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dynamical systems are mathematical models describing the evolution of systems in terms of equation of motion with sensitive initial values
. the application areas of dynamical systems can be found e.g. in many disciplines in physics @xcite , population dynamics in biology @xcite , and chemical kinetics in chemistry @xcite .
dynamic systems theory has also found its way into subjects outside the fields of mathematics and natural sciences , exemplarily we would like to mention mathematical economy and finance @xcite .
poincar made a kick start to the subject of dynamical systems in his pioneering work in 1890 @xcite .
later , in the 1920 s , @xcite and @xcite introduced the dynamics of complex analytic maps .
on the one hand , studies like @xcite , @xcite , cartwright and littlewood @xcite find their origin in physics , like e.g. the three body problem in astronomy , whereas on the other hand stephan smale provided a purely mathematics motivated approach @xcite .
e. n. lorenz @xcite observed that very simple differential equations become chaotic under certain circumstances .
the system proposed by lorenz shows a very complex dynamical behaviour and displays the well - known two - scroll butterfly - shape .
the dynamic equations of the lorenz system are given as @xmath3 where the parameters @xmath4 , @xmath5 , and @xmath6 are assumed to be positive .
lorenz used exemplarily the values @xmath7 , @xmath8 and @xmath9 to demonstrate the systems chaotic behaviour .
the study on `` the equation for continuous chaos '' by rssler @xcite can be seen as another landmark in the discussion of 3d dynamic systems .
the rssler system and the burke shaw system @xcite have the property of two unstable saddle foci in common .
other chaotic systems exhibiting a similar simple structure as the lorenz system , without being topologically equivalent , are proposed by chen @xcite and l and chen @xcite .
@xcite discusses the transition between lorenz and chen attractors .
furthermore , yang and chen introduced another chaotic system with three fixed points : one saddle and two stable fixed points .
yang et al .
@xcite and pehlivan et al .
@xcite introduced and analysed chaotic systems similar to the lorenz , chen , and yang - chen systems , with two different fixed points , i.e. two stable node - foci .
chaotic systems have found their ways also into many applications in engineering , such as electronic circuits @xcite-@xcite .
modern pacemakers actually rely nowadays on these chaotic circuits .
dynamical systems have also been discussed in the framework of various non - newtonian calcului , like fractional calculus , geometric multiplicative calculus , and bigeometric multiplicative calculus .
jun guo lu transformed the l system into fractional calculus @xcite , investigating chaotic behaviour of fractional - order of the l system numerically .
aniszewska applied multiplicative calculus to the rssler system and showed chaotic behaviour of multiplicative rssler@xcite .
as an application of the bigeometric multiplicative runge - kutta method , the bigeometric multiplicative rssler system was solved in @xcite .
this study introduces a new chaotic attractor , found by modification of the lorenz system by a quadratic term .
detailed numerical and theoretical analysis reveals that the proposed system shows chaotic behaviour and the property of a two - scroll attractor like the lorenz attractor .
section [ sec:1 ] introduces the modified quadratic lorenz attractor , discusses and analyses its properties by determining the equilibria points , and lyapunov exponents theoretically as well as numerically . in section 3
, the modified quadratic lorenz attractor is translated into geometric and bigeometric calculus , and the solutions of the the system are obtained using the corresponding multiplicative runge - kutta methods @xcite and @xcite . finally , this paper closes with the summary of the obtained results .
this paper presents a new chaotic system derived from the lorenz system .
the system is generated by the following simple three - dimensional system :    @xmath10    where @xmath11 , @xmath12 , and @xmath13 are variables and @xmath4 , @xmath5 , and @xmath6 are real parameters . in the new proposed chaotic system ,
all the equations have some differences compared to the original lorenz system . in order to see the differences between the two systems we can compare the equations - and - one by one .
evidently , in the lorenz system equation is linear , whereas equation is non - linear .
furthermore , equations and are both non - linear , where the @xmath12-dependence of equation is cancelled .
