ... , h n (x n)) T , K ∈ ℝ n × n , and h i (x i) are odd power functions; i = 1 , ... , n. (In other words, a state feedback x → u (x) is introduced into the right side of the system under study: ẋ = f (x) → ẋ = f (u (x)).) As a result, the newly obtained system generates new chaotic attractors, which are topologically not equivalent (generally speaking) to the attractor . In addition, for an antisymmetric neural ODE system with a homoclinic orbit connected at a saddle point, the conditions for the occurrence of chaotic dynamics are found. [ABSTRACT FROM AUTHOR]

The bifurcations and chaos in autonomous systems of two- and three-coupled ramp-type neurons are considered. An asymmetric piecewise linear function is employed for the output function of neurons in order to examine changes in the bifurcations from a sigmoid output function to a ramp output function. Steady solutions in the systems are obtained exactly and they undergo discontinuous bifurcations because the systems are piecewise linear. Periodic solutions and homoclinic/heteroclinic orbits in the systems are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. The bifurcations of the periodic solutions are calculated with the Poincaré maps and the Jacobian matrices, which are also derived rigorously. A stable periodic solution in a two-neuron oscillator of the Wilson–Cowan type with three couplings remains in the case of a ramp neuron. A chaotic attractor of Rössler type emerges in a network of three ramp neurons with six couplings, which is due to two saddle-focuses. The network consists of the two-neuron oscillator and one bypass neuron connected through three couplings. One-dimensional Poincaré maps show the generation of the chaotic attractor through a cascade of period-doubling bifurcations. Further, multiple homoclinic orbits based at a saddle are generated from the destabilization of two focuses when asymmetry in the output function is large. This homoclinicity causes qualitative change in the bifurcations of the periodic solutions as the output function of neurons changes from sigmoid to ramp. [ABSTRACT FROM AUTHOR]

We study the existence of fast homoclinic orbits for the following damped vibration system u ¨ (t) + q (t) u ˙ (t) + ∇ V (t , u (t)) = 0 ; where q ∈ C (R , R) and V ∈ C 1 (R × R N , R) is of the type V(t,x)=-K(t,x)+W(t,x). A map K is not a quadratic form in x and W(t, x) is superquadratic in x. [ABSTRACT FROM AUTHOR]

Accurate predictions for the spread and evolution of epidemics have significant societal and economic impacts. The temporal evolution of infected (or dead) persons has been described as an epidemic wave with an isolated peak and tails. Epidemic waves have been simulated and studied using the classical SIR model that describes the evolution of susceptible (S), infected (I), and recovered (R) individuals. To illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions, we propose a Korteweg–de Vries (KdV)–SIR equation and obtain its analytical solutions. Among classical and simplified SIR models, our KdV–SIR equation represents the simplest system that produces a solution with both exponential and oscillatory components. The KdV–SIR model is mathematically identical to the nondissipative Lorenz 1963 model and the KdV equation in a traveling-wave coordinate. As a result, the dynamics of an epidemic wave and its predictability can be understood by applying approaches used in nonlinear dynamics, and by comparing the aforementioned systems. For example, a typical solitary wave solution is a homoclinic orbit that connects a stable and an unstable manifold at the saddle point within the I – I ′ space. The KdV–SIR equation additionally produces two other types of solutions, including oscillatory and unbounded solutions. The analysis of two critical points makes it possible to reveal the features of solutions near a turning point. Using analytical solutions and hypothetical observed data, we derive a simple formula for determining predictability horizons, and propose a method for predicting timing for the peak of an epidemic wave. [ABSTRACT FROM AUTHOR]

The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices. [ABSTRACT FROM AUTHOR]

This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize the dynamics of the four-dimensional phase space of a Hamiltonian system with non-semisimple 0:1 resonance related to the reversible butterfly catastrophe. Through the normalization procedure, the Hamiltonian can be easily represented as the integrable part N (1 2 (x 2 + y 2) , p , q ; λ) plus a small perturbation P (x , y , p , q ; λ) . Firstly, we give the bifurcation diagram of the integrable part N on each level of the integral I 1 = 1 2 (x 2 + y 2) in detail, which contains three subordinate local codimension-one bifurcations (subcritical/supercritical Hamiltonian pitchfork bifurcation and Hamiltonian centre-saddle bifurcation) as well as a subordinate global codimension-one connection bifurcation. Then, we prove the persistence of typical bifurcation scenarios (e.g., two-dimensional invariant tori and homoclinic orbits) under the small perturbation P , as described by Broer et al. (Z Angew Math Phys 44:389–432, 1993). [ABSTRACT FROM AUTHOR]

We obtain an existence theorem of nonzero solution for a class of bounded selfadjoint operator equations. The main result contains as a special case the existence result of a nontrivial homoclinic orbit of a class of Hamiltonian systems by Coti Zelati, Ekeland and Séré. We also investigate the existence of nontrivial homoclinic orbit of indefinite second order systems as another application of the theorem.

