László Hatvani

• DSc, member of HAS
• Professor Emeritus at University of Szeged

The equation \[x''(t)=a(t,x(t))+b(t,x)+d(t,x)e(x'(t))\] is considered, where $a:\mathbb{R}^2\to\mathbb{R}$, $b,d:\mathbb{R}\times C(\mathbb{R},\mathbb{R})\to\mathbb{R}$, $e:\mathbb{R}\to\mathbb{R}$ are continuous, and $a,b,d$ are $T$-periodic with respect to $t$. Using the Leray–Schauder degree theory we prove that a sign condition, in which $a$ do...

By the use of interval methods it is proven that there exists an unstable periodic solution to the damped and periodically forced pendulum around the upper equilibrium. It is also proved that this solution can be stabilized by a control which does not need the knowledge of values of the state variables but of the unstable periodic solution.

The equation x′′+h(t,x,x′)x′+f(x)=0(x∈R,xf(x)≥0,t∈[0,∞))is considered, where the damping coefficient h allows an estimate a(t)|x′|αw(x,x′)≤h(t,x,x′)≤b(t)W(x,x′).Sufficient conditions on the lower and upper control functions a, b are given guaranteeing that along every motion the total mechanical energy tends to zero as t→∞. The key condition in the...

The outstanding Russian mathematician Aleksandr M. Lyapunov passed away one hundred years ago, on November 6, 1918. Honouring his memory, we recall the main events of his life when he was a student, then from the years in Saint Petersburg until 1885, from the Kharkov period, finally from his second period in Saint Petersburg from 1902. We recount t...

The second order nonlinear differential equation \begin{equation*} x''+h(t,x,x')x'+f(x)=0 \qquad \bigl(x\in\mathbb{R},\ t\in\mathbb{R}_+:=[0,\infty),\ ()':=\tfrac{\text{d}}{\text{d}t}()\bigr), \end{equation*} and a sequence $\{I_n\}_{n=1}^\infty$ of non-overlapping intervals are given, where the damping coefficient $h$ admits an estimate \begin{equ...

Using purely elementary methods, necessary and sufficient conditions are given for the existence of 2T-periodic and 4T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating with period 2T. The equation of the motion is of the form $$\ddot{\theta}-\frac{1}{l}(g+a(t)) \theta=0,$$where l, g...

Conditions guaranteeing asymptotic stability for the differential equation $$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$are studied, where the damping coefficient \(h:[0,\infty )\rightarrow [0,\infty )\) is a locally integrable function, and the frequency \(\omega >0\) is constant. Our conditions need neither t...

We consider the integro differential equation $$ x'(t)=-a(t)x(t)+b(t)\int^t_{t-h} \lambda(s)x(s)\,ds,\quad o\leq a(t),\; 0\le t

We make more realistic our model [Nonlinear Anal. 73(2010), 650-659] on the coexistence of fishes and plants in Lake Tanganyika. The new model is an asymptotically autonomous system whose limiting equation is a Lotka-Volterra system. We give conditions for the phenomenon that the trajectory of any solution of the original non-autonomous system "rol...

The autonomous system of differential equations x¹ = f (x), (x = (x1, x2)T ∈ R², f (x) = (f1(x), f2(x))T), is considered, and sufficient conditions are given for the global attractivity of the unique equilibrium x = 0. This property means that all solutions tend to the origin as t → ∞. The two cases (a) div f (x) ≤ 0 (x ∈ R²) and (b) div f (x) ≥ 0...

Sufficient conditions for uniform equi-asymptotic stability and uniform asymptotic stability of the zero solution of the retarded equation \[x'(t) = f(t, x_t), \qquad (x_t(s):= x(t+s),\ -h\le s\le 0)\] are given. In the stability theory of non-autonomous differential equations a result is of Marachkov type if it contains some kind of boundedness or...

An elementary geometric method is established to study non-linear second order differential equations with step function coefficient (Equation Presented), where (Equation Presented). The equation is rewritten into a discrete dynamical system on the plane. The method is applied to the excited pendulum equation when g(x) = sin x. Starting from the us...

