Hyperbolic Equilibrium Research Articles

Two geometrical invariants for three‐dimensional systems

The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.

Transitions in echo index and dependence on input repetitions

The echo index counts the number of simultaneously stable asymptotic responses of a nonautonomous (i.e. input-driven) dynamical system. It generalises the well-known echo state property for recurrent neural networks — this corresponds to the echo index being equal to one. In this paper, we investigate how the echo index depends on parameters that govern typical responses to a finite-state ergodic external input that forces the dynamics. We consider the echo index for a nonautonomous system that switches between a finite set of maps, where we assume that each map possesses a finite set of hyperbolic equilibrium attractors. We find the minimum and maximum repetitions of each map are crucial for the resulting echo index. Casting our theoretical findings in the RNN computing framework, we obtain that for small amplitude forcing the echo index corresponds to the number of attractors for the input-free system, while for large amplitude forcing, the echo index reduces to one. The intermediate regime is the most interesting; in this region the echo index depends not just on the amplitude of forcing but also on more subtle properties of the input.

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

On non-conservative perturbations of three-dimensional integrable systems

At present, non-conservative perturbations of two-dimensional nonlinear Hamiltonian systems have been studied quite fully. The purpose of the study is to generalize this theory to the three-dimensional case, when the unperturbed system is nonlinear, integrable and has a region filled with closed phase trajectories. In this paper, autonomous perturbations are considered and the main attention is paid to the problem of limit cycles. Methods. The study is based on the construction of special coordinates in which the variables are divided into two slow and one fast, and in the first approximation with respect to a small parameter the equations for the slow variables are separated. Results. It is shown that hyperbolic equilibrium states of a truncated system determine closed phase trajectories, in the vicinity of which cycles appear under the perturbation. Conclusion. Thus, the problem is reduced to the study of solutions of the “generating” system of two algebraic or transcendental equations, similar to the generating Poincare–Pontryagin equation for two-dimensional systems. As examples, we considere a three-imensional van der Pol type system and the Lorentz system in the case of large Rayleigh numbers.

Distorted stability pattern and chaotic features for quantized prey-predator-like dynamics.

Nonequilibrium and instability features of prey-predator-like systems associated to topological quantum domains emerging from a quantum phase-space description are investigated in the framework of the Weyl-Wigner quantum mechanics. Reporting about the generalized Wigner flow for one-dimensional Hamiltonian systems, H(x,k), constrained by ∂^{2}H/∂x∂k=0, the prey-predator dynamics driven by Lotka-Volterra (LV) equationsis mapped onto the Heisenberg-Weyl noncommutative algebra, [x,k]=i, where the canonical variables x and k are related to the two-dimensional LV parameters, y=e^{-x} and z=e^{-k}. From the non-Liouvillian pattern driven by the associated Wigner currents, hyperbolic equilibrium and stability parameters for the prey-predator-like dynamics are then shown to be affected by quantum distortions over the classical background, in correspondence with nonstationarity and non-Liouvillianity properties quantified in terms of Wigner currents and Gaussian ensemble parameters. As an extension, considering the hypothesis of discretizing the time parameter, nonhyperbolic bifurcation regimes are identified and quantified in terms of z-y anisotropy and Gaussian parameters. The bifurcation diagrams exhibit, for quantum regimes, chaotic patterns highly dependent on Gaussian localization. Besides exemplifying a broad range of applications of the generalized Wigner information flow framework, our results extend, from the continuous (hyperbolic regime) to discrete (chaotic regime) domains, the procedure for quantifying the influence of quantum fluctuations over equilibrium and stability scenarios of LV driven systems.

The role of tumor activation and inhibition with saturation effects in a mathematical model of tumor and immune system interactions undergoing oncolytic viral therapy

We propose and study a mathematical model governing interactions between cancer and immune system with an oncolytic viral therapy (OVT), wherein cancer cells can activate and inhibit immune cells simultaneously with saturations. When the therapy is not applied, it is shown that the interaction can support at most three hyperbolic positive equilibria where two of them are always asymptotically stable and the other is a saddle point. The reachable stable tumor burden can be either small or large depending on initial tumor size. We analyze the full model by proving global asymptotic stability of the virus‐free equilibrium that corresponds to OVT failure. Sufficient conditions based on model parameters are derived under which the model is uniformly persistent. The proposed system is validated using a mouse model of human pancreatic cancer carried out by Koujima et al. Global sensitivity analysis indicates that the rates of tumor‐mediated killing and immune cell exhaustion are critical for tumor progression and therapy success. Numerical bifurcation analysis reveals that the saddle point can be utilized to estimate the maximum tumor load for eradication by OVT. Moreover, an immunosuppressive microenvironment may enhance viral therapy efficacy.

