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\begin{document}
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	\title{Investigation of Novel Modified ABC Differential Operator in Chaotic System Application with Deep Neural Networks }
		\author[]{Atul Kumar} 

	\affil[]{Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, SIMATS, Chennai, India,\ Email: atul1230123@gmail.com}
	
	\date{\today}
	\maketitle
	\begin{abstract}
		The main focus of the inquiry is the Newton-Leipnik system in piecewise approach. Piecewise differential and integral operator theory, as well as its applications, have grown significantly. By assessing the state of the art and approaching the subject in a methodical manner, this essay aims to provide an outcome for this circumstance.
		We might observe an attractor in this system. Using piecewise modelling, this system illustrates Newton-Leipnik's method. The piecewise derivatives are introduced in this study together with the Newton-Leipnik system. It is possible to approximate these piecewise derivatives using the Adams-Bashforth approach. Using detailed numerical simulations, the numerical results have been confirmed and demonstrated. Our investigation focusses on the piecewise Newton-Leipnik system with two strange attractors and its chaotic dynamics. The Newton-Leipnik system behaves in real life with piecewise patterns shown in the numerical simulations.\\
		
		
		\textbf{Keywords:}  Modified ABC Differential Operator, Modified Newton Leipnik systems, Nonlinear Equations, Piecewise Models, Chaos and Cross-over Behaviors.
	\end{abstract}
	\section{Introduction}	
	
	Fractional integral and differential operators, as well as their applications, have received a lot of attention. The primary cause is the rapid expansion of the theory of fractional differential and integral, which finds extensive application in mechanics, environmental science, biology, chemistry, physics, control theory, medicine, biology, and other fields. Fractional integral and differential operators are arbitrary order, generalised integral and differential operators that can be obtained in space and time using a power law memory kernel of the nonlocal relationships. These operators are an effective means of elucidating the inheritance and memory characteristics of various substances. These findings all have a transparent physical base, which leads to new avenues for scientific investigation, such as theoretical analysis and novel numerical techniques for fractional models. Numerous disciplines of study have effectively employed the numerical outcomes of differential and fractional integral operators. Recently, there has been increased interest in fractional differential operators because of their use in fractional-order dynamical systems. These types of integral and differential operators define many mathematical fractional models in science and engineering. In the fractional order dynamical systems, we therefore require the memory terms. The history and its influence on the present and future are guaranteed by such memory terms. Something like this is crucial, particularly for dynamical systems with fractional orders. Examining dynamical systems' chaos behaviours has attracted a lot of interest. As a result of their extreme sensitivity to initial circumstances, chaotic dynamical systems pose a challenging problem when it comes to understanding their chaotic behaviours. There are a plethora of techniques available for identifying chaos in fractional-order dynamical systems that exhibit chaotic behaviour.
	\cite{Bukhari}-\cite{Zhou}.
	One of the fractional differentional operators is modofied differential operator.
	There are some studies regarding modified ABC differential operator such as 
	fractional order forestry resource conservation model featuring chaos control and simulations for toxin activity and human-caused fire through MABC operator \cite{Farman}. 
	 Dynamics of the  MABC fractional-order leukemia model with symmetric numerical simulations \cite{Alfwzan}. 
	The criteria for solution of a MABC fractional order smoking model \cite{Alkhazan}.
	 The solutions and a numerical method for a generalized hybrid class of MABC-fractional differential equations \cite{Alobaidi}. 
	

	This article explores chaotic behaviour from the viewpoint of the Newton–Leipnik system. To stabilise chaotic behaviour to unstable equilibria, its parameters and states variables are taken into consideration.  
	There are two attractors in the dynamic system. After modifying Euler's rigid body equations, Newton and Leipnik constructed a set of differential operators in 1981 \cite{Leipnik RB}. Chen et al.\cite{Chen S} for the first time investigated chaos behaviors of the 	Newton–Leipnik system in 2002 applying a stable-manifold-based method. Later, Richter further investigated the stabilization of a desired motion having the system from one attractor to another using the Lyapunov function method \cite{Richter H}. Recently some studies have been investigated by some limited researchers such as Wang et al.\cite{Wang X} suggested the bifurcation analysis of the Newton–Leipnik system. Ghanbari B, et al.\cite{Ghanbari} suggested the two efficient numerical schemes for simulating dynamical systems and capturing chaotic behaviors with Mittag–Leffler memory. Khan A et al. \cite{Khan} proposed the fractional order disturbance observer-based adaptive sliding mode hybrid projective synchronization of fractional order Newton–Leipnik chaotic system.  Zhang K et al.\cite{Zhang}  proposed the feedback control and hybrid projective synchronization of a fractional-order Newton–Leipnik system.
 Khan MA.\cite{Khan MA} proposed the adaptive synchronization of two coupled Newton-Leipnik systems with uncertain parameters.
	
