Hyperbolic Equilibrium Research Articles

Nonlinear Compartment Models with Time-Dependent Parameters

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

Estabilidad en sistemas cuadráticos del tipo Kolmogorov describiendo interacciones entre dos especies. Una breve revisión

Population dynamics is a relevant topic in Biomathematics, being the study of the long-term behavior of interaction models between species, one of its central problems. A large part of these relationships are described by ordinary differential equations (ODE), having as main objectives the study of the stability of their solutions. In this document we mainly describe the dynamic behavior of the Volterra predation model. In addition, we make a review of some derived predation models and a brief review of the dynamical properties of models describing other interactions between species such as: competition, mutualism, amensalism, and commensalism; also described by nonlinear ODE systems of the second order of Kolmogorov-type. For each of these models, the non-existence of limit cycles can be demonstrated and in most of them, there is a globally stable equilibrium point. In one of them, there are conditions in the parameters for which the only positive equilibrium point is a center, as in the original Lotka-Volterra model. The methodology used is the usual one for the analysis of models with hyperbolic equilibrium points, but it can guide the analysis of other more complicated models.

Dynamical systems under random perturbations with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles

This work is devoted to the study of long-term qualitative behavior of randomly perturbed dynamical systems. The focus is on certain stochastic differential equations (SDE) with Markovian switching, when the switching is fast varying and the diffusion (white noise) is slowly changing. Consider the system d X ε , δ ( t ) = f ( X ε , δ ( t ) , α ε ( t ) ) d t + δ σ ( X ε , δ ( t ) , α ε ( t ) ) d W ( t ) , X ε , δ ( 0 ) = x , α ε ( 0 ) = i , where α ε ( t ) is a finite state Markov chain with irreducible generator Q = ( q ι ℓ ) . The relative changing rates of the switching and the diffusion are highlighted by two small parameters ε and δ . Associated with the above stochastic differential equation, there is an averaged ordinary differential equation (ODE) d X ‾ ( t ) = f ‾ ( X ‾ ( t ) ) d t , X ‾ ( 0 ) = x , where f ‾ ( ⋅ ) = ∑ ι = 1 m 0 f ( ⋅ , ι ) ν ι and ( ν 1 , … , ν m 0 ) is the unique stationary distribution of the Markov chain with generator Q . Suppose that for each pair ( ε , δ ) , the process has an invariant probability measure μ ε , δ , and that the averaged ODE has a limit cycle in which there is an averaged occupation measure μ 0 for the averaged equation. It is proved in this paper that under weak conditions, if f ‾ has finitely many stable or hyperbolic fixed points, then μ ε , δ converges weakly to μ 0 as ε → 0 and δ → 0 . Our results generalize to the setting where the switching process α ε is state-dependent in that P { α ε ( t + Δ ) = ℓ | α ε = ι , X ε , δ ( s ) , α ε ( s ) , s ≤ t } = q ι ℓ ( X ε , δ ( t ) ) Δ + o ( Δ ) , ι ≠ ℓ as long as the generator Q ( ⋅ ) = ( q ι ℓ ( ⋅ ) ) is locally bounded, locally Lipschitz, and irreducible for all x ∈ R d . Finally, we provide two examples in two and three dimensions to showcase our results.

Linear and center manifold analysis of FRW cosmological model with variable equation of state in Lyra geometry

The present work deals with the dynamical evolution of flat FRW cosmological model with variable equation of state in Lyra geometry. We investigate the phase-plane analysis of our constructed model. The evolution of cosmological solutions is studied by using dynamical system techniques. Stability and viability of the model are discussed using linear Jacobian stability analysis for hyperbolic equilibrium points and center manifold theory for non-hyperbolic equilibrium points.

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

A New Five-Dimensional Hyperchaotic System with Six Coexisting Attractors

This article presents a hyperchaotic system of five-dimensional autonomous ODEs that has five cross-product nonlinearities. Under certain parametric conditions, it exhibits three different types of hyperchaotic and chaotic systems which correspond to six hyperchaotic attractors with a non-hyperbolic equilibrium line, four chaotic attractors with seventeen hyperbolic equilibria, and four chaotic attractors with only one hyperbolic equilibrium, respectively. The fundamental dynamics are analyzed theoretically and numerically, such as the onset of hyperchaos and chaos, routes to chaos, persistence of chaos, coexistence of attractors, periodic windows and bifurcations. It is particularly shown that the coexisting attractors of the 5D system inside the hypercone are symmetric. Some dynamical characteristics of these attractors are illustrated.