A New Simple 6D Hyperchaotic System with Hyperbolic Equilibrium and Its Electronic Circuit

This paper represents a sequential design of controlling from a 3D chaotic system to a simple 6D hyperchaotic system by adding three linear controllers via state feedback control and coupling strategies. The new 6D system has a unique non-hyperbolic equilibrium point with four positive Lyapunov exponents (LE). Moreover, some dynamic properties are analyzed which inclined equilibrium and their stability, dissipative, conservative, and multistability. In addition, the corresponding analog electronic circuit is designed and implemented to verify the new simple 6D hyperchaotic system. Finally, the NI Multisim simulation results observed from the digital oscilloscope are consistent with the mathematical simulation based on the Matlab R2021a.

Invariant family of leaf measures and the Ledrappier–Young property for hyperbolic equilibrium states

AbstractThe manifold M is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$ , and f is a $C^{1+\beta }$ ( $\kern0.3pt\beta>0$ ) diffeomorphism of M. $\varphi $ is a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi $ ), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_\chi (p)$ , we prove that $\varphi $ admits a local equilibrium state on $H_\chi (p)$ if and only if $\varphi $ is ‘recurrent on $H_\chi (p)$ ’ (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitute as its conditional measures. An immediate corollary is the local product structure of hyperbolic equilibrium states. Second, we prove a Ledrappier–Young property for hyperbolic equilibrium states: if $\varphi $ admits a conformal family of leaf measures and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi $ ) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic Sinai–Ruelle–Bowen measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure zero) has conditional measures on local unstable leaves which are absolutely continuous with respect to the Riemannian volume of these leaves.

Stability indices of non-hyperbolic equilibria in two-dimensional systems of ODEs

We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number , we show that it is possible to choose a parameter in these equations such that the stability index is precisely . In contrast to that, for a hyperbolic equilibrium x it is known that either or . Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.

Stable and unstable flow regimes for active fluids in the periodic setting

Depending on the involved physiobiological parameters, stable or unstable behavior in active fluids is observed. In this paper a rigorous analytical justification of (in-)stability within the corresponding regimes is given. In particular, occurring instability for the manifold of ordered polar states caused by self-propulsion is proved. This represents the prerequisite for active turbulence patterns as observed in a number of applications. The approach is carried out in the periodic setting and is based on the generalized principle of linearized (in)-stability related to normally stable and normally hyperbolic equilibria.

Sturm attractors for fully nonlinear parabolic equations

We explicitly construct global attractors of fully nonlinear parabolic equations in one spatial dimension. These attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and forwards in time). In particular, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria, which is accompanied by a method that computes such conditions.

Equilibrium points in restricted problems on S2 and H2

In this paper, we consider the restricted four-body problem on S2 and the restricted three-body problem on H2. In the first case, the primary particles are considered to be rotating around the vertical axis. In the latter case, the primaries move at a hyperbolic relative equilibrium. If the primary particles locate at height z0 = 0 and they form an isosceles triangle on S2, then the number of symmetric equilibrium points of the restricted four-body problem depends on the angular velocity, the configuration, and the masses of the primaries. For this case, the number of equilibrium points is 0, 2, 4, 6, or 8. If the primaries are at height z0 ∈ (0, 1), then the primaries form an equilateral triangle and the number of symmetric equilibrium points is 8, 11, 14, 17, or 20. The number depends on the position of the primaries, and it is higher than the previous case due to the higher number of symmetry axes of the triangle formed by the primaries. On the other hand, for the restricted three-body problem on H2, the number of equilibrium points is 3, considering that the primaries move at a hyperbolic relative equilibrium. We analyzed the general (non-symmetric) case, showing the locations of the equilibrium points for the given positions of the primaries. We also studied the stability of these equilibrium points.

Estimating rate-induced tipping via asymptotic series and a Melnikov-like method* *CK was partly supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges partial support of the EU within the TiPES project funded the European Unions Horizon 2020 research and innovation programme under Grant Agreement No. 820970. IPL was partly supported by MICIIN/FEDER project RTI2018-096523-B-100 and by European Union’s

The paper deals with the study of rate-induced tipping in asymptotically autonomous scalar ordinary differential equations. We prove that, in such a tipping scenario, a solution which limits at a hyperbolic stable equilibrium of the past limit-problem loses uniform asymptotic stability and coincides with a solution which limits at a hyperbolic unstable equilibrium of the future limit-problem. We use asymptotic series to approximate such pairs of solutions and characterize the occurrence of a rate-induced tipping by using only solutions calculable on finite time intervals. Moreover, we show that a Melnikov-inspired method employing the asymptotic series allows to asymptotically approximate the tipping point.

