1 Introduction

Chaos can be described as confusion or disorder that occurs in events so that they appear erratic and unpredictable. Chaos is an interesting as well as important phenomenon due its emergent properties or surprises. An emergent property is something that emerges from a system that is unexpected. It is a surprise. We can’t predict it, and we can’t even judge what surprise will come. We can’t even know if a surprise will happen. We can only make a guess. Whereas, chaotic behavior is a common feature of nonlinear dynamics, as well as hyper-chaos in high-dimensional systems. Recently, hyper-chaos due to its attractive features like high security, high efficiency and high capacity has been deeply investigated in many application fields such as secure communications [1], lasers [2], nonlinear circuits [3], control [4] and synchronization [5]. Hyper-chaotic systems can be defined as chaotic systems with more than one positive Lyapunov exponents i.e the number of directions of spreading is greater than one, which results the system to show the behavior of high disorder and randomness. Every day the number of articles that relates to introducing new hyper chaotic system is increasing. Generally, creating a hyper chaotic system with a more complicated topological structure in particular, creating attractors with multi scrolls or multi-wings, is an interesting topic for research, and therefore, becomes a desirable task and sometimes a key issue for many engineering applications [6]. The first classical hyper-chaotic system is the well known hyper-chaotic Rössler system [7]. After that, many hyper-chaotic systems have been developed and the applications of these models have been enhanced recently. Over the past decades, many other hyper-chaotic systems have been introduced,such as hyper-chaotic Chen system [8], hyper-chaotic Chen system [9], hyper-chaotic Lü system [10], hyper-chaotic Nikolov system [11], hyper-chaotic Lorenz system [12], and hyper-chaotic Lorenz system [13].

In recent years, the control of chaotic systems has received great attention due to its potential applications in physics, chemical reactor, biological networks, artificial neural networks, telecommunications, etc. Basically, chaos controlling is the stabilization of an unstable periodic orbit or equilibria by means of tiny perturbations of the system. For chaos control, some useful and powerful methods have been developed for sustained development of humanity. These may include optimal control [14–16], synchronization [15], adaptive control [17], state-feedback control [18], sliding mode control [19], time-delayed feedback control [20], etc. In present manuscript, basic dynamical characters of the new 4-D hyper-chaotic system are investigated by means of dissipation, equilibrium, stability, time series, phase portraits, Lyapunov exponents and Poincaré maps. Highly computational strategy is proposed for hyper-chaos and optimal control. To obtain the optimal controllers for the new 4-D hyper-chaotic system Pontryagins minimum principle (PMP) [21] has been applied. Furthermore, an adaptive control law is introduced to stabilize the new 4-D hyper-chaotic system with unknown parameters. The adaptive control results derived in this paper are established by using the Lyapunov stability theory [22]. The numerical simulation results are strong enough to show the high efficiency and accuracy of the proposed technique.

The paper is organised as: Sect. 2 contains the formulation of a new 4-D hyper-chaotic system. Section 3 is about the analysis of the presented new hyper-chaotic system. In Sect. 4, Hyper-chaos and optimal control law are formulated for new 4-D hyper-chaotic system along with the simulations. In Sect. 5, an adaptive control law is devised to stabilize the new 4-D hyper-chaotic system. The computational studies of the unknown parameters have also been preformed in this section. Finally, in the last section conclusions are drawn.

2 Formulation of a new 4-D hyper-chaotic system

Our new 4-D hyper-Chaotic system is based on a 3-D four wings chaotic attractor presented by Dong et al. [23], which can be given as:

$$\begin{aligned} \dot{x}= & {} ax-byz, \nonumber \\ \dot{y}= & {} -cy+xz, \nonumber \\ \dot{z}= & {} kx-dz+xy. \end{aligned}$$
(1)

where x, y and z are the state variables and \(a,b,c,d,k\in R^{+}\) The above system shows chaotic behaviour with wide range parameters a and k. It is well known that, to generate hyper-chaos from the dissipative autonomously systems, the state equation must satisfy the following two basic conditions:

  • The dimension of the state equations should to be at least 4 and the order of the state equations should be at least 2.

  • The system must have at least two positive Lyapunov exponents, one null Lyapunov exponent and one negative Lyapunov exponent ,with the condition that the sum of all Lyapunov exponents is less than zero.