The equation x-Œ-Œ + a2(t)x = 0 with is considered, where g and l denote the constant of gravity and the length of the pendulum, respectively; e > 0 is a parameter measuring the intensity of swinging. Concepts of solutions going away from the origin and approaching to the origin are introduced. Necessary and sufficient conditions are given in terms...

Sufficient conditions are given for the stability of the upper equilibrium of the mathematical pendulum (inverted pendulum) when the suspension point is vibrating vertically with high frequency. The equation of the motion is of the form $$ \ddot{\theta}-\frac{1}{l}\bigl(g+a(t)\bigr) \theta=0, $$ where l,g are constants and a is a periodic step fu...

A geometric method is presented to describe the dynamics of the linear second order differential equation with step function coefficient x″ + a2(t)x = 0, a(t) := ak if tk-1 ≤ t < tk (k ∈ ℕ), where ak > 0, t0 = 0, tk ↗ ∞ as k → ∞. We rewrite this equation into a discrete dynamical system on the plane. The method is applied to the Meissner equation x...

We give a sufficient condition guaranteeing asymptotic stability with respect to x for the zero solution of the half-linear differential equation x′′|x′|n-1 + q(t)|x|n-1x = 0, 1≤n ∈ R, with step function coefficient q. The geometric method of the proof can be applied also to two dimensional systems of linear non autonomous difference equations. The...

We prove that (λ∗,C/λ∗) is an eventually uniform-asymptotically stable point in the large of the system L̇=C−LG,Ġ=(L−λ(t))G. on the quadrant {(L,G):L≥0,G>0}. Here function λ(t) is positive and λ(t)→λ∗>0 as t→∞. The study was inspired by observations of distributions of peculiar carnivore and herbivore fish species in Lake Tanganyika.

The equationx″+a2(t)x=0,a(t):=ak>0if tk−1⩽ttk(k∈N) is considered where {ak}k=1∞ is given and {tk}k=1∞ is a random sequence. Sufficient conditions are proved which guarantee either stability or instability for the zero solution. Stability means that all solutions almost surely tend to zero as t→∞. By instability we mean that the sequence of the expe...

In this paper the second order liner differential equation \begin{equation*} \left\{\begin{array}{l} x'' + a^2 (t) x=0,\\ a(t) = \left\{\begin{array}{ll} \pi+\varepsilon, &\textrm{if\ $2nT\le t0$ ($T:=(T_1+T_2)/2$) and $\varepsilon \in [0,\pi)$. We say that a parametric resonance occurs in this equation if for every $\varepsilon >0$ sufficiently sm...

We construct a new continuous time selection-mutation-recombination model for population dynamics, which describes the development of the distribution of the different gametes in the population. We show that cyclic mutation rates can result in stable and unstable limit cycles due to Hopf bifurcation. In addition, we give a qualitative characterizat...

Small oscillations of an undamped holonomic mechanical system with varying parameters are described by the equations ∑ k=1 n (a ik (t)q ¨ k +c ik (t)q k )=0(i=1,2,⋯,n)·(*) A nontrivial solution q 1 0 ,⋯,q n 0 is called small if lim t→∞ q k (t)=0(k=1,2,⋯,n)· It is known that in the scalar case (n=1, a 11 (t)≡1, c 11 (t)=:c(t)) there exists a small s...

The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If the coefficient is a step function, then small oscillations are described by the equation $$ \ddot x + a^2 (t)x = 0,a(t): = a_k ift_{k - 1} \leqslant t < t_k ,k = 1,2,.... $$ Using a probability approach, we assume that (a k...

The present paper is devoted to studying Hubbard's pendulum equation ¨ x + 10 1 ú x + sin(x) = cos(t) . By rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is lifted from the level of experimentally observed facts to the level of a theorem completely proved. A distinguished family o...

The second order linear differential equation is considered, where 𝑞 : [0, ∞) → (0, ∞) is continuous, piecewise continuously differentiable, non-decreasing, and lim 𝑡→∞ 𝑞(𝑡) = ∞. A solution 𝑥 0 is called small if lim 𝑡→∞ 𝑥 0 (𝑡) = 0. It is known that the equation always has at least one nontrivial small solution, but, in general, it can have also...

