Quartic Map Research Articles

A new chaotic map derived from the Hermite–Kronecker–Brioschi characterization of the Bring-Jerrard quintic form

Abstract The Bring-Jerrard normal form, achieved by Tschirnhaus transformation of a regular quintic, is a reduced type of the general quintic equation with quartic, cubic and quadratic terms omitted. However, the form itself is an equation opposing the mandatory characteristics of the iterative chaotic maps. Given the form represents the fixed-point equations, it is possible to turn it into a map of iterations. Under specific conditions, the quartic map achieved by transformation from the quintic normal form exhibits chaotic behavior for real numbers. Depending on the system parameters, the new map causes period-doubling until a complete chaos within a very short range. Basically, in this paper, we present a new one-dimensional chaotic map derived from the Hermite–Kronecker–Brioschi characterization of the Bring-Jerrard normal form, which exhibits chaotic behavior for negative initial points. We also included the brief analysis of the Bring-Jerrard generalized case which is the parent system of the chaotic map we proposed in this paper.

Perturbations of graphs for Newton maps I: bounded hyperbolic components

AbstractWe consider graphs consisting of finitely many internal rays for degenerating Newton maps and state a convergence result. As an application, we prove that a hyperbolic component in the moduli space of quartic Newton maps is bounded if and only if every element has degree $2$ on the immediate basin of each root. This provides the first complete description of bounded hyperbolic components in a complex two-dimensional moduli space.

General system of cubic–quartic functional equations in quasi-β-normed spaces

In the current article, we define the multicubic–quartic mappings and describe them as an equation. We also study n-variable mappings, which are mixed type cubic–quartic in each variable and then give a characterization of such mappings. Indeed, we unify the general system of cubic–quartic functional equations defining a multimixed cubic–quartic mapping to a single equation, say, the multimixed cubic–quartic functional equation. Furthermore, we show under what conditions every multimixed cubic–quartic mapping can be multicubic, multiquartic and multicubic–quartic. In addition, by means of a known fixed-point result, we prove the Găvrua stability of multimixed cubic–quartic functional equations in the setting of quasi-β-normed spaces. One of the important results is that every multimixed cubic–quartic functional equation on a quasi-β-normed space is the Hyers–Ulam stable. Lastly, we investigate the hyperstability of multicubic -derivations on -algebras.

The Stability Analysis of A-Quartic Functional Equation

In this paper, we study the general solution of the functional equation, which is derived from additive–quartic mappings. In addition, we establish the generalized Hyers–Ulam stability of the additive–quartic functional equation in Banach spaces by using direct and fixed point methods.

Some hyperstability results of a p-radical functional equation related to quartic mappings in non-archimedean Banach spaces

The aim of this paper is to introduce and solve the following p-radical functional equation related to quartic mappings where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzd¸ek’s fixed point theorem [13], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to quartic mapping.

A quartic polynomial chaotic map with its application in S-box generation

A quartic polynomial chaotic map with its application in S-box generation

Boundedness of hyperbolic components of Newton maps

We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we characterize the possible points on the boundary at infinity for some other types of hyperbolic components. For general maps, we prove hyperbolic components whose elements have fixed superattracting basins mapping by degree at least three are unbounded.

Approximate Quartic Lie *-Derivations with Perturbing Terms

The aim of this paper is to investigate stable approximation of almost quartic Lie $*$-derivationsassociated with approximate quartic homogeneous mappings by quartic Lie $*$-derivationson $rho$-complete convex modular algebras $chi_rho$ by using $Delta_2$-condition via convex modular $rho.$

Fractal structures in the parameter space of nontwist area-preserving maps.

Fractal structures are very common in the phase space of nonlinear dynamical systems, both dissipative and conservative, and can be related to the final state uncertainty with respect to small perturbations on initial conditions. Fractal structures may also appear in the parameter space, since parameter values are always known up to some uncertainty. This problem, however, has received less attention, and only for dissipative systems. In this work we investigate fractal structures in the parameter space of two conservative dynamical systems: the standard nontwist map and the quartic nontwist map. For both maps there is a shearless invariant curve in the phase space that acts as a transport barrier separating chaotic orbits. Depending on the values of the system parameters this barrier can break up. In the corresponding parameter space the set of parameter values leading to barrier breakup is separated from the set not leading to breakup by a curve whose properties are investigated in this work, using tools as the uncertainty exponent and basin entropies. We conclude that this frontier in parameter space is a complicated curve exhibiting both smooth and fractal properties, that are characterized using the uncertainty dimension and basin and basin boundary entropies.

Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings

Abstract The “oldest quartic” functional equation f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 4 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] - 6 ⁢ f ⁢ ( x ) + 24 ⁢ f ⁢ ( y ) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2 ⁢ f ⁢ ( 2 ⁢ x + y ) + 2 ⁢ f ⁢ ( 2 ⁢ x - y ) + f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 20 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] + 90 ⁢ f ⁢ ( x ) . 2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.