	%	 At recent, Scientists (Abdon Atangana and Seda İğret Araz) suggested a concept of piecewise systems of differential operators. There are some limited investigations such as Atangana et al.\cite{Atangana4} proposed the new concept in calculus: piecewise differential and integral operators. Atangana et al.\cite{Atangana3} proposed the modeling third waves of Covid-19 spread with piecewise differential and integral operators. Atangana et al.\cite{Atangana6} proposed the  rhythmic behaviors of the human heart with piecewise derivative. Atangana et al.\cite{Atanganaji} proposed the modeling the Spread of Tuberculosis with Piecewise Differential Operators. Li XP et al.\cite{Li XP} proposed the application of piecewise fractional differential equation to COVID-19 infection dynamics. Such investigation of piecewise modelling has driven us to inject it into the piecewise Newton–Leipnik system and capture the chao and crossover behaviors.
	%	
	
	Piecewise derivatives are involved with novel fractional differential operators. Indeed, piecewise modelling with novel fractional differential operators saw substantial advancements. Numerous fields, including finance, biology and medicine, engineering, physics, and biology, have benefited from these developments. Piecewise concept with novel fractional differential operators are more broadly applicable across a wide range of disciplines thanks to developments in theory and numerical structure for solving them. 
	Piecewise modelling has been the subject of relatively research. Atangana 
	et al.\cite{Atangana4} introduced piecewise differential and integral operators as a novel idea in calculus. Kumar et al.	\cite{Arora3} suggested the fractional Stochastic Van der Pol Oscillator. Atangana et al.\cite{Atangana3} presented the use of piecewise differential and integral operators in the modelling of the third waves of the Covid-19 outbreak.

		Kumar A.\cite{Kumar A} suggested utilising piecewise derivative to create Rossler's system.
	Atangana et al.\cite{Atanganaji} suggested the use of piecewise differential operators to predict the spread of tuberculosis.
	Li XP et al.\cite{Li XP} suggested using a piecewise fractional differential equation to analyse the kinetics of COVID-19 infection.
		Atangana et al.\cite{Atangana6} suggested the piecewise derivative of the human heart's rhythmic behaviours.
 We have been motivated to incorporate piecewise modelling into the piecewise Newton-Leipnik system in order to represent its crossover and chaotic behaviours.
	
	
	This is how the examination is set up. Section 2 provided an introduction to the basic definitions of piecewise differential and integral operators. Section 3 presents the modified piecewise Newton-Leipnik system. Section 4 presents the numerical scheme results achieved using the considered model at arbitrary $e$ and parameter values, while Section 5 concludes the study.
	
	
	\section{Definitions}
	\textbf{Definition 1:} The $X$ be a function, and $e \in (0,1)$. Next, the order $\alpha$ Caputo derivative\cite{Owolabi} will be provided by
	\begin{equation*}	
		_0^CD^{e}_tX(t) = 
		\frac{1}{\Gamma{(e)}}\int_0^t(t-s)^{-e}  { X'(s)} d{s}.
	\end{equation*}
	\textbf{Definition 2:} Let $e \in (0, 1]$ and let $X \in H^{1}(c_{1}, c_{2})$, for $c_{2}>c_{1}$, be a  function. Next, the Atangana–Baleanu derivative\cite{Owolabi} with order $\alpha$ is defined as
	\begin{equation*}	
		_0^{ABC}D^{e}_tX(t) = 
		\frac{AB(e)}{1-e}\int_0^t  E_{e}\left[-\frac{e}{1-e}(t-s)^{e}\right] X'(s) ds,
	\end{equation*}
	where the normalizing function is indicated by the notation $AB(e)=1-e+\frac{e}{\Gamma(e)}$.
	