In this paper, we analytically and geometrically investigate the complexity of Maxwell–Bloch system by giving new insight. In the first place, the existence of homoclinic orbits is rigorously proved by means of the generalized Melnikov method. More precisely, for 6a−2b>c and d>0, it is certified analytically that Maxwell–Bloch system has two nontransverse homoclinic orbits. Secondly, Jacobi stability on the orbits of Maxwell–Bloch system is examined in view point of Kosambi–Cartan–Chern theory (KCC-theory). In other words, in the light of the deviation curvature tensor of the five corresponding invariant associated to the reformulated Maxwell–Bloch system, we further proved that Jacobi stability of all equilibria under appropriate parameters. Moreover, the deviation vector, as well as the curvature of the deviation vector near equilibrium points, is focused to interpret the chaotic behavior of Maxwell–Bloch system in Finsler geometry. [ABSTRACT FROM AUTHOR]

The object of this paper is to study the existence and nonexistence of an important orbit in a generalized Liénard type system. This trajectory is doubly asymptotic to an equilibrium solution, i.e., an orbit which lies in the intersection of the stable and unstable manifolds of a critical point. Such an orbit is called a homoclinic orbit.

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system. • Periodic perturbations of single-degree-of-freedom Hamiltonian systems are studied. • They are assumed to have homo- or heteroclinic orbits. • Their real-meromorphic nonintegrability near the orbits is discussed. • They are shown to be nonintegrable if the Melnikov functions are not constant. • The result is applied to three examples including the Duffing oscillators. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation with nonlinear dispersion is investigated. By using the bifurcation-theoretic method of planar dynamical systems, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons as well as compacton solutions for this planar dynamical system are obtained. Under different parameter conditions, solutions can be exactly obtained. Fifteen exact explicit solutions of the traveling wave system are derived. [ABSTRACT FROM AUTHOR]

This paper presents new geometric viewpoints to Chen chaotic system. Firstly, the existence of two nontransverse homoclinic orbits in Chen system is rigorously proved beyond the classical parameters. Secondly, combined with the theory of tangent bundle, a new geometric viewpoint is given to explore chaos mechanism of Chen system. The fundamental geometric definition of tangent bundle and the essential role of nonlinear connection between the tangent space and the base space are described. By introducing the geometrical viewpoints of second order system governed by Lie-Poisson equation, some geometric invariants of Chen system can be obtained. Furthermore, the torsion tensor as one of the geometric invariants is obtained, and it gives the geometrical interpretation to the chaotic behaviour of Chen system. Finally, the torsion tensor of Chen system and Lorenz system are also compared. The obtaining results show that torsion tensor change will lead the Chen system from periodic to chaotic, which is not found in Lorenz system. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation { u t t − u x x − α u + β f ∗ u | u | 2 − 2 u v − τ u ( | u | 2) x = 0 , v t t + v x x − β ( | u | 2) x x = 0. We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method. Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Fredholm theory. According to the relationship between solitary wave and homoclinic orbit, the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equations with perturbation also possess homoclinic orbit, and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation. [ABSTRACT FROM AUTHOR]

This paper aims at investigating the existence of localized stationary waves in the shallow subsurface whose constitutive behavior is governed by the hyperbolic model, implying non-polynomial nonlinearity and strain-dependent shear modulus. To this end, we derive a novel equation of motion for a nonlinear gradient elasticity model, where the higher-order gradient terms capture the effect of small-scale soil heterogeneity/micro-structure. We also present a novel finite-difference scheme to solve the nonlinear equation of motion in space and time. Simulations of the propagation of arbitrary initial pulses clearly reveal the influence of the nonlinearity: strain-dependent speed in general and, as a result, sharpening of the pulses. Stationary solutions of the equation of motion are obtained by introducing the moving reference frame together with the stationarity assumption. Periodic (with and without a descending trend) as well as localized stationary waves are found by analyzing the obtained ordinary differential equation in the phase portrait and integrating it along the different trajectories. The localized stationary wave is in fact a kink wave and is obtained by integration along a homoclinic orbit. In general, the closer the trajectory lies to a homoclinic orbit, the sharper the edges of the corresponding periodic stationary wave and the larger its period. Finally, we find that the kink wave is in fact not a true soliton as the original shapes of two colliding kink waves are not recovered after interaction. However, it may have high amplitude and reach the surface depending on the damping mechanisms (which have not been considered). Therefore, seismic site response analyses should not a priori exclude the presence of such localized stationary waves. [ABSTRACT FROM AUTHOR]