Based on system (1) and above two basic conditions, we construct a new 4-D hyper-chaos system defined by:

$$\begin{aligned} \dot{x}= & {} ax-byz,\nonumber \\ \dot{y}= & {} -cy+xz, \nonumber \\ \dot{z}= & {} kx-dz+xy, \nonumber \\ \dot{w}= & {} hw+xy. \end{aligned}$$
(2)

where \([x,y,z,w]^{T}\in R^{4}\) is the state vector, and a, b, c, d, k and h are positive constant parameters of the system.

3 Analysis of the new 4-D hyper-chaotic system

3.1 Phase portraits and time responses

For the parameters a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5 , k \(=\) 3.5 and h \(=\) .04, the system shows hyper-chaos. The new system is Hyper chaotic having the Lyapunov exponents \(LE_{1}=1.5278,\,LE_{2}=.041041,\,LE_{3}= 0.0023108\) and \(LE_{4}= -12.5454\). It is clear that \(LE_{3}= 0.0023108\) is close to 0. Thus for the values of chosen parameters, the system exhibits hyper-chaos. The corresponding phase portraits, time series and Lyapunov exponents displayed in Figs. 1, 2 and 3 respectively.

Fig. 1
figure 1

Phase portrait of a new 4-D hyper-chaotic systems

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Fig. 2
figure 2

Time series of a new 4-D hyper-chaotic systems

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3.2 Dissipation

The divergence of a vector field \(\mathbf{F }\) of the system (2) can be obtained as:

$$\begin{aligned} \bigtriangledown \cdot \mathbf{F }=\frac{\partial f_{1}}{\partial x}+\frac{\partial f_{2}}{\partial y}+\frac{\partial f_{3}}{\partial z}+\frac{\partial f_{4}}{\partial w}= a-c-d+h, \end{aligned}$$
(3)

where

$$\begin{aligned} \mathbf F= & {} (f_{1},f_{2},f_{3},f_{4})\nonumber \\= & {} (ax-byz,-cy+xz,kx-dz+xy,hw+xy). \end{aligned}$$
(4)

For a system to be dissipative it is required that \(\bigtriangledown \cdot \mathbf F <0\). When we substitute the parameter values as a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5 , k \(=\) 3.5 and h \(=\) .04, we get \(\bigtriangledown \cdot \mathbf F =-11.01<0\). Thus, the system (2) is dissipative and converges with the rate \(e^{-11.01t}\). Which means, each volume containing the system orbit shrinks to zero as \(t \rightarrow \infty \) with an exponential rate \(\bigtriangledown \cdot \mathbf F =-11.01\) and is independent of system states. Consequently, all system orbits will ultimately be confined to a specific subset of zero volume and the asymptotic motion dies onto an attractor. It proves, the existence of an attractor.

3.3 Symmetry

The new 4-D hyper-chaotic system (2) is invariant under the coordinate transformation

$$\begin{aligned} (x, y, z,w)\rightarrow (-x,y,-z,-w). \end{aligned}$$

Thereby the system (2) is symmetrical about the y-axis.

Fig. 3
figure 3

Lyapunov exponents graph for new 4-D hyper-chaotic systems

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3.4 Equilibrium points

The equilibria of new 4-D system (2) can be calculated by solving the following equations:

$$\begin{aligned} ax-byz= & {} 0,\nonumber \\ -cy+xz= & {} 0, \nonumber \\ kx-dz+xy= & {} 0, \nonumber \\ hw+xy= & {} 0. \end{aligned}$$
(5)

On solving above equation and substituting the value parameters a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5, k \(=\) 3.5 and h \(=\) 0.04 ,we get five Equilibrium points for new 4-D hyper-chaotic system which are given by:

$$\begin{aligned} E_{1}= & {} (0,0,0,0),\nonumber \\ E_{2}= & {} (11.349,-6.15424,-5.47693,1746.12),\nonumber \\ E_{3}= & {} (-11.349,-6.15424,5.47693,-1746.12),\nonumber \\ E_{4}= & {} (4.89469,2.65424,5.47693,-324.792),\nonumber \\ E_{5}= & {} (-4.89469,2.65424,-5.47693,324.792). \end{aligned}$$
(6)

3.5 Stability analysis of equilibrium points

The Jacobian matrix for the hyper-chaotic system (2) is given by

$$\begin{aligned} \mathbf{J }=\begin{bmatrix} a&-bz&-by&0\\ z&-c&x&0\\ k+y&x&-d&0\\ y&x&0&h\\ \end{bmatrix} \end{aligned}$$
(7)

with the parameter a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5, k \(=\) 3.5 and h \(=\) .04, the system (2) has \(E_{1},E_{2},E_{3},E_{4}\), and \(E_{5}\) equilibrium points given as in (6).