	\textbf{Definition 3:} For every $c_{2}>c_{1}$, let $e \in (0, 1]$ and $X \in H^{1}(c_{1}, c_{2})$. For order $\alpha$, the Caputo–Fabrizio fractional derivative\cite{Owolabi} is determined by
	\begin{equation*}	
		_0^{CF}D^{e}_tX(t) = 
		\frac{M(e)}{1-e}\int_0^t \exp\left[-\frac{e}{1-e}(t-s)\right] X'(s) ds
	\end{equation*}
	considering the normalization function $M(e)$ in order to get M(0)=M(1)=1.
	
	\textbf{Definition 4: }
	We create a piecewise integral for a continuous function $X$, $_0^{PCF}J^{e}_tX(t)$, as follows: we use the Caputo-Fabrizio integral and traditional one \cite{Atangana4}.
	\begin{equation}
		_0^{PCF}J^{e}_tX(t) =
		\begin{cases}
			\displaystyle\int_0^{t_{1}} X(s) d{s} & 0\leq t\leq t_{1} \\
			\frac{1-e}{M(e)}X(t)+\frac{e}{M(e)}\displaystyle\int_{t_{1}}^t X(s) d{s}	& t_{1}\leq t\leq T
		\end{cases}
	\end{equation}
	\textbf{Definition 5: }
	We create a piecewise integral for a continuous function $X$, that is, $_0^{PAB}J^{e}_tX(t)$. We do this by combining the traditional and Atangana-Baleanu integrals \cite{Atangana4}.
	\begin{equation}
		_0^{PAB}J^{e}_tX(t) =
		\begin{cases}
			\displaystyle\int_0^{t_{1}} X(s) d{s} & 0\leq t\leq t_{1} \\
			\frac{1-e}{AB(e)}X(t)+\frac{e}{AB(e)}\frac{1}{\Gamma{(e)}}\displaystyle\int_{t_{1}}^t X(s)(t-s)^{e-1}    d{s}	& t_{1}\leq t\leq T
		\end{cases}
	\end{equation}
	
	\textbf{Definition 6: } For function $X$, defined on $[0,T]$, we define its piecewise derivative with a power-law kernel\cite{Atangana4}. 
	\begin{equation}\label{p_caputo}
		_0^{PC}D^{e}_tX(t) = 
		\begin{cases}
			X'(t) & 0\leq t\leq t_{1} \\
			_0^{C}D^{e}_tX(t)
			& t_{1}\leq t\leq T
		\end{cases}
	\end{equation}
	\textbf{Definition 7:} 
	For each $c_{2}>c_{1}$, let $X \in H^{1}(c_{1}, c_{2})$. With the help of the modified ABC derivative and traditional one, we define a piecewise derivative of $X$, represented by $_0^{PAB}D^{e}_tX(t)$. 
	\begin{equation}\label{p_atangana_balaneu}
		_0^{PMABC}D^{e}_tX(t) = 
		\begin{cases}
				X^{'}(t) & 0\leq t\leq t_{1} \\
			_{}^{MABC}D^{e}_t X(t)
			& t_{1}\leq t\leq T
		\end{cases}
	\end{equation}
	\textbf{Definition 8: }
	For each $c_{2}>c_{1}$, let $X \in H^{1}(c_{1}, c_{2})$. Using the Caputo-Fabrizio derivative and traditional one, we define a piecewise derivative of $X$, denoted by $_0^{PCF}D^{e}_tX(t)$ \cite{Atangana4}. 
	\begin{equation}\label{p_caputo_fabrizio}
		_0^{PCF}D^{e}_tX(t) = 
		\begin{cases}
			X'(t) & 0\leq t\leq t_{1} \\
			_0^{CF}D^{e}_tX(t)
			& t_{1}\leq t\leq T
		\end{cases}
	\end{equation}
	
