Second-order Nonlinear Dynamic Equations Research Articles

A Near-Infrared Retinomorphic Device with High Dimensionality Reservoir Expression.

Physical reservoir-based reservoir computing (RC) systems for intelligent perception have recently gained attention because they require fewer computing resources. However, the system remains limited in infrared (IR) machine vision, including materials and physical reservoir expression power. Inspired by biological visual perception systems, the study proposes a near-infrared (NIR) retinomorphic device that simultaneously perceives and encodes narrow IR spectral information (at ≈980nm). The proposed device, featuring core-shell upconversion nanoparticle/poly (3-hexylthiophene) (P3HT) nanocomposite channels, enables the absorption and conversion of NIR into high-energy photons to excite more photo carriers in P3HT. The photon-electron-coupled dynamics under the synergy of photovoltaic and photogating effects influence the nonlinearity and high dimensionality of the RC system under narrow-band NIR irradiation. The device also exhibits multilevel data storage capability (≥8 levels), excellent stability (≥2000 s), and durability (≥100 cycles). The system accurately identifies NIR static and dynamic handwritten digit images, achieving recognition accuracies of 91.13% and 90.07%, respectively. Thus, the device tackles intricate computations like solving second-order nonlinear dynamic equations with minimal errors (normalized mean squared errorof 1.06 × 10⁻3 during prediction).

Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term

Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term

Oscillation Criteria of Second-Order Dynamic Equations on Time Scales

In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.

Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order

In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.

Oscillatory behavior of solutions of dynamic equations of higher order on time scales

We study the n -th-order nonlinear dynamic equations x [ n ] ( t ) + p ( t ) ϕ α n − 1 [ ( x [ n − 2 ] ( t ) ) Δ σ ] + q ( t ) ϕ γ ( x ( g ( t ) ) ) = 0 on an unbounded time scale 𝕋 , where n ≥ 2 and for i = 1 , … , n − 1 x [ i ] ( t ) : = r i ( t ) ϕ α i [ ( x [ i − 1 ] ( t ) ) Δ ] , with r n = α n = 1 and x [ 0 ] = x ; here the constants α i and the functions r i , i = 1 , … , n − 1 , are positive and p , q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator σ and g do not commute.

A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.

Oscillation for Forced Second-Order Impulsive Nonlinear Dynamic Equations on Time Scales

As the unification and development of impulsive differential equations and difference equations, impulsive dynamic equations on time scales are a powerful tool to simulate the natural and social phenomena. In this paper, we study the interval oscillation of a type of forced second-order nonlinear impulsive dynamic equations with changing signs coefficients. By using the Riccati transformation technique, we obtain some new interval oscillation criteria, based only on information of a sequence of subintervals of positive axis. In addition, we provide an example to illustrate the use of our oscillatory results.

New oscillation criteria of special type second-order non-linear dynamic equations on time scales

By the Riccati transformation technique, we study some new oscillatory properties for the second-order dynamic equation on an arbitrary time scale mathbb {T}. We also establish the Kamenev-type and Philos-type oscillation criteria. At the end, we give examples which illustrate our main results.

Oscillation constants for second-order nonlinear dynamic equations of Euler type on time scales

We are concerned with the oscillation problem for second-order nonlinear dynamic equations on time scales of the form , where f(x) satisfies if . By means of Riccati technique and phase plane analysis of a system, (non)oscillation criteria are established. A necessary and sufficient condition for all nontrivial solutions of the Euler–Cauchy dynamic equation to be oscillatory plays a crucial role in proving our results.

Oscillation Criteria of Second-Order Non-Linear Dynamic Equations with Integro Forcing Term on Time Scales

Oscillation Criteria of Second-Order Non-Linear Dynamic Equations with Integro Forcing Term on Time Scales

Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the formrtg1xt,xΔtΔ+p(t)g2(x(t),xΔ(t))xΔ(t)+q(t)f(x(τ(t)))=e(t), on a time scaleT. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results.

New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales

New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales

Oscillation Criteria for Functional Nonlinear Dynamic Equations with $${\gamma}$$ γ -Laplacian, Damping and Nonlinearities Given by Riemann–Stieltjes Integrals

In this paper, we present oscillation criteria for the second-order nonlinear dynamic equation \({[a(t)\phi_{\gamma} (x^{\Delta}(t))]^{\Delta} + p(t)\phi_{\gamma}(x^{\Delta^{\sigma}}(t)) + q_{0}(t) \phi_{\gamma}(x(g_{0}(t)))+\sum_{i=1}^{2}\int_{a_{i}}^{b_{i}}q_{i}(t,s)\phi_{\alpha_{i}(s)}(x(g_{i}(t,s))) \Delta \zeta_{i}(s)=0}\) on a time scale \({\mathbb{T}}\) which is unbounded above. Our results generalize and improve some known results for oscillation of second-order nonlinear dynamic equation. Some examples are given to illustrate the main results.

Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales

This paper is concerned with oscillations of the second-order delay nonlinear dynamic equation on a time scale , where a and q are real-valued rd-continuous positive functions on , α and β are ratios of odd positive integers, , , , and . We establish some new sufficient conditions for this equation. MSC:34K11, 39A10, 39A99.

New Oscillatory Behavior of Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

We establish four new oscillation criteria of Grace-type for the second-order nonlinear dynamic equations with damping. These criteria extend known criteria for corresponding dynamic equations. Our results are new even in the continuous and the discrete cases.

Forced oscillation of second-order nonlinear functional dynamic equations on time scales

In this paper, we establish some new oscillation criteria for the second-order nonlinear functional dynamic equation with forced term (r(t)x Δ (t)) Δ ± p(t)f(x(τ(t))) = e(t), on a time scale T. No restriction is imposed on the forcing term e(t) to satisfy the Kartsatos condition. p(t) and r(t) are real-valued rd-continuous functions defined on T. There are many cases have been taken into consideration: (i) p(t) > 0, τ(t) t( t) and τ(t) σ(t)( σ(t)) (ii) p(t) changes its sign, τ(t)t(t), τ : T →T is a strictly increasing differentiable function and limt→∞ τ(t )= ∞. Our results not only generalize and extend some existing results but also can be applied to the oscillation problems that are not covered in literature. Finally, we give some examples to illustrate our main results.

Oscillation Results for Second-Order Nonlinear Damped Dynamic Equations on Time Scales

This paper is concerned with second-order nonlinear damped dynamic equations on time scales of the following more general form(p(t)k1(x(t),xΔ(t)))Δ+r(t)k2(x(t),xΔ(t))xΔ(t)+f(t,x(σ(t)))=0. New oscillation results are given to handle some cases not covered by known criteria. An illustrative example is also presented.

New Kamenev-type oscillation criteria for second-order nonlinear advanced dynamic equations

This paper presents a new Kamenev-type oscillation criterion for a second-order advanced dynamic equation on a time scale. Two examples are included to show that the result obtained complements and improves those reported in the literature.

Oscillation Theorems for Second-Order Nonlinear Dynamic Equation on Time Scales

This paper concerns the oscillation of solutions to the second order non-lin ear dynamic equation (r(t)x Δ (t)) Δ + p(t) f(x σ (t))g(x Δ (t)) = 0 on a time scale T which is unbounded above. By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates.

Oscillation of second-order nonlinear dynamic equations with positive and negative coefficients

The paper considers the oscillation of a second-order nonlinear dynamic equation with positive and negative coefficients of the form