The eigenvalues of the jacobian matrix \(\mathbf{J }\) at the equilibrium point \(E_{1}\), are given by: \(\lambda _{1}^{1} = -10.1, \lambda _{2}^{1} = -5.5\), \(\lambda _{3}^{1} = 4.55, \lambda _{4}^{1}=.04\). It is observed that two eigenvalues \(\lambda _{3}^{1}\) and \(\lambda _{4}^{1}\) are positive. Thus, according to Lyapunov stability theory, the equilibrium point \(E_{1}\) is unstable.

The eigenvalues of the jacobian matrix \(\mathbf{J }\) at the equilibrium point \(E_{2}\) are given by: \(\lambda _{1}^{2} = -19.3295, \lambda _{2}^{2} = 4.13977 + 8.34574 i\) , \(\lambda _{3}^{2} = 4.13977 - 8.34574 i, \lambda _{4}^{2} = .04\). By the Lyapunov stability theory, the equilibrium point \(E_{2}\) is unstable as \(\lambda _{2}^{2}\), \(\lambda _{3}^{2}\) have eigenvalues with positive real parts and \(\lambda _{4}^{2}\) is positive.

The eigenvalues of the jacobian matrix \(\mathbf{J }\) at the equilibrium point \(E_{3}\) are identical to the eigenvalues of the equilibrium point \(E_{2}\). Thus, by using the same arguments the equilibrium point \(E_{3}\) becomes unstable.

Now the eigenvalues of the jacobian matrix \(\mathbf{J }\) at the equilibrium point \(E_{4}\), are: \(\lambda _{1}^{4} = -12.933\), \( \lambda _{2}^{4} = 0.941475 +7.42009 i\), \(\lambda _{3}^{4} =0.941475 - 7.42009 i\), \(\lambda _{4}^{4}=.04\). Here, also \(\lambda _{2}^{4}\), \(\lambda _{3}^{4}\) have eigenvalues with positive real parts and \(\lambda _{4}^{4}\) is positive. On again applying the Lyapunov stability theory, the equilibrium point \(E_{4}\) is unstable.

The eigenvalues of the jacobian matrix \(\mathbf J \) at the equilibrium point \(E_{5}\) is same as for the equilibrium point \(E_{4}\), which means that the the equilibrium point \( E_{5}\) is also an unstable equilibrium point.

It is observed that, all the five equilibrium points \(E_{1},E_{2}, E_{3},E_{4}\) and \(E_{5}\) of new 4-D hyper-chaotic system (2) are unstable.

3.6 Poincaré mapping

The Poincaré map is one such analytical technique which helps to visualise the folding properties of chaos. This also provides the idea of the bifurcation. When a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5, k \(=\) 3.5 and h \(=\) .04 and on taking the different crossing planes such as y \(=\) 0, x \(=\) 0. The corresponding Poincaré maps on the x–z and y–w planes are displayed in Fig. 4 that, system (2) has a self-similar structure and some sheets are folded and wing type structure is also visualized.

Fig. 4
figure 4

Poincare section for new hyper-chaotic systems (2) with the parameters a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5, k \(=\) 3.5 and h \(=\) .04. a Projection on x–z plane with y=0; b projection on y–w plane with x \(=\) 0

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Fig. 5
figure 5

Bifurcation diagram of new hyper chaotic system (2) versus the parameter ‘a’ \(\in [2,6]\) when b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5, k \(=\) 3.5 and h \(=\) .04

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3.7 Bifurcation

When the parameter ‘a’ is varied, the corresponding bifurcation diagram of state y with respect to ‘a’ is obtained as shown in Fig. 5. It is easy to see the chaotic behavior of new 4-D hyper-chaotic system when the parameter ‘a’ \(\in [2,6]\).