	
	\section{New Newton–Leipnik system \cite{Petráš1}}
	\begin{align}\label{linear_caputo}
		\begin{cases}
			& x'(t) = -\chi_1 x+y+10yz \\
			& y'(t) =-x-0.4y+5xz \\
			& z'(t) = \chi_2 z-5xy \\
		\end{cases}
	\end{align}
	This dynamic system displays crossover and chaos tendencies, and it features two unusual attractors.$u$, $v$, and $w$ represent the state variables. $\chi_1$ and $\chi_2$ are the two parameters. Numerical examples seem to suggest that the system of differential equations has two odd attractors when the parameters $\chi_1$ and $\chi_2$ take on values near $0.4$ and $0.175$. The primary source of the significance of the original explanation was the behavior of that attractor as determined by numerical analysis. We shall assume certain things regarding the circumstances in which a fresh Newton-Leipnik system exhibits differential operators with different piecewise patterns in order to achieve this. We introduced piecewise patterns in this study to illustrate the chaotic and crossover properties of the Newton-Leipnik system. In this work, we presented piecewise patterns to illustrate the crossover and chaos properties of the Newton-Leipnik system. The system behaves chaotically when the starting values of the system state variable and parameters are selected from a predefined range. In the piecewise Newton-Leipnik system, the behavior of the fractional order and integer order Newton-Leipnik systems is comparable. The crossover and chaos tendencies of the Newton-Leipnik system are shown in the figures.
	
	
		\section{The Mathematical Model}
	The model showed above has been a subject of several studies as authors aimed to explore the relationship of these species from different point of view. Novel types of differential operators shall be introduced to accommodate the aforementioned processes through some modifications. 
	\begin{align}\label{linear_caputo}
		\begin{cases}
					&_{T_{1}}^{MABC}D^{e}_t u(t)=-\chi_1x+y+10yz\\
		&_{T_{1}}^{MABC}D^{e}_t v(t)=-x-0.4y+5xz\\
		&_{T_{1}}^{MABC}D^{e}_t w(t)= \chi_2z-5xy 	
		\end{cases}
	\end{align}
	\section*{ Classical-Modified ABC-randomness :}
	The piecewise expression is established in \cite{Atangana3}, then similarly we established for classical derivative,  Modified ABC derivative, and stochastic one is established using equation \eqref{p_atangana_balaneu} as below.
	\begin{align}\label{linear_caputo_mittag}
		\begin{cases}
			&\frac{d X_{i}(t)}{dt} = g(t, X_{i}), 0 \le t\le T_{1} ,  \\
			& X_{i}(0)=X_{i,0}, i=1,2,3,..,k  \\
			&_{T_{1}}^{mABC}D^{e}_t X_{i}(t) = g(t, X_{i}), T_{1} \le t\le T_2  \\
			& X_{i}(T_{1})= X_{i,1}, i=1,2,3...k  \\
			&dX_{i}(t)=g(t,X_{i})dt+\sigma_{i} X_{i}d \mathbb{B}_{i}(t),  T_{2}< t\le T,\\
			& X_{i}(T_{2})=X_{i,2}, i=1,2,3...k.
		\end{cases}
	\end{align}
	The mathematical expression of \eqref{linear_caputo_mittag} for the piecewise modelling of three-species food chain with modified ABC derivative is given as following\\
	\begin{align}\label{extendedmittag}
		\begin{cases}
				& x'(t) = -\chi_1 x+y+10yz \\
				& y'(t) =-x-0.4y+5xz \\
				& z'(t) = \chi_2 z-5xy \\
			&_{T_{1}}^{mABC}D^{e}_t x(t)=-\chi_1 x+y+10yz\\
			&_{T_{1}}^{mABC}D^{e}_t y(t)=-x-0.4y+5xz\\
			&_{T_{1}}^{mABC}D^{e}_t z(t)=  \chi_2 z-5xy \\
			& dx(t)= \bigg( -\chi_1 x+y+10yz \bigg)dt+\sigma_{1}xd \mathbb{B}_{1}(t) \\
			& d y(t)=\bigg(  -x-0.4y+5xz  \bigg)dt+\sigma_{2} yd \mathbb{B}_{2}(t)\\
			& d z(t)=\bigg(  \chi_2 z-5xy  \bigg)dt+\sigma_{3} zd \mathbb{B}_{3}(t)
		\end{cases}
	\end{align}			
			\section{Numerical solution}
			Numerical approaches for the systems under consideration are developed in this part. For all circumstances, Newton interpolation is used. The following is how we divided the interval $[0,T]$:
			\begin{align*}
				0\leq t_{0} \leq t_{1} \leq t_{2}...\leq t_{k_{1}} =\\ T_{1}\leq t_{k_{1}+1} \leq t_{k_{1}+2} \leq t_{k_{1}+3}...\leq t_{k_{2}}=T
			\end{align*}
			Next, Newton interpolation\label{Newton_method} $Q(t)$, which is defined as follows, approximates $\psi(t,X)$.
			\begin{align}\label{Newton_method}
				\psi(t,X) \approx Q(t)=\psi(t_{j-1},X_{j-1})+ \frac{\psi(t_{j-1},X_{j-1})-\psi(t_{j-2},X_{j-2})}{\Delta t}(t-t_{j-2})+\\ \frac{\psi(t_{j},X_{j})-2\psi(t_{j-1},X_{j-1})+\psi(t_{j-2},X_{j-2})}{2(\Delta t)^2}(t-t_{j-1})(t-t_{j-2})  \nonumber
			\end{align} 
			