We report on the first steps made towards the computational proof of the chaotic behaviour of the forced damped pendulum. Although, chaos for this pendulum was being conjectured for long, and it has been plausible on the basis of numerical simulations, there is no rigorous proof for it. In the present paper we provide computational details on a fit...

The linear system of difference equations x(n+1) = M(n)x(n) n = 0,1,2,... is considered. A non-trivial solution {x(n)}(n=0)(infinity) is small if lim(n ->infinity)x(n) = 0. Conditions on the sequence of matrices {M-n}(n=0)(infinity) are given guaranteeing that Pi(infinity)(n=0)vertical bar det M-n vertical bar = 0 is necessary and sufficient for th...

Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovskiĭ functionals whose derivatives with respect to the equations are negative semidefinite and can van...

A review of recent results on the different properties of stability and stability with respect to a part of variables for a damped oscillator is presented. Asymptotic stability with respect to velocities is guaranteed for the equilibrium of Lagrange systems acted upon by friction with unlimited damping factors. Instances of the scalar equations are...

Sucient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theo- rems are based upon Lyapunov{Krasovski functionals whose derivatives with respect to the equations are negative semidenite and can vanish...

Stability properties of the solutions of the half-linear differential equation $$x''|x'|^{n-1}+a_k|x|^{n-1}x=0,\quad (t_k\le t

We developed and tested a novel quantitative method for the quantification of film autoradiographs, involving a mathematical model and a dot-blot-based membrane standard scale. The exponential model introduced here, ROD = p1(1 - exp[p2x]), appropriately (r2>0. 999), describes the relation between relative optical density (ROD) and radioactivity (x)...

The equation x " + a(t)x: = 0 is considered, where a : [0, infinity) --> (0, infinity) is a nondecreasing step function tending to infinity as t goes to infinity. It is proved by a geometric method that the equation has a nontrivial small solution, i.e., a solution x(0) with lim(t-->infinity) x(0)(t) = 0. The method is generalized to the case of no...

. Sufficient conditions are given guaranteeing that all solutionsof the equationx00+ a(t)f(x) = 0 (xf(x) ? 0)tend to zero as t goes to infinity. The conditions contain integrals insteadof maxima and minima in earlier results. Finally, a probabilisticgeneralization of Armellini-Tonelli-Sansone theorem is formulated.reziume moKvanilia sakmarisi pirob...

The oscillator $$ x''+h(t)x'+x=0 $$ is considered, where the damping $h:{\Bbb R}_+\to{\Bbb R}_+$ is piecewise continuous and large in the sense $$ \liminf_{t\to\infty}\int _t^{t+\delta}h>0 \quad \text{ for every }\ \delta>0. $$ The problem of intermittent damping, initiated by P\. Pucci and J\. Serrin, is investigated. Let a sequence $\{I_n=[\alpha...

In the first part of the paper sufficient conditions for the asymptotic stability of the zero solution of a non-autonomous functional differential equation with finite delay are given by using Lyapunov functionals with negative semidefinite derivatives. In the second part we formulate theorems on the asymptotic stability of the equilibrium of linea...

The equation x + h ( t ) x ′ + k 2 x = 0 x+h(t)x’+k^2x=0 is considered under the assumption 0 ≤ h ( t ) ≤ h ¯ > ∞ 0\le h(t)\le \overline {h}>\infty ( t ≥ 0 ) (t\ge 0) . It is proved that lim sup t → ∞ ( t − 2 / 3 ∫ 0 t h ) > 0 \limsup _{t \to \infty }\left (t^{-2/3}\int _0 ^t h\right )>0 is sufficient for the asymptotic stability of x = x ′ = 0 x=x...

Conditions are given guaranteeing the property x(t)→0, x ˙(t)→0 (t→∞) for every solution of the equation x ¨+h(t)x ˙+k 2 x=0 (t≥0,0<k=const.), where h is a nonnegative function. It is known that this property requires that in the average the damping coefficient h is not “too small” or “too large”. In the first part we give a necessary and sufficien...