4 Hyper-chaos and optimal control of a new 4-d hyper-chaos system

In this section, we use Pontryagin minimum principle (PMP) to achieve optimal hyper-chaos control of the new 4-D hyper-chaotic system (2) at its equilibrium points.

4.1 Methodology

For the purpose of optimal hyper-chaos control, we add controllers \(U_{1}, U_{2}, U_{3}\) and \(U_{4}\) to the new 4-D hyper-chaotic system (2). The controlled new 4-D hyper-chaotic system is defined as:

$$\begin{aligned} \dot{x}= & {} ax-byz+U_{1},\nonumber \\ \dot{y}= & {} -cy+xz+U_{2}, \nonumber \\ \dot{z}= & {} kx-dz+xy+U_{3}, \nonumber \\ \dot{w}= & {} hw+xy+U_{4}. \end{aligned}$$
(8)

where \(U_{1},U_{2},U_{3}\) and \(U_{4}\) are the control inputs which should be satisfied by the optimal conditions at its equilibrium points obtained by PMP with respect to the cost function \({{\varvec{J}}}\). The main strategy to control the system is to design the optimal control inputs \(U_{1},U_{2},U_{3}\) and \(U_{4}\) such that the state trajectories tends to the unstable equilibrium points in a given finite time interval \([0,t_{f}]\). Thus,the boundary conditions are:

$$\begin{aligned} x(0)= & {} x_0,\, x(t_{f})= \overline{x},\nonumber \\ y(0)= & {} y_0,\, y(t_{f})= \overline{y},\nonumber \\ z(0)= & {} z_0,\, z(t_{f})= \overline{z},\nonumber \\ w(0)= & {} w_0,\, w(t_{f})= \overline{w} \end{aligned}$$
(9)

where \(\overline{x}, \overline{y}, \overline{z}\) and \(\overline{w}\), denote the coordinates of the equilibrium points. The objective functional to be minimized is defined as:

$$\begin{aligned} {{\varvec{J}}}= & {} 1/2\int _{0}^{t_{f}}(\alpha _{1}(x-\overline{x})^{2}+ \alpha _{2}(y-\overline{y})^{2} \nonumber \\&+\,\alpha _{3}(z-\overline{z})^{2}+ \alpha _{4}(w-\overline{w})^{2} \nonumber \\&+\,\beta _{1}U_{1}^2+\beta _{2}U_{2}^2+\beta _{3}U_{3}^2+\beta _{4}U_{4}^2)dt, \end{aligned}$$
(10)

where \(\alpha _{i}\) and \(\beta _{i}(i=1,2,3,4)\) are positive constants. Now, we will derive the optimality conditions as a nonlinear two point boundary value problem(TPBVP) arising in the Pontryagin minimum principle(PMP). The corresponding Hamiltonian function \(\textit{H}\) will be:

$$\begin{aligned}&H=\lambda _{1}[ax-byz+U_{1}]+\lambda _{2}[-cy+xz+U_{2}] \nonumber \\&~~~~~~~+\,\lambda _{3}[kx-dz+xy+U_{3}]+\lambda _{4}[hw+xy+U_{4}], \nonumber \\&~~~~~~~-\,1/2\left[ \alpha _{1}(x-\overline{x})^{2}+\alpha _{2}(y-\overline{y})^{2}+\alpha _{3}(z-\overline{z})^{2} \right. \nonumber \\&~~~~~~\left. +\,\alpha _{4}(w-\overline{w})^{2}+\beta _{1}U_{1}^2+\beta _{2}U_{2}^2+\beta _{3}U_{3}^2+\beta _{4}U_{4}^2\right] \end{aligned}$$
(11)

where \(\lambda _{1},\lambda _{2},\lambda _{3}\) and \(\lambda _{4}\) are the costate variables. On applying the Pontryagin minimum principle(PMP), we obtain the Hamiltonian equations:

$$\begin{aligned} \dot{\lambda _{1}}= & {} -\frac{\partial \textit{H}}{\partial x},\nonumber \\ \dot{\lambda _{2}}= & {} -\frac{\partial \textit{H}}{\partial y},\nonumber \\ \dot{\lambda _{3}}= & {} -\frac{\partial \textit{H}}{\partial z},\nonumber \\ \dot{\lambda _{4}}= & {} -\frac{\partial \textit{H}}{\partial w}. \end{aligned}$$
(12)