						\subsection{Case 1: Numerical Approach and Simulation Outcomes
			}
			This section addresses the case 1. The numerical technique will be provided using Newton interpolation \eqref{Newton_method} as
			\begin{align}
				\begin{cases}
					&X^{k} = X(0)+\frac{1}{12}\sum_{j = 2}^{k}\bigg[23\psi(t_{j}, X(t_{j}))-16\psi(t_{j - 1}, X(t_{j - 1}))+5\psi(t_{j - 2}, X(t_{j - 2})) \bigg]* \Delta t
				\end{cases}
			\end{align}
			\subsection*{ Piecewise Newton–Leipnik system with classical derivative  }  
			It illustrates how the classical derivative can be used to find numerical solutions to the piecewise Newton-Leipnik system. The initial values are regarded as
			$t(0)=0$, $h=0.01$, tfinal=80, $u(0)=0.19$, $v(0)=0$, $w(0)=-0.16$ and $\chi_2=0.175$.
			Refer to figures of the dynamics and chaotic dynamics of with aspect of classical derivative \ref{fig:fig1b}(a)- \ref{fig:fig1b}(b), \ref{fig:fig2e}(a)- \ref{fig:fig2e}(e), and \ref{fig:fig3k}(a)- \ref{fig:fig3k}(k).
						\begin{figure}[H]
				\centering
				\begin{subfigure}{0.53\textwidth}
					\includegraphics[width=\textwidth]{CLASS17.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.46\textwidth}
					\includegraphics[width=\textwidth]{CLASS18.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{The dynamics of the Newton-Leipnik model based on classical derivative}
				\label{fig:fig1b}
			\end{figure}
									\begin{figure}[H]
				\centering
				\begin{subfigure}{0.49\textwidth}
					\includegraphics[width=\textwidth]{CLASS12.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.5\textwidth}
					\includegraphics[width=\textwidth]{CLASS13.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:second}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.5\textwidth}
					\includegraphics[width=\textwidth]{CLASS14.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\centering
				\begin{subfigure}{0.49\textwidth}
					\includegraphics[width=\textwidth]{CLASS15.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.52\textwidth}
					\includegraphics[width=\textwidth]{CLASS16.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}	
				\caption{The dynamics of the Newton-Leipnik model based on classical derivative}
				\label{fig:fig2e}
			\end{figure}
									\begin{figure}[H]
				\centering
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{CLASS1.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{CLASS2.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:second}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{CLASS3.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\centering
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS4.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS5.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS6.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS7.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS8.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS9.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS10.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{CLASS11.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{Chaos and system dynamics  based on classical derivative}
				\label{fig:fig3k}
			\end{figure}
			
			
			