Suppression of chaos in nonlinear oscillators using a linear vibration absorber

This paper proposes a nonfeedback control to suppress the chaotic response of nonlinear oscillators. A linear vibration absorber is used as nonfeedback method, whose key idea is to find conditions under which the nonlinear oscillator response converges to an equilibrium point or stay oscillating around it. Theoretical results show that if the ratio between the natural frequencies of the primary system $$(\omega _1)$$ and the undamped absorber $$(\omega _2)$$ , that is, $$\omega _r=\frac{\omega _2}{\omega _1}$$ is tuned to be equal to the excitation frequency $$(\Omega )$$ , then the chaotic behavior of a nonlinear oscillator is driven to a stable hyperbolic equilibrium point. Numerical results are presented for the Duffing oscillator shown that if $$\omega _r$$ is tuned close to excitation frequency, then the chaotic response of the Duffing oscillator is driven to periodic orbits. In the case of damped absorber, $$\omega _r$$ to be tuned close to the excitation frequency does not guarantee that the response of the primary system is periodic, it is also necessary to take into account the amplitude of the excitation.

Overdetermined constraints and rigid synchrony patterns for network equilibria

In network dynamics, synchrony between nodes defines an equivalence relation, usually represented as a colouring. If the colouring is balanced, meaning that nodes of the same colour have colour-isomorphic inputs, it determines a subspace that is flow-invariant for any ODE compatible with the network structure. Therefore any state lying in such a subspace has the synchrony pattern determined by that balanced colouring. In 2005 Golubitsky and coworkers proved a strong converse for synchronous equilibria: every rigid synchrony colouring for a hyperbolic equilibrium is balanced, where rigidity means that the pattern persists under small admissible perturbations. We give a different proof of this theorem, based on overdetermined constraint equations, Sard’s Theorem, bump functions, and groupoid symmetrisation.

Local regularization of a restricted problem on [formula omitted] with primaries on hyperbolic motion

In this paper we perform a local regularization of singularities in the restricted (n+1)−body problem on a two-dimensional hyperbolic space using different coordinate and time transformations. The motion of the primary particles is known and their orbit is given by a hyperbolic relative equilibrium. We show some numerical examples of the orbit of the massless particle.

Chazy-Type Asymptotics and Hyperbolic Scattering for the n-Body Problem

We study solutions of the Newtonian $n$-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t \rightarrow +\infty$ or as $t \rightarrow -\infty$. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold "at infinity". We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy's classical asymptotic formulas. We also address the scattering problem, namely, for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering "near infinity", that is, when the bodies stay far apart and interact only weakly.

Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark–Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearance, studying its dynamics between these events, and to (B) study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range, focusing on their role as transport barriers and their participation in global bifurcations. Taken together the results highlight the main features of the global dynamics of the system.

Higher order normal modes

Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.

Identification of trapping in a peristaltic flow: A new approach using dynamical system theory

In this paper, we propose a new approach for the identification of characteristic peristaltic flow features such as “bolus” and “trapping.” Using dynamical system analysis, we relate the occurrence of a bolus to the existence of a center (an elliptic equilibrium point). Trapping occurs when centers exist under the wave crests along with a pair of saddles (hyperbolic equilibrium points) lying on the central line. For an augmented flow, centers form under the wave crests, whereas saddles lie above (below) the central line. The proposed approach works much better than the presently adopted approach in two ways: (1) it does not require random testing and (2) it characterizes the qualitative flow behavior for the complete range of parameter values. The literature is somewhat inconsistent with regard to the terminologies used for describing characteristic flow behaviors. We have addressed this issue by explicitly defining quantities such as “bolus,” “backward flow,” “trapping,” and “augmented flow.”