From (11) and (12), we have:

$$\begin{aligned} \dot{\lambda _{1}}= & {} \alpha _{1}(x-\overline{x})-a\lambda _{1}-z\lambda _{2}-(k+y)\lambda _{3}-y\lambda _{4},\nonumber \\ \dot{\lambda _{2}}= & {} \alpha _{2}(y-\overline{y})+bz\lambda _{1}+c\lambda _{2}-x\lambda _{3}-x\lambda _{4},\nonumber \\ \dot{\lambda _{3}}= & {} \alpha _{3}(z-\overline{z})+by\lambda _{1}-x\lambda _{2}+d\lambda _{3},\nonumber \\ \dot{\lambda _{4}}= & {} \alpha _{4}(w-\overline{w})-h\lambda _{4}. \end{aligned}$$
(13)

The optimal control functions that have to be used are determined from the condition \(\frac{\partial \textit{H}}{\partial U_{i}}=0(i=1,2,3,4)\). Hence,we get

$$\begin{aligned} U^{*}_{i}=\frac{\lambda _{i}}{\beta _{i}}\quad (i=1,2,3,4). \end{aligned}$$
(14)

After substituting the value from (14) into (9), we get the controlled non linear state equations:

$$\begin{aligned} \dot{x}= & {} ax-byz+\frac{\lambda _{1}}{\beta _{1}},\nonumber \\ \dot{y}= & {} -cy+xz+\frac{\lambda _{2}}{\beta _{2}}, \nonumber \\ \dot{z}= & {} kx-dz+xy+\frac{\lambda _{3}}{\beta _{3}}, \nonumber \\ \dot{w}= & {} hw+xy+\frac{\lambda _{4}}{\beta _{4}}. \end{aligned}$$
(15)

The above system of nonlinear ordinary differential equations together with the (13) forms a complete system for solving the optimal control of the new hyper-chaotic system. The boundary conditions for this system have been given in (9). On solving the above nonlinear two point boundary value problem, we can obtain the optimal control law and the optimal state trajectories.

4.2 Numerical simulation and Discussions

In this section we demonstrate the effectiveness and feasibility of the proposed optimal control scheme by using the MATLAB’s bvp4c in-built solver. We solve the system (15) along with (13) and using the boundary conditions given in (9). For solving we choose the finite time interval as: [0,6], initial values as: \(x(0)= -2, y(0)=4, z(0)=2, w(0)=-3\), final values as: \(x(6)= \overline{x}, y(6)= \overline{y}, z(6)= \overline{z}, w(6)=\overline{w}\) where, \(\overline{x}, \overline{y}, \overline{z}\) and \(\overline{w}\) are the coordinates of the equilibrium points, and system parameters for the new 4-d hyper-chaotic system as: \(a=4.55, b=1.532, c=10.1, d=5.5, k=3.5 \) and \( h=0.04\). Also,the positive constants in cost function J for all the five equilibrium points are chosen as: \(\alpha _{i}=50\) and \(\beta _{i}=25\) for \(i=1,2,3\) and 4. Figures 6, 7, 8, 9 and 10 exhibits the controlled behaviour of the states variables (x, y, z, w) and controllers(\(U_{1},U_{2},U_{3},U_{4}\)) for its equilibrium points \(E_{1},E_{2},E_{3},E_{4}\) and \(E_{5}\) of the controlled new hyper-chaotic system.

Fig. 6
figure 6

The stabilized behaviour of states and control functions for the equilibrium points \(E_{1}\)

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Fig. 7
figure 7

The stabilized behaviour of states and control functions for the equilibrium points \(E_{2}\)

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Fig. 8
figure 8

The stabilized behaviour of states and control functions for the equilibrium points \(E_{3}\)

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Fig. 9
figure 9

The stabilized behaviour of states and control functions for the equilibrium points \(E_{4}\)

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Fig. 10
figure 10

The stabilized behaviour of states and control functions for the equilibrium points \(E_{5}\)

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5 Adaptive control approach for new 4-D hyper-chaos system with unknown parameters

This section aims to find an adaptive control law along with a parameter estimation update law for the new 4-D hyper-chaotic system (2), such that all the state variable x, y, z and w converges to its equilibrium points as ‘t’ approaches to infinity.