the most significant difference can be observed comparing equations and . in the term @xmath0
is squared compared to equation .
the initial values and the parameters of the system are chosen as @xmath14 and @xmath15 , @xmath16 with varying @xmath6 .
then it has been observed that the behavior of the new chaotic system changes depending on the different values of @xmath6 as shown in figure [ fig : three graphs ] .    0.4 , @xmath16 and various parameter values , title="fig : " ]    0.4 , @xmath16 and various parameter values , title="fig : " ]    0.4 , @xmath16 and various parameter values , title="fig : " ]    0.4 , @xmath16 and various parameter values , title="fig : " ]    0.4 , @xmath16 and various parameter values , title="fig : " ]    as a result of the calculations , as it can be seen in figure [ fig : three graphs ]
, it has been observed that the new system is chaotic for the parameters @xmath17 the analysis of the new chaotic system will be done according to those parameters .
the first step to analyze a chaotic system is to find the equilibrium points . in order to determine the equilibrium points of the proposed system -
, we need to solve the system    @xmath18    thus , the solution of the system with respect to @xmath11 , @xmath12 , @xmath13 give the equilibria points as :    @xmath19{\beta\,\rho^3},\sqrt[4]{\frac{\beta}{\rho}},\rho\right ) \\
\it e^-&=&\left(-\sqrt[4]{\beta\,\rho^3},-\sqrt[4]{\frac{\beta}{\rho}},\rho\right)\end{aligned}\ ] ]    for the parameters chosen in , the numerical values of the equilibria points are    @xmath20    in order to decide on the stability of the new proposed system , the eigenvalues of the jacobian matrix must be analyzed .
the jacobian matrix for this system - can be easily obtained as @xmath21    the expressions for the eigenvalues of the jacobian matrix are very long and complicated .
as we are only interested in the numerical values of the eigenvalues at the equilibria points - for the given parameters , the eigenvalues are stated in the table below :    .eigenvalues of the jacobian at the equilibrium points [ cols="<,^,^,^ " , ]     as the eigenvalues @xmath22 and @xmath23 for the equilibrium point @xmath24 are both negative , the system is unstable at this equilibrium point .
the eigenvalues corresponding to the equilibrium point @xmath25 will be the same with the eigenvalues of @xmath26 , because of the quadratic nature of the system . since @xmath27 is a negative real number and @xmath22 and @xmath23 are two complex conjugate eigenvalues with positive real parts , equilibrium points @xmath26 and @xmath25 are unstable according to @xcite .
the system - has a natural symmetry and is invariant under the coordinate transformation @xmath28 which persists for all values of the system parameters .
so , system - has rotation symmetry about the @xmath29 .
let , @xmath30 , @xmath31 and @xmath32 in the system - .
then we get for the vector field    @xmath33    consequently the divergence of the vector field @xmath34 yields to :    @xmath35    note that @xmath36 is a negative value , so the system is a dissipative system and an exponential rate is :    @xmath37    from , it can be seen that a volume element @xmath38 is contracted by the flow into a volume element @xmath39 at the time @xmath40 .
the lyapunov exponents generally refer to the average exponential rates of divergence or convergence of nearby trajectories in the phase space .
the important part is that if there is at least one positive lyapunov exponent , the system can be defined to be chaotic .
according to the detailed numerical and theoretical analysis , the lyapunov exponents are found to be @xmath41 , @xmath42 , and @xmath43 .
therefore , the lyapunov dimension of this system is :    @xmath44    this result is consistent with the findings in @xcite , i.e. that the lyapunov dimension is in the range 2 - 3 .
equation means that the system - is a dissipative system , and the lyapunov dimensions of the system are fractional . having a strange attractor and positive lyapunov exponent , it is obvious that the system is a 3d chaotic system .