			
	\subsection{Case 2: Numerical Approach and Simulation Outcomes
			}
			This section addresses the case 1. The numerical technique will be provided using Newton interpolation \eqref{Newton_method} as
			\begin{align}
				\begin{cases}
					&X^{k} = X(T_{2})+ \frac{1}{12} \sum_{j = 2}^{k}\bigg[23\psi(t_{j}, X(t_{j}))- 16\psi(t_{j - 1}, X(t_{j - 1})) + 5\psi(t_{j - 2}, X(t_{j - 2})) \bigg]*\Delta t+ \\
					&~\sigma_{i} \sum_{j =2}^{k}X(\mathbb{B}_i(t_{j})-\mathbb{B}_i(t_{j-1}))
				\end{cases}
			\end{align}
			\subsection*{ Piecewise Newton–Leipnik system with stochastic type derivative }  
			It illustrates how the classical Mittag-Leffler law can be used to find numerical solutions to the piecewise Newton-Leipnik system. The initial values are regarded as
			$t(0)=0$, $h=0.01$, tfinal=80, $u(0)=0.19$, $v(0)=0$, $w(0)=-0.16$ and $\chi_2=0.175$.
			Refer to figures of the dynamics and chaotic dynamics of with aspect of stochastic type derivative \ref{fig:fig4b}(a)- \ref{fig:fig4b}(b), \ref{fig:fig5e}(a)- \ref{fig:fig5e}(e), and \ref{fig:fig6k}(a)- \ref{fig:fig6k}(k).
						\begin{figure}[H]
				\centering
				\begin{subfigure}{0.52\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC17.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.47\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC18.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{The dynamics of the Newton-Leipnik model based on stochastic type derivative}
				\label{fig:fig4b}
			\end{figure}
									\begin{figure}[H]
				\centering
				\begin{subfigure}{0.49\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC12.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.5\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC13.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:second}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.49\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC14.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\centering
				\begin{subfigure}{0.5\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC15.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.52\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC16.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{The dynamics of the Newton-Leipnik model based on stochastic type derivative}
				\label{fig:fig5e}
			\end{figure}
			
									\begin{figure}[H]
				\centering
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC1.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC2.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:second}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC3.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\centering
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC4.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC5.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC6.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC7.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC8.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC9.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC10.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{STOCHASTIC11.eps}
					\caption{ $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{Chaos and system dynamics  based on stochastic type derivative}
				\label{fig:fig6k}
			\end{figure}
			
			
			\subsection{Case 3: Numerical Approach and Simulation Outcomes
			}
			This section addresses the case 1. The numerical technique will be provided using Newton interpolation \eqref{Newton_method} as
			\begin{align}
	\begin{cases}
	&X^{k_{1}} = X(0)+\frac{1}{12}\sum_{j_1 = 2}^{k_{1}}\bigg[23\psi(t_{j_1}, X(t_{j_1}))-16\psi(t_{j_1 - 1}, X(t_{j_1 - 1}))+5\psi(t_{j_1 - 2}, X(t_{j_1 - 2})) \bigg]* \Delta t, 0 \le t\le T_{1}  \\
	&X^{k_{2}} = X(T_{1})+
	\frac{1-e}{AB(e)} \psi(t_{k_2}, X(t_{k_2}))+\\
	&+\frac{e}{AB(e)} \frac{h^e}{\Gamma{(e)}} \sum_{j_2 = k_{1}+3}^{k_{2}}\bigg(\psi(t_{j_2 - 1}, X(t_{j_2 -1}))\bigg[\frac{(k_2-j_2+1)^{e}-(k_2-j_2)^{e} }{e} \bigg]\\
	&+(\psi(t_{j_2 - 1}, X(t_{j_2 - 1}))-\psi(t_{j_2 - 2}, X(t_{j_2 - 2}))) \bigg[(k_2-j_2+2)\frac{(k_2-j_2+1)^{e}-(k_2-j_2)^{e}}{e} 
	-\frac{(k_2-j_2+1)^{e+1}-(k_2-j_2)^{e+1} }{e+1}\bigg]\\
	&+\frac{\psi(t_{j_2}, X(t_{j_2}))-2\psi(t_{j_2-1 }, X(t_{j_2-1 }))+\psi(t_{j_2 -2}, X(t_{j_2 -2}))}{2} \bigg[(k_2-j_2+2)(k_2-j_2+1)\frac{(k_2-j_2+1)^{e}-(k_2-j_2)^{e} }{e}\\
	&-(2k_2-2j_2+3)\frac{(k_2-j_2+1)^{e+1}-(k_2-j_2)^{e+1} }{e+1}
	+\frac{(k_2-j_2+1)^{e+2}-(k_2-j_2)^{e+2} }{e+2}\bigg]\bigg)-\frac{1-e}{AB(e)} \psi({0}, X({0})) \bigg(1+\frac{\xi_{e}t^{e}}{\Gamma(e+1)}\bigg),\\ &T_{1} \le t\le T_2\\
	&X^{k_{3}} = X(T_{2})+ \frac{1}{12} \sum_{j_3 = k_{2}+3}^{k_{3}}\bigg[23\psi(t_{j_3}, X(t_{j_3}))- 16\psi(t_{j_3 - 1}, X(t_{j_3 - 1})) + 5\psi(t_{j_3 - 2}, X(t_{j_3 - 2})) \bigg]*\Delta t+ \\
	&~\sigma_{i} \sum_{j_3 = k_{2}+3}^{k_{3}}X(\mathbb{B}_i(t_{j_3})-\mathbb{B}_i(t_{j_3-1})), T_{2} \le t\le T \\
\end{cases}
			\end{align}
			\subsection*{ Piecewise Newton–Leipnik system with modified ABC derivative }  
			It illustrates how the modified ABC differential operator can be used to find numerical solutions to the piecewise Newton-Leipnik system. The initial values are regarded as
			$t(0)=0$, $h=0.01$, tfinal=80, $u(0)=0.19$, $v(0)=0$, $w(0)=-0.16$ and $\chi_2=0.175$.
			Refer to figures of the dynamics and chaotic dynamics of with aspect of Atangana-Baleanu derivative \ref{fig:fig7b}(a)- \ref{fig:fig7b}(b), \ref{fig:fig8e}(a)- \ref{fig:fig8e}(e), and \ref{fig:fig9k}(a)- \ref{fig:fig9k}(k).
			\begin{figure}[H]
				\centering
				\begin{subfigure}{0.482\textwidth}
					\includegraphics[width=\textwidth]{MABC17.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.46\textwidth}
					\includegraphics[width=\textwidth]{MABC18.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{The dynamics of the Newton-Leipnik model based on MABC derivative}
				\label{fig:fig7b}
			\end{figure}
						\begin{figure}[H]
				\centering
				\begin{subfigure}{0.5\textwidth}
					\includegraphics[width=\textwidth]{MABC12.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.49\textwidth}
					\includegraphics[width=\textwidth]{MABC13.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:second}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.49\textwidth}
					\includegraphics[width=\textwidth]{MABC14.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\centering
				\begin{subfigure}{0.5\textwidth}
					\includegraphics[width=\textwidth]{MABC15.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.52\textwidth}
					\includegraphics[width=\textwidth]{MABC16.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{The dynamics of the Newton-Leipnik model based on MABC derivative}
				\label{fig:fig8e}
			\end{figure}
			