5.1 Design of the adaptive controllers

Let’s assume that the controlled system can be written in the form:

$$\begin{aligned} \dot{x}= & {} ax-byz+V_{1},\nonumber \\ \dot{y}= & {} -cy+xz+V_{2}, \nonumber \\ \dot{z}= & {} kx-dz+xy+V_{3}, \nonumber \\ \dot{w}= & {} hw+xy+V_{4}. \end{aligned}$$
(16)

where x, y, z and w are the states of the system and a, b, c, d, k, h are the unknown parameters of the system and \(V_{1},V_{2},V_{3},V_{4}\) are the adaptive controllers to be designed.

Theorem 5.1

The new 4-D hyper-chaos system (2) with unknown parameters is asymptotically and globally stabilized for all initial values of states \((x(0),y(0),z(0),w(0))\in R^4\) by the following adaptive control law:

$$\begin{aligned} V_{1}= & {} -a_{1}x + b_{1}yz - \ell _{1}(x-\overline{x}),\nonumber \\ V_{2}= & {} c_{1}y- xz - \ell _{2}(y-\overline{y}), \nonumber \\ V_{3}= & {} -k_{1}x+d_{1}z -xy - \ell _{3}(z-\overline{z}), \nonumber \\ V_{4}= & {} -h_{1}w- xy- \ell _{4}(w-\overline{w}). \end{aligned}$$
(17)

and the parameter estimation update law

$$\begin{aligned} \dot{a_{1}}= & {} (x-\overline{x})x +\ell _{5}(a-a_{1}),\nonumber \\ \dot{b_{1}}= & {} -(x-\overline{x})yz+\ell _{6}(b-b_{1}), \nonumber \\ \dot{c_{1}}= & {} -(y-\overline{y})y+\ell _{7}(c-c_{1}), \nonumber \\ \dot{d_{1}}= & {} -(z-\overline{z})z+\ell _{8}(d-d_{1}), \nonumber \\ \dot{k_{1}}= & {} (z-\overline{z})x+\ell _{9}(k-k_{1}), \nonumber \\ \dot{h_{1}}= & {} (w-\overline{w})w+\ell _{10}(h-h_{1}). \end{aligned}$$
(18)

where \(a_{1},b_{1},c_{1},d_{1},k_{1}\) and \( h_{1}\) are estimated values of uncertain parameters a, b, c, d, k, h and \(\ell _{i}(i=1,\ldots \ldots ,10)\) are the positive constants.

Proof

After substituting (17) into (16), we get the closed-loop system as:

$$\begin{aligned} \dot{x}= & {} (a-a_{1})x-(b-b_{1})yz- \ell _{1}(x-\overline{x}),\nonumber \\ \dot{y}= & {} -(c-c_{1})y-\ell _{2}(y-\overline{y}), \nonumber \\ \dot{z}= & {} (k-k_{1})x-(d-d_{1})z-\ell _{3}(z-\overline{z}), \nonumber \\ \dot{w}= & {} (h-h_{1})w-\ell _{4}(w-\overline{w}). \end{aligned}$$
(19)

For the derivation of the update law for adjusting the parameter estimates, the lyapunov approach is used. We define the lyapunov function as:

$$\begin{aligned} {{\varvec{V}}}(x,y,z,w,\tilde{a},\tilde{b},\tilde{c},\tilde{d},\tilde{k},\tilde{h})= & {} 1/2((x-\overline{x})^2+(y-\overline{y})^2 \nonumber \\&+\,(z-\overline{z})^2 \nonumber \\&+\,(w-\overline{w})^2\,{+}\,\tilde{a}^2\,{+}\,\tilde{b}^2\,{+}\,\tilde{c}^2 \nonumber \\&+\,\tilde{d}^2+\tilde{k}^2+\tilde{h}^2) \end{aligned}$$
(20)

where \(\tilde{a}=a-a_{1}, \tilde{b}=b-b_{1}, \tilde{c}=c-c_{1}, \tilde{d}=d-d_{1}, \tilde{k}=k-k_{1}\) and \(\tilde{h}= h-h_{1}\).