[ h ]          time series analysis of the system - according to @xmath45 axes are listed in the figure [ fig : wf ] . for the solutions of the system
the runge - kutta method was employed .    0.4 , @xmath46 , @xmath47 respectively , title="fig : " ]    0.4 , @xmath46 , @xmath47 respectively , title="fig : " ]    0.4 , @xmath46 , @xmath47 respectively , title="fig : " ]
obviously the time series show that the functions @xmath48 , @xmath46 , and @xmath47 are not periodic , indicating that the system is chaotic .
the projections of the system - , on the various axes is given in the figure [ fig : pp ] .    0.4     0.4     0.4
the chaotic attractors are displayed in the figure [ fig : pp ] .
it appears that the new attractor exhibits an interesting complex chaotic dynamics behavior .
as an application of the dynamical systems , the newly formed chaotic system can also be written in the sense of geometric and the bigeometric calculus .      the geometric chaotic system corresponding to system - , can be written as    @xmath49    on the other hand , the bigeometric chaotic system corresponding to system - will be    @xmath50    in order to solve the system , one can use the geometric runge - kutta method defined in @xcite , while the system can be solved by using the bigeometric runge - kutta method which was defined in @xcite . by choosing the same values for the parameters , such as @xmath15 , @xmath16 with varying @xmath6 and the initial values as @xmath51
, we can see that the solutions of the geometric chaotic system and the bigeometric chaotic system will be similar to the ones that we get from the solutions of the chaotic system - .
the following figure shows the chaotic behaviour of the geometric and the bigeometric dynamic systems when the parameters are chosen as @xmath15 , @xmath16 and @xmath52 .
, width=302 ]
the proposed modified quadratic lorenz attractor was analysed theoretically and numerically showing that this system is chaotic .
following the standardized analysis method for chaotic systems , we determined first the equilibria points and the eigenvalues of the jacobian matrix at these equilibria points to get a first indication about the stability of the proposed system .
as the eigenvalues are either non - positive real numbers , or complex numbers with positive real parts , we can conclude that this system is not stable at the equilibria points .
moreover , we could identify that the proposed system has a rotational symmetry about the @xmath13-axis , and shows dissipative behaviour contracting the volume element @xmath38 to @xmath53 .
the lyapunov exponents yield to @xmath41 , @xmath42 and @xmath43 , showing the chaotic nature of the system .
the fractional dimension of the system @xmath54 has also been given .
overall , the analysis shows that this is a new chaotic system with two scrolls .
32 [ 1]#1 [ 1]`#1 ` urlstyle [ 1]doi : # 1          ml  cartwright and je  littlewood . on non - linear differential equations of the second order : ii .
the equation y+kf(y,y+g(y , k)=p(t)=p_1(t)+kp_2(t);k>0,f(y)@xmath55underline>1 . _
annals of mathematics _ , pages 472494 , 1947 .
a.  a.  n. kolmogorov . .
in g.  casati and j.  ford , editors , _ stochastic behavior in classical and quantum hamiltonian systems _ , volume  93 of _ lecture notes in physics , berlin springer verlag _ , pages 5156 , 1979 .
doi : 10.1007/bfb0021737 .
ihsan pehlivan and yilmaz uyarolu . a new chaotic attractor from general lorenz system family and its electronic experimental implementation .
_ turkish journal of electrical engineering & computer sciences _ , 180 ( 2):0 171184 , 2010 .
simin yu , jinhu l , wallace  ks tang , and guanrong chen . a general multiscroll lorenz system family and its realization via digital signal processors .
_ chaos : an interdisciplinary journal of nonlinear science _ , 160 ( 3):0 033126 , 2006 .
this study introduces a modified quadratic lorenz attractor . 
 the properties of this new chaotic system are analysed and discussed in detail , by determining the equilibria points , the eigenvalues of the jacobian , and the lyapunov exponents . 
 the numerical simulations , the time series analysis , and the projections to the @xmath0-plane , @xmath1-plane , and @xmath2-plane are conducted to highlight the chaotic behaviour . 
 the multiplicative form of the new system is also presented and the simulations are conducted using multiplicative runge - kutta methods .