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in [formula omitted] with [formula omitted]-symmetry and integral of motion

We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

Morse Index of Saddle Equilibria of Gradient-Like Flows on Connected Sums of $$\mathbb S^{n-1}\times \mathbb S^1$$

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.

Isolation and Characterization of Leishmanial Adenine Aminohydrolase as a Drug Target.

Search for new drug targets is becoming imperative these days, given that marketed chemotherapeutic drugs have lost their efficacy against harmful agents because of adaptability to climatic changes and co-evolving vectors to new hosts. In the wake of such a challenge, the prominence of biochemical studies is increasing by way of exploring selective enzymes and investigating their structural and functional properties through biochemical kinetic parameter Km for the application of IC50 using designed drugs. Recently, discovered Adenine Aminohydrolase (EC 3.5.4.2) in Leishmania has been found to be absent in mammalian purine salvage pathway and thus considered as a promising drug target against infectious agents. The objective of this study is to isolate and characterize AAH by learning its kinetic mode of action using preferred substrate Adenine and additives estimated through expected product formation Hypoxanthine. Bioassays designed to measure exact Enzyme kinetic parameter Km value through establishing hyperbolic curve of an enzyme reaction with the use of exact values of cellular quantities for IC50 application under experimental conditions devised by presteady state approach for SSA validity. Following saturation kinetic, the plot of hyperbolic equilibrium curve developed using initial rates of product formation as a function of (Si) through forward shift under circumstance dG0 the system allows product and reactant favored reactions in relation to (Ef) ≈ [E0 = KM] until complete saturation and estimates Km and Vmax of enzyme system under applied conditions. M-M equation used to assess experimental initial rate data for estimation of Km on excel using Solver and nonlinear least square coefficient correlation "R2" using logarithmic equation for nonlinear curve assessment. UV/Vis spectrophotometer selectively analyzed reacting components confirming Enzyme characteristic reaction constant Km equal toi15. 0 ± 2 μ mol acquired from the Hyperbolic curve developed through the use of exact (Si) ranges at selected parameter Km and Vmax. The curve assessed by Michaelis Menten equation provides Km value=14.99 μmol and non-linear least square coefficient correlation "R2" value equal to 0.9895, along with that optimized lysis buffer formulation. In the docked complexes, the interactive amino acids identified were MSE441, ALA 364, GLN363, MSE518, VAL362, GLY517, ASP538, ALA445, TYR521, and TYR444. 2D interactions revealed hydrophobic and alkyl interactions at the noncompetitive binding site of the enzyme and therefore recommended as potential inhibitors against 3ICS protein. This study encourages biochemical analysis of the novel enzymes with the use of presteady state rationale in association with the computational tools as an effective way of designing drugs in a short time against selective enzymes to meet the current challenge efficiently.

Some Classical Methods in the Analysis of an Aedes aegypti Model


 
 
 
 The Taylor series approximation is often used to convert non-linear dynamical systems to linear systems, while the Hartman-Großman theorem analyzes the local qualitative behavior of the non-linear system around a hyperbolic equilibrium point. The global stability of an equilibrium point in the Lyapunov sense is based on the principle that if the equilibrium point is disturbed and the flow of the system is dissipative, then the system must be stable. This article applies these methods to an ecological Aedes aegypti model, whose local and global stability are characterized by a population growth threshold. In conclusion, the classical theory of dynamical systems, validated computationally, yields theoretical results in favor of controlling the local population of Aedes aegypti. It becomes usable if the proposed model is reinforced with the estimation of the parameters that describe the relationships between stages (aquatic and aerial) of the mosquito population and the inclusion of vector control strategies to protect people from the viruses transmitted by Aedes aegypti.
 
 
 

The turnpike property in nonlinear optimal control—A geometric approach

This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike behavior to occur. The approach taken is geometric and gives insights for the behaviors of controlled trajectories as well as links between the turnpike property and stability and/or stabilizability in nonlinear control theory. It also allows us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target states are also discussed based on the geometry of (un)stable manifolds and a recent result on exponential stabilizability of nonlinear control systems obtained by one of the authors.

The set of hyperbolic equilibria and of invertible zeros on the unit ball is computable

In this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow f defined on the closed unit disk, outputs the exact number of the (hyperbolic) equilibrium points and their locations with arbitrary accuracy. By arbitrary accuracy it is meant that any accuracy required by the input can be achieved. The algorithm can be further extended to a root-finding algorithm that computes the exact number of zeros as well the location of each zero of a continuously differentiable function f defined on the closed unit ball of Rd, provided that the Jacobian of f is invertible at each zero of f; moreover, the computation is uniform in f.

Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold.Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system.In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics is quantified.We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.