On taking the time derivative of the lyapunov function V, we obtain

$$\begin{aligned} \dot{{{\varvec{V}}}}= & {} (x-\overline{x})\dot{x}+ (y-\overline{y})\dot{y}+(z-\overline{z})\dot{z} \nonumber \\&+\,(w-\overline{w})\dot{w}+\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}} \nonumber \\&+\,\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{k}\dot{\tilde{k}}+\tilde{h}\dot{\tilde{h}}. \end{aligned}$$
(21)

On substituting (18)and (19) in (21), the derivative of lyapunov function with respect to time becomes:

$$\begin{aligned} \dot{{{\varvec{V}}}}= & {} -\ell _{1}(x-\overline{x})^2-\ell _{2}(y-\overline{y})^2-\ell _{3}(z-\overline{z})^2 \nonumber \\&-\,\ell _{4}(w-\overline{w})^2-\ell _{5}(a-a_{1})^2-\ell _{6}(b-b_{1})^2 \nonumber \\&-\,\ell _{7}(c-c_{1})^2-\ell _{8}(d-d_{1})^2 \nonumber \\&-\,\ell _{9}(k-k_{1})^2-\ell _{10}(h-h_{1})^2. \end{aligned}$$
(22)

Since the lyapunov function V is a positive definite function on \(R^{10}\) and clearly its derivatives \(\dot{{{\varvec{V}}}}\) on \(R^{10}\) is negative definite function, then by using lyapunov stability theory, the controlled system (16) converge asymptotically and globally for all initial values to its equilibrium points with the adaptive control law (17) and the parameter update law (18). This completes the proof. \(\square \)

Fig. 11
figure 11

Time history of states function and parameter estimates for the equilibrium points \(E_{1}\)

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Fig. 12
figure 12

Time history of states function and parameter estimates for the equilibrium points \(E_{2}\)

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Fig. 13
figure 13

Time history of states function and parameter estimates for the equilibrium points \(E_{3}\)

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Fig. 14
figure 14

Time history of states function and parameter estimates for the equilibrium points \(E_{4}\)

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Fig. 15
figure 15

Time history of states function and parameter estimates for the equilibrium points \(E_{5}\)

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5.2 Numerical simulation and discussions

Numerical results are presented to demonstrate the effectiveness of the proposed adaptive control technique. For simulation we solve the controlled hyper-chaotic system (16) with the adaptive control law (17) and the parameter update law (18) by using MATLAB’s ode45 in-built solver. The initial and parameters values are respectively chosen as: \(\hbox {x}(0)=-2\), y(0) \(=\) 4, z(0) \(=\) 2, \(\hbox {w}(0)=-3\), a \(=\) 4.55, b \(=\) 1.532, c \(=\) 10.1, d \(=\) 5.5, k \(=\) 3.5 and h \(=\) 0.04 for the controlled new 4-D hyper-chaotic system. Also, for adaptive and update laws,we take \(\ell _{i}=0, i=1,\ldots .10\). Further, the initial values for parameter estimates are chosen to equal zero. From the Figs. 11, 12, 13, 14 and 15, it is clear that the trajectories of the controlled new 4-D hyper-chaotic system (16) converges asymptotically to \(E_{i}(i=1,2,\ldots ,5)\) with time ‘t’ and these figures also shows that the parameter estimates \(a_{1}(t), b_{1}(t), c_{1}(t), d_{1}(t), k_{1}(t)\) and \(h_{1}(t)\) converge to the system chosen parameter values asymptotically with time.

6 Conclusion

In present manuscript, we have introduced a new 4-D hyper-chaotic system which has been successfully validated by the system analysis by means of dissipation, equilibrium, stability, time series, phase portraits, Lyapunov exponents, bifurcation and Poincaré maps. Also, an optimal control law has been formulated for the new 4-D hyper-chaotic system, which is based on the Pontryagin minimum principle(PMP). Furthermore, an adaptive control law has been devised to stabilize the new 4-D hyper-chaotic system with unknown parameters. Effectiveness and feasibility of results are validated in numerical simulations which are performed by using MATLAB’S bvp4c and ode-45 in-built solver. Remarkably, our analytic and computational results are in an excellent agreement.