						\begin{figure}[H]
				\centering
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{MABC1.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{MABC2.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:second}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.325\textwidth}
					\includegraphics[width=\textwidth]{MABC3.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\centering
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC4.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:first}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC5.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC6.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC7.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC8.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC9.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC10.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\hfill
				\begin{subfigure}{0.32\textwidth}
					\includegraphics[width=\textwidth]{MABC11.eps}
					\caption{$e=0.8$, $\chi_1=0.4$.}
					%		\label{fig:third}
				\end{subfigure}
				\caption{Chaos and system dynamics based on modified ABC derivative}
				\label{fig:fig9k}
			\end{figure}
			
			
			
			
			
			
			

			
			
			
		
			
			\section{Conclusion} 
			The generated formula for piecewise integral and differential operator model solution demonstrations and application is used in this article to attain the desired results. The results are illustrated with examples. This article offered new piecewise integration and differentiation operators. These new operators were offered with various properties. In this study, we examined the piecewise Newton-Leipnik system with numerous attractors in terms of its crossover and chaos behaviours. Using piecewise derivatives, we established in this study a generalisation of the piecewise Newton-Leipnik system. Piecewise models admit chaotic solutions, and when the arbitary order falls below a certain threshold, the chaotic character vanishes. As a result, the model's order serves as a chaotic governing parameter. 
			Therefore, research into chaos and arbitrary order models is crucial. The investigation of the arbitary order model is still in its early stages, with many facets needing to be explored. The outcomes are compared using numerical simulations and theoretical analysis. We investigate the effectiveness of the piecewise Newton-Leipnik system through numerical simulations. Different behaviours may be represented by simulation experiments for piecewise Newton-Leipnik systems. It might exhibit certain unpredictable behaviours with varying settings and $e$ values. Numerical simulations are utilised to present the results in depth. Our study confirms the piecewise Newton-Leipnik system's cross-over and chaos behaviours, which are defined by its attractors. For creative researchers, the idea of piecewise modelling offers a wealth of opportunities.
			
			
			
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