A novel 4D chaotic system with nonhyperbolic hyperbolic shape equilibrium points: Analysis, circuit implementation and color image encryption

This paper presents a new 4D chaotic system by extending a 3D system to investigate chaotic systems2 with hidden attractors further and explore their engineering applications in color image encryption. The system has hyperbolic shape equilibriums and the equilibriums are nonhyperbolic which is interesting and rarely mentioned before. The system has been investigated deeply by phase plot, time series, the largest Lyapunov exponent, bifurcation diagram, Poincaré map and 0–1 test. A coexisting hidden attractor has been found in this new system and the circuit simulation model of the system is built by Power Simulation (PSIM). Circuit simulation results are consistent with results of numerical simulations which verify its dynamic behavior further. Then, by combining the generalized Arnold transform with the new chaotic system, a new color image encryption algorithm has been proposed which has large key space and can encrypt either square images or nonsquare images with different size. Finally, the results of encryption experiment and security analysis prove the effectiveness of the algorithm.

Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System

We consider a Hamiltonian system depending on a parameter which slowly changes with rate e ≪ 1. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order e. We prove a partial analog of Neishtadt’s result for a system with n degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order e at time intervals of order ∣ln e∣, so the energy may grow with rate e/∣ln e∣. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order e.

Local stable manifolds for nonlinear planar fractional differential equations with order 1<α<2

This paper reports the local stable manifolds near hyperbolic equilibria for nonlinear planar fractional differential equations of order 1<α<2. By using several useful estimates of Mittag‐Leffler function and fractional calculus technique, we construct two suitable Lyapunov‐Perron operators and set up their fixed points as the desired stable manifolds. We further present a specific example to compute explicitly the corresponding stable manifold as the application.

Complex Dynamical Behaviors in a 3D Simple Chaotic Flow with 3D Stable or 3D Unstable Manifolds of a Single Equilibrium

This paper shows some examples of chaotic systems for the six types of only one hyperbolic equilibrium in changed chameleon-like chaotic system. Two of the six cases have hidden attractors. By adjusting the parameters in the system and controlling the stability of only one equilibrium, we can further obtain chaos with four kinds of conditions: (1) index-0 node; (2) index-3 node; (3) index-0 node foci; (4) index-3 node foci. Based on the method of focus quantities, we study three limit cycles (the outmost and inner cycles are stable, and the intermediate cycle is unstable) bifurcating from an isolated Hopf equilibrium. In addition, one periodic solution can be obtained from a nonisolated zero-Hopf equilibrium. The system may help us in better understanding, revealing an intrinsic relationship of the global dynamical behaviors with the stability of equilibrium point, especially hidden chaotic attractors.

Самопересечение комплексных асимптотических многообразий гиперболического положения равновесия и несуществование первых интегралов в системах типа гамильтоновых с двумя степенями свободы.

Самопересечение комплексных асимптотических многообразий гиперболического положения равновесия и несуществование первых интегралов в системах типа гамильтоновых с двумя степенями свободы.

Constructing a New 3D Chaotic System with Any Number of Equilibria

In the chaotic polynomial Lorenz-type systems (including Lorenz, Chen, Lü and Yang systems) and Rössler system, their equilibria are unstable and the number of the hyperbolic equilibria are no more than three. This paper shows how to construct a simple analytic (nonpolynomial) chaotic system that can have any preassigned number of equilibria. A special 3D chaotic system with no equilibrium is first presented and discussed. Using a methodology of adding a constant controller to the third equation of such a chaotic system, it is shown that a chaotic system with any preassigned number of equilibria can be generated. Two complete mathematical characterizations for the number and stability of their equilibria are further rigorously derived and studied. This system is very interesting in the sense that some complex dynamics are found, revealing many amazing properties: (i) a hidden chaotic attractor exists with no equilibria or only one stable equilibrium; (ii) the chaotic attractor coexists with unstable equilibria, including two/five unstable equilibria; (iii) the chaotic attractor coexists with stable equilibria and unstable equilibria, including one stable and two unstable equilibria/94 stable and 93 unstable equilibria; (iv) the chaotic attractor coexists with infinitely many nonhyperbolic isolated equilibria. These results reveal an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of some unusual chaotic systems.

Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovský and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equations by Lappicy (2017). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is computed yielding the well known Chafee–Infante attractor.