Bifurcation Diagram of a Map with Multiple Critical Points

In this work a conjecture to draw the bifurcation diagram of a map with multiple critical points is enunciated. The conjecture is checked by using two quartic maps in order to verify that the bifurcation diagrams obtained according to the conjecture contain all the periodic orbits previously counted by Xie and Hao for maps with four laps. We show that a map with split bifurcation contains more periodic orbits than those counted by these authors for a map with the same number of laps.

Homoclinic tangle of the primary separatrix in the compact and closed versus open and unbounded magnetic topologies for divertor tokamaks

ABSTRACTThe simple map and the symmetric quartic map are the generic representation of the open-unbounded and closed-compact magnetic topologies for single-null divertor tokamaks, respectively. The map parameter represents the magnetic asymmetries. The two maps are used to calculate the homoclinic tangles in the two topologies in divertor tokamaks. It is found that as the magnetic asymmetries become larger, the homoclinic tangle of the simple map becomes extremely elongated radially as opposed to the homoclinic tangle of the symmetric quartic map which becomes wider poloidally.

Identification of Chaos-Periodic Transitions, Band Merging, and Internal Crisis Using Wavelet-DFA Method

Detrended Fluctuation Analysis (DFA) is a scaling analysis method that can identify intrinsic self-similarity in any nonstationary time series. In contrast, Wavelet Transform (WT) method is widely used to investigate the self-similar processes, as the self-similarity properties exist within the subbands. Therefore, a combination of these two approaches, DFA and WPT, is promising for rigorous investigation of such a system. In this paper a new methodology, so-called wavelet DFA, is introduced and interpreted to evaluate this idea. This approach, further than identifying self-similarity properties, enable us to detect and capture the chaos-periodic transitions, band merging, and internal crisis in systems that become chaotic through period-doubling phenomena. Changes of wavelet DFA exponent have been compared with that of Lyapunov and DFA through Logistic, Sine, Gaussian, Cubic, and Quartic Maps. Furthermore, the potential capabilities of this new exponent have been presented.

Fixed points and generalized stability for quadratic and quartic mappings in $${C^*}$$ C ∗ -algebras

We consider the following quadratic and quartic functional equations: $$\begin{array}{l}\qquad \qquad f(\sqrt{xx^* + yy^*}) = f(x) + f(y),\\ f(\sqrt{xx^* + yy^*}) + f(\sqrt{|xx^* + yy^*|}) = 2f(x) + 2f(y)\end{array}$$ in \({C^*}\)-algebras. We also prove the stability of these functional equations in \({\beta}\)-normed spaces by using the Banach fixed point theorem and a direct method.

Breaking Points in Quartic Maps

Dynamical systems, whether continuous or discrete, are used by physicists in order to study nonlinear phenomena. In the case of discrete dynamical systems, one of the most used is the quadratic map depending on a parameter. However, some phenomena can depend alternatively on two values of the same parameter. We use the quadratic map [Formula: see text] when the parameter alternates between two values during the iteration process. In this case, the orbit of the alternate system is the sum of the orbits of two quartic maps. The bifurcation diagrams of these maps present breaking points at which there is an abrupt change in their evolution.

On the generalized Hyers-Ulam stability of quartic mappings in non-Archimedean Banach spaces

On the generalized Hyers-Ulam stability of quartic mappings in non-Archimedean Banach spaces

Fixed points and stability for quartic mappings in β-Normed left Banach modules on Banach algebras

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equation ∑k=2n (∑i1 =2k ∑i2 =i1 +1k+1…∑in −k+1=in −k+1n )f(∑i=1,i≠i1 ,…,in −k+1nxi −∑r=1n−k+1xir ) + f(∑i=1nxi ) = 2n−2∑1≤i<j≤n (f(xi + xj )+f(xi −xj ))−2n−5(n−2)∑i=1nf(2xi )(n ∈ ℕ, n ≥ 3)in β-Banach modules on Banach algebras.The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

Correction: Nearly Quartic Mappings in β-Homogeneous F-Spaces [Results Math. 63 (2013) 529–541

In Chang et al. (Results Math. 63:529–541, 2013), Eshaghi Gordji et al. proved the Hyers-Ulam stability of a quartic functional equation in β-homogeneous F-spaces. In the main step of the proof of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2), there is a fatal error. We correct the statement of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2).

Approximation of radical functional equations related to quadratic and quartic mappings

In this paper, we introduce and solve of the radical quadratic and radical quartic functional equations: f(ax2+by2)=af(x)+bf(y),f(ax2+by2)+f(|ax2−by2|)=2a2f(x)+2b2f(y). We also establish some stability results in 2-normed spaces and then the stability by using subadditive and subquadratic functions in p-2-normed spaces for these functional equations.

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ \(({n \in \mathbb{N}, n \geq 3})\) in β-homogeneous F-spaces.