On considere le systeme x'(t)=F(t,x t ) ou x t est le segment de x(s) sur [t-h,t] decale en [−h,0], ou h>0 est une constante fixee. On donne des conditions sur des fonctionnelles de Lyapunov pour assurer la stabilite des solutions

We study the asymptotic behavior of the solutions of a broad class of second order nonlinear differential equations, namely, (E)(a(t)x ' ) ' +h(t,x,x ' )+q(t)f(x)=e(t,x,x ' )· Equation (E) can be interpreted as the equation of the motion of a mechanical system with one degree of freedom having kinetic energy a(t)[x ' ] 2 /2 and potential energy q(t...

By analogy with the division of energy into kinetic and potential types, it is proposed that a Liapunov function (LF) be constructed as the sum of two auxiliary scalar functions. Attention is given to the case when the derivative of the LF can take positive values, and when the comparison equation arising from the estimation of the LF does not admi...

Si danno condizioni sufficienti per la stabilità e la instabilità della posizione di equilibrio x=y=z=0 nel sistema meccanico che consiste di un punto materiale vincolato a muoversi sulla superficie mobile z=−λ(t)(x2+y2) (λ(t)>0) in un campo di gravità costante (l'asse 0z è diretto verticalmente e orientato verso l'alto) sotto l'azione di attriti v...

The extensions of the Barbashin-Krasovskij theorem to the partial asymptotic stability of the zero solution of a differential system require the boundedness of the uncontrolled coordinates along the solutions. In this paper the Barbashin-Krasovskij method is generalized without supposing a priori knowledges on the solutions. At the same time, the r...

On donne des conditions suffisantes pour la stabilite asymptotique partielle de la solution nulle d'un systeme non autonome en utilisant une fonction auxiliaire additionnelle au lieu de la decrescence de la fonction de Ljapunov

In this note we prove a continuation theorem applicable also when the estimate of the derivative of the vector Ljapunov function contains the phase coordinates explicitly. Our theorem combines and strengthens several earlier continuation results including a recent theorem of T. A. Burton, who conjectured that the monotonicity assumption on a functi...

In this note we prove a continuation theorem applicable also when the estimate of the derivative of the vector Ljapunov function contains the phase coordinates explicitly. Our theorem combines and strengthens several earlier continuation results including a recent theorem of T. A. Burton, who conjectured that the monotonicity assumption on a functi...

In connection with a conjecture of J. M. Bownds, conditions will be given on the fundamental system of the solutions of the unstable differential equation $y'' + a(t)y = 0$ which assure that the differential equation $x'' + a(t)x = g(t, x, x')$ has a solution with the property $$\lim \sup(|x(t)| + |x'(t)|) = \infty\quad\text{as}\quad t \rightarrow...

In connection with a conjecture of J. M. Bownds, conditions will be given on the fundamental system of the solutions of the unstable differential equation y + a ( t ) y = 0 y + a(t)y = 0 which assure that the differential equation x + a ( t ) x = g ( t , x , x ′ ) x + a(t)x = g(t,x,x’) has a solution with the property \[ lim sup ( | x ( t ) | + | x...

We construct a new continuous time selection-mutation-recombi-nation model for population dynamics, which describes the development of the distribution of the different gametes in the population. We show that cyclic mutation rates can result in stable and unstable limit cycles due to Hopf -bifurcation. In addition, we give a qualitative characteriz...

An annulus argument is a method of proof detecting that a curve in ℝ n crosses an annulus around the origin from inside to outside infinitely many times. We give the abstract formulation of a new annulus argument not supposing the boundedness of the derivatives of the functions involved. We use this argument to establish theorems on the asymptotic...

First order systems and a scalar second order equation with unbounded delays are considered. By the use of the Leray-Schauder continuation method, sign conditions are established guaranteeing the existence of periodic solutions.

Másodrendű közönséges differenciálegyenletek és funkcionál differenciálegyenletek megoldásainak aszimptotikus viselkedésére bizonyítottunk eredményeket. Többek között egy periodikusan perturbált fékezett inga mozgását leíró differenciálegyenlet megoldásainak a kaotikus viselkedését igazoltuk analitikus módszerek, topológiai eszközök és megbízható n...

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