In this article, we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative (CFFD). The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii. Also, we enriched our work by establishing a stable result based on the Ulam-Hyers (U-H) concept. Also, the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method. We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders. The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier. Also, the suggested method results for the proposed system under the mentioned derivative are new. Furthermore, the adopted technique has some useful features, such as the lack of prior discrimination required by wavelet methods. our proposed method does not depend on auxiliary parameters like the homotopy method, which controls the method. Our proposed method is rapidly convergent and, in most cases, it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.

Fractional calculus has gotten considerable attention in the last few decades. This is because of numerous applications in various fields of science and technology. Many real-world problems where hereditary properties and memory characteristics are involved can be comprehensively explained by using fractional calculus. For recent applications and interesting results, we refer to [

In recent times, some new types of fractional differential operators have been introduced. The concerned definitions have been obtained, preserving the regular kernel instead of the singular one. In this regard, in 2015, Caputo et al. [

Keeping the importance of FDEs in mind, various real-world problems have been investigated by using concepts of fractional calculus. Because fractional differential operators geometrically provide accumulation for a function, which includes its integer counter part as a special case. Also, in various cases, it has been found that fractional order derivative is more powerful than classical and describes the dynamics of various real world phenomena with more details (see [

The said famous classical Lorenz system has been described in [

Keeping in mind the importance of the said model, it has never been investigated till now by using CFFD.

Instead of ordinary calculus fractions, order derivatives and integrals are more practical in nature and preserve a greater degree of freedom. Using this type of operator, additional short and long-memory terms are better explained. Because power law singular kernels are used in Caputo and Reimann-Liouville operators. in numerical discretization, it causes difficulties. Therefore, by using those differential operators which involve exponential type kernels, the descriptions of some problems are more easily understood. Therefore, in this regard, the first one, which is increasingly used as CFFD, The Lorenz model has been investigated under various fractional order concepts by using different techniques. In most cases, researchers have investigated the approximate solution of the Lorenz model by using differential transform techniques [

Also, to the best of our knowledge, the Lorenz system under CFFD for semi-analytical solutions by using Laplace Adomian decomposition has never been investigated. Therefore, we update the model

It is a tedious job to deal with problems under the concept of fractional calculus for their exact or numerical solutions. Several algorithms, tools, and procedural theories have been established during the past few decades. A dynamical problem should be first treated for the criteria of its existence. Because, without knowing about its existence, we do not implement other techniques to compute numerical or semi-analytical results. For the existence theory, various tools have been developed. For instance, fixed point approaches, coincidence degree theories due to Mawhin and Schauder, etc., have been used very well. The most powerful one is the use of the fixed point approach to investigate a dynamical problem whether it has a solution or not (see [

Inspired by the aforesaid work, we are going to derive some adequate results for the existence of approximate solutions to the nonlinear system given in

Our work is organized as: We first provide some literature and refer to it in

This portion is devoted to the first part of our main results. Here we establish the existence criteria for our adopted model

Further, we can write

We need the following hypothesis to be exist for onward analysis.

(A1) Subject constants

(A2) For constants

Using

Hence

For

Hence

Since at

Therefore

If

As a result,

Here we recollect basic notions for U-H stability from [

Also the integral

The given remark is needed.

Then the solution of

Hence using Remark 1,

Moreover, the approximate solution of the model

From

Thus

Obviously in

We first develop a general algorithms for approximate solution to

Using initial condition,

The solution we are computing can be expressed as

Also, the nonlinear terms can be decomposed as

Here few initial terms of Adomian polynomials are computed from

Thus we calculate few terms

Comparing terms on both sides of

Applying inverse Laplace transform to both sides of

Using

Using Banach theorem

Also

A phase portrait is a geometric description of a dynamical system’s paths in the phase plane. The collection of initial conditions is represented by a separate curve or point. Phase portraits are an immensely valuable tool in the study of dynamical systems. They are comprised of a structure of common state-space trajectories. This shows whether the selected parameter values have an attractor, repeller, or limit cycle. Phase portraits of a dynamical system can be used to study the directed characteristics of that system. In

A time series is a collection of data points that are indexed (listed or graphed)

When a system is chaotic in its nature, it shows sensitive dependence on initial conditions. A very small change results in a great change in the dynamics of the system when it has chaotic behavior. Therefore, we present the dependence of our considered system on initial conditions. In

In the present work, we have derived some theoretical results based on some fixed point theorems due to Banach and Krassnoselskii for the existence and uniqueness of approximate solutions and their computation corresponding to the famous Lorenz nonlinear dynamical system. Sufficient conditions have been developed for the existence and uniqueness of solutions to the proposed model. Also, utilizing U-H and generalized U-H concepts, we have derived a few results for stability under some conditions for the considered system. Further, using a hybrid technique based on the Laplace transform and the Adomian decomposition method, we have also established an algorithm for approximate solutions. Some chaotic behaviors of the Lorenz system have been presented under the given fractional order by using five terms of approximate solution. Also, convergence and sensitivity of the model have been discussed. The proposed method has some features like being easy to implement, no need for prior discretization, and neither depends on auxiliary parameters like the homotopy analysis method. Also, the method is rapidly convergent in many cases. In future, we will investigate the aforesaid Lorenz model under piece-wise equations with